Singular Bott-Chern Classes

and the Arithmetic Grothendieck

Riemann Roch Theorem for Closed Immersions

José I. Burgos Gil^{1}and Răzvan Liţcanu^{2}

Received: May 13, 2009

Communicated by Peter Schneider

Abstract. We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soulé. Singular Bott-Chern classes are the main ingredient to define direct images for closed immersions in arithmetic K-theory. In this paper we give an axiomatic definition of a theory of singular Bott-Chern classes, study their properties, and classify all possible theories of this kind. We identify the theory defined by Bismut, Gillet and Soulé as the only one that satisfies the additional condition of being homogeneous. We include a proof of the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions that generalizes a result of Bismut, Gillet and Soulé and was already proved by Zha. This result can be combined with the arithmetic Grothendieck-Riemann-Roch theorem for submersions to extend this theorem to arbitrary projective morphisms. As a byproduct of this study we obtain two results of independent interest. First, we prove a Poincaré lemma for the complex of currents with fixed wave front set, and second we prove that certain direct images of Bott-Chern classes are closed.

2010 Mathematics Subject Classification: 14G40 32U40 Keywords and Phrases: Arakelov Geometry, Closed immersions, Bott-Chern classes, Arithmetic Riemann-Roch theorem, currents, wave front sets.

1Partially supported by Grant DGI MTM2006-14234-C02-01.

2Partially supported by CNCSIS Grant 1338/2007 and PN II Grant ID 2228 (502/2009)

Documenta Mathematica 15 (2010) 73–176

Contents

0 Introduction 74

1 Characteristic classes in analytic Deligne cohomology 79

2 Bott-Chern classes 87

3 Direct images of Bott-Chern classes 102

4 Cohomology of currents and wave front sets 107

5 Deformation of resolutions 111

6 Singular Bott-Chern classes 116

7 Classification of theories of singular Bott-Chern classes 122

8 Transitivity and projection formula 127

9 Homogeneous singular Bott-Chern classes 145

10 The arithmetic Riemann-Roch theorem for regular closed immersions 158

0 Introduction

Chern-Weil theory associates to each hermitian vector bundle a family of closed characteristic forms that represent the characteristic classes of the vector bundle. The characteristic classes are compatible with exact sequences. But this is not true for the characteristic forms. The Bott-Chern classes measure the lack of compatibility of the characteristic forms with exact sequences. The Grothendieck-Riemann-Roch theorem gives a formula that relates direct images and characteristic classes. In general this formula is not valid for the characteristic forms. The singular Bott-Chern classes measure, in a functorial way, the failure of an exact Grothendieck-Riemann-Roch theorem for closed immersions at the level of characteristic forms. In the same spirit, the analytic torsion forms measure the failure of an exact Grothendieck-Riemann-Roch theorem for submersions at the level of characteristic forms. Hence singular Bott-Chern classes and analytic torsion forms are analogous objects, the first for closed immersions and the second for submersions. Let us give a more precise description of Bott-Chern classes and singular Bott-Chern classes. Let X be a complex manifold and let φ be a symmetric power series in r variables with real coefficients. Let \overline{E} = (E, h) be a rank r holomorphic vector bundle provided with a hermitian metric. Using Chern-Weil theory, we can associate to \overline{E} a differential form φ(\overline{E}) = φ(−K), where K is the curvature tensor of E viewed as a matrix of 2-forms. The differential form φ(\overline{E}) is closed and is a sum of components of bidegree (p, p) for p ≥ 0. If

\overline{ξ}: 0 ⟶ \overline{E}′ ⟶ \overline{E} ⟶ \overline{E}′′ ⟶ 0

is a short exact sequence of holomorphic vector bundles provided with hermitian metrics, then the differential forms φ(\overline{E}) and φ(\overline{E}′ ⊕ \overline{E}′) may be different, but they represent the same cohomology class. The Bott-Chern form associated to \overline{ξ} is a solution of the differential equation

− 2∂ ¯∂φ(\overline{ξ}) = φ(\overline{E}′ ⊕ \overline{E}′) − φ(\overline{E})  (0.1)

obtained in a functorial way. The class of a Bott-Chern form modulo the image of ∂ and \overline{∂} is called a Bott-Chern class and is denoted by φ̃(\overline{ξ}). There are three ways of defining the Bott-Chern classes. The first one is the original definition of Bott and Chern [7]. It is based on a deformation between the connection associated to \overline{E} and the connection associated to \overline{E}′ ⊕ \overline{E}′′. This deformation is parameterized by a real variable. In [17] Gillet and Soulé introduced a second definition of Bott-Chern classes that is based on a deformation between \overline{E} and \overline{E}′ ⊕ \overline{E}′′ parameterized by a projective line. This second definition is used in [4] to prove that the Bott-Chern classes are characterized by three properties

(i) The differential equation (0.1).

(ii) Functoriality (i.e. compatibility with pull-backs via holomorphic maps).

(iii) The vanishing of the Bott-Chern class of a orthogonally split exact sequence.

In [4] Bismut, Gillet and Soulé have a third definition of Bott-Chern classes based on the theory of superconnections. This definition is useful to link Bott-Chern classes with analytic torsion forms. The definition of Bott-Chern classes can be generalized to any bounded exact sequence of hermitian vector bundles (see section 2 for details). Let

\overline{ξ}: 0 ⟶ (E_{n}, h_{n}) ⟶ … ⟶ (E_{1}, h_{1}) ⟶ (E_{0}, h_{0}) ⟶ 0

be a bounded acyclic complex of hermitian vector bundles; by this we mean a bounded acyclic complex of vector bundles, where each vector bundle is equipped with an arbitrarily chosen hermitian metric. Let

r = ∑_{i even} rk(E_{i}) = ∑_{i odd} rk(E_{i}).

As before, let φ be a symmetric power series in r variables. A Bott-Chern class associated to \overline{ξ} satisfies the differential equation

−2∂ ¯∂φ̃(\overline{ξ}) = φ(⨁_{k} \overline{E}_{2k}) − φ(⨁_{k} \overline{E}_{2k+1}).

In particular, let “ch” denote the power series associated to the Chern character class. The Chern character class has the advantage of being additive for direct sums. Then, the Bott-Chern class associated to the long exact sequence \overline{ξ} and to the Chern character class satisfies the differential equation

−2∂ ¯∂c̃h(\overline{ξ}) = − ∑_{k=0}^{n}(−1)^{i} ch(\overline{E}_{k}).

Let now i: Y ⟶ X be a closed immersion of complex manifolds. Let \overline{F} be a holomorphic vector bundle on Y provided with a hermitian metric. Let \overline{N} be the normal bundle to Y in X provided also with a hermitian metric. Let

0 ⟶ \overline{E}_{n} ⟶ \overline{E}_{n−1} ⟶ … ⟶ \overline{E}_{0} ⟶ i_{∗}F ⟶ 0

be a resolution of the coherent sheaf i_{∗}F by locally free sheaves, provided with hermitian metrics (following Zha [32] we shall call such a sequence a metric on the coherent sheaf i_{∗}F). Let Td denote the Todd characteristic class. Then the Grothendieck-Riemann-Roch theorem for the closed immersion i implies that the current i_{∗}(Td(\overline{N})^{−1} ch(\overline{F})) and the differential form ∑_{k}(−1)^{k} ch(\overline{E}_{k}) represent the same class in cohomology. We denote \overline{ξ} the data consisting in the closed embedding i, the hermitian bundle \overline{N}, the hermitian bundle \overline{F} and the resolution \overline{E}_{∗} ⟶ i_{∗}F. In the paper [5], Bismut, Gillet and Soulé introduced a current associated to the above situation. These currents are called singular Bott-Chern currents and denoted in [5] by T (\overline{ξ}). When the hermitian metrics satisfy a certain technical condition (condition A of Bismut) then the singular Bott-Chern current T (\overline{ξ}) satisfies the differential equation

−2∂ ¯∂T (\overline{ξ}) = i_{∗}(Td(\overline{N})^{−1} ch(\overline{F})) − ∑_{i=0}^{n}(−1)^{i} ch(\overline{E}_{i}).

These singular Bott-Chern currents are among the main ingredients of the proof of Gillet and Soulé’s arithmetic Riemann-Roch theorem. In fact it is the main ingredient of the arithmetic Riemann-Roch theorem for closed immersions [6]. This definition of singular Bott-Chern classes is based on the formalism of superconnections, like the third definition of ordinary Bott-Chern classes. In his thesis [32], Zha gave another definition of singular Bott-Chern currents and used it to give a proof of a different version of the arithmetic Riemann-Roch theorem. This second definition is analogous to Bott and Chern’s original definition. Nevertheless there is no explicit comparison between the two definitions of singular Bott-Chern currents. One of the purposes of this note is to give a third construction of singular Bott-Chern currents, in fact of their classes modulo the image of ∂ and \overline{∂}, which could be seen as analogous to the second definition of Bott-Chern classes. Moreover we will use this third construction to give an axiomatic definition of a theory of singular Bott-Chern classes. A theory of singular Bott-Chern classes is an assignment that, to each data \overline{ξ} as above, associates a class of currents T (\overline{ξ}), that satisfies the analogue of conditions (i), (ii) and (iii). The main technical point of this axiomatic definition is that the conditions analogous to (i), (ii) and (iii) above are not enough to characterize the singular Bott-Chern classes. Thus we are led to the problem of classifying the possible theories of Bott-Chern classes, which is the other purpose of this paper. We fix a theory T of singular Bott-Chern classes. Let Y be a complex manifold and let \overline{N} and \overline{F} be two hermitian holomorphic vector bundles on Y. We write P = ℙ(N ⊕ 1) for the projective completion of N. Let s: Y ⟶ P be the inclusion as the zero section and let π_{P} : P ⟶ Y be the projection. Let \overline{K}_{∗} be the Koszul resolution of s_{∗}O_{Y} endowed with the metric induced by \overline{N}. Then we have a resolution by hermitian vector bundles

K(\overline{F} , \overline{N}): \overline{K}_{∗} ⊗ π_{P}^{∗} \overline{F} ⟶ s_{∗}F.

To these data we associate a singular Bott-Chern class T (K(\overline{F} , \overline{N})). It turns out that the current

\frac{1}{(2πi)^{rk N}} ∫_{π_{P}} T (K(\overline{F} , \overline{N})) = (π_{P})_{∗}T (K(\overline{F} , \overline{N}))

is closed (see section 3 for general properties of the Bott-Chern classes that imply this property) and determines a characteristic class C_{T} (F, N) on Y for the vector bundles N and F. Conversely, any arbitrary characteristic class for pairs of vector bundles can be obtained in this way. This allows us to classify the possible theories of singular Bott-Chern classes:

Claim (theorem 7.1). The assignment that sends a singular Bott-Chern class T to the characteristic class C_{T} is a bijection between the set of theories of singular Bott-Chern classes and the set of characteristic classes.

The next objective of this note is to study the properties of the different theories of singular Bott-Chern classes and of the corresponding characteristic classes. We mention, in the first place, that for the functoriality condition to make sense, we have to study the wave front sets of the currents representing the singular Bott-Chern classes. In particular we use a Poincaré Lemma for currents with fixed wave front set. This result implies that, in each singular Bott-Chern class, we can find a representative with controlled wave front set that can be pulled back with respect certain morphisms. We also investigate how different properties of the singular Bott-Chern classes T are reflected in properties of the characteristic classes C_{T}. We thus characterize the compatibility of the singular Bott-Chern classes with the projection formula, by the property of C_{T} of being compatible with the projection formula. We also relate the compatibility of the singular Bott-Chern classes with the composition of successive closed immersions to an additivity property of the associated characteristic class.

Furthermore, we show that we can add a natural fourth axiom to the conditions analogue to (i), (ii) and (iii), namely the condition of being homogeneous (see section 9 for the precise definition).

Claim (theorem 9.11). There exists a unique homogeneous theory of singular Bott-Chern classes.

Thanks to this axiomatic characterization, we prove that this theory agrees with the theories of singular Bott-Chern classes introduced by Bismut, Gillet and Soulé [6], and by Zha [32]. In particular this provides us a comparison between the two definitions. We will also characterize the characteristic class C_{T^{h}} for the theory of homogeneous singular Bott-Chern classes. The last objective of this paper is to give a proof of the arithmetic Riemann-Roch theorem for closed immersions. A version of this theorem was proved by Bismut, Gillet and Soulé and by Zha. Next we will discuss the contents of the different sections of this paper. In section §1 we recall the properties of characteristic classes in analytic Deligne cohomology. A characteristic class is just a functorial assignment that associates a cohomology class to each vector bundle. The main result of this section is that any characteristic class is given by a power series on the Chern classes, with appropriate coefficients. In section §2 we recall the theory of Bott-Chern forms and its main properties. The contents of this section are standard although the presentation is slightly different to the ones published in the literature. In section §3 we study certain direct images of Bott-Chern forms. The main result of this section is that, even if the Bott-Chern classes are not closed, certain direct images of Bott-Chern classes are closed. This result generalizes previous results of Bismut, Gillet and Soulé and of Mourougane. This result is used to prove that the class C_{T} mentioned previously is indeed a cohomology class, but it can be of independent interest because it implies that several identities in characteristic classes are valid at the level of differential forms. In section §4 we study the cohomology of the complex of currents with a fixed wave front set. The main result of this section is a Poincaré lemma for currents of this kind. This implies in particular a ∂ ¯∂-lemma. The results of this section are necessary to state the functorial properties of singular Bott-Chern classes. In section §5 we recall the deformation of resolutions, that is a generalization of the deformation to the normal cone, and we also recall the construction of the Koszul resolution. These are the main geometric tools used to study singular Bott-Chern classes. Sections §6 to §9 are devoted to the definition and study of the theories of singular Bott-Chern classes. Section §6 contains the definition and first properties. Section §7 is devoted to the classification theorem of such theories. In section §8 we study how properties of the theory of singular Bott-Chern classes and of the associated characteristic class are related. And in section §9 we define the theory of homogeneous singular Bott-Chern classes and we prove that it agrees with the theories defined by Bismut, Gillet and Soulé and by Zha.

Finally in section §10 we define arithmetic K-groups associated to a D_{log}- arithmetic variety (X , C) (in the sense of [13]) and push-forward maps for closed immersions of metrized arithmetic varieties, at the level of the arithmetic Kgroups. After studying the compatibility of these maps with the projection formula and with the push-forward map at the level of currents, we prove a general Riemann-Roch theorem for closed immersions (theorem 10.28) that compares the direct images in the arithmetic K-groups with the direct images in the arithmetic Chow groups. This theorem is compatible, if we choose the theory of homogeneous singular Bott-Chern classes, with the arithmetic Riemann-Roch theorem for closed immersions proved by Bismut, Gillet and Soulé [6] and it agrees with the theorem proved by Zha [32]. Theorem 10.28, together with the arithmetic Grothendieck-Riemann-Roch theorem for submersions proved in [16], can be used to obtain an arithmetic Grothendieck-Riemann-Roch theorem for projective morphisms of regular arithmetic varieties. Acknowledgements: This project was started during the Special Year on Arakelov Theory and Shimura Varieties held at the CRM (Bellaterra, Spain). We would like to thank the CRM for his hospitality during that year. We would also like to thank the University of Barcelona and the University Alexandru Ioan Cuza of Iaşi for their hospitality during several visits that allowed us to finish the project. We would also like to thank K. Köhler, J. Kramer, U. Kühn, V. Maillot, D. Rossler, and J. Wildeshaus with whom we have had many discussions on the subject of this paper. Our special thanks to G. Freixas and Shun Tang for their careful reading of the paper and for suggesting some simplifications of the proofs. Finally we would like to thank the referee for his excellent work.

1 Characteristic classes in analytic Deligne cohomology

A characteristic class for complex vector bundles is a functorial assignment which, to each complex continuous vector bundle on a paracompact topological space X, assigns a class in a suitable cohomology theory of X. For example, if the cohomology theory is singular cohomology, it is well known that each characteristic class can be expressed as a power series in the Chern classes. This can be seen for instance, showing that continuous complex vector bundles on a paracompact space X can be classified by homotopy classes of maps from X to the classifying space BGL_{∞}(ℂ) and that the cohomology of BGL_{∞}(ℂ) is generated by the Chern classes (see for instance [28]). The aim of this section is to show that a similar result is true if we restrict the class of spaces to the class of quasi-projective smooth complex manifolds, the class of maps to the class of algebraic maps and the class of vector bundles to the class of algebraic vector bundles and we choose analytic Deligne cohomology as our cohomology theory. This result and the techniques used to prove it are standard. We will use the splitting principle to reduce to the case of line bundles and will then use the projective spaces as a model of the classifying space BGL_{1}(ℂ). In this section we also recall the definition of Chern classes in analytic Deligne cohomology and we fix some notations that will be used through the paper.

Definition 1.1. Let X be a complex manifold. For each integer p, the analytic real Deligne complex of X is

ℝ_{X,D}(p) = (\underline{ℝ}(p) ⟶ O_{X} ⟶ Ω_{X}^{1} ⟶ … ⟶ Ω_{X}^{p−1}) ≅ s(\underline{ℝ}(p) ⊕ F^{p}Ω_{X}^{∗} ⟶ Ω_{X}^{∗}),

where \underline{ℝ}(p) is the constant sheaf (2πi)^{p}\underline{ℝ} ⊆ \underline{ℂ}. The analytic real Deligne cohomology of X, denoted H_{D^{an}}^{∗}(X, ℝ(p)), is the hyper-cohomology of the above complex.

Analytic Deligne cohomology satisfies the following result.

Theorem 1.2. The assignment X ⟼ H_{D^{an}}^{∗}(X, ℝ(∗)) = ⨁_{p} H_{D^{an}}^{∗}(X, ℝ(p)) is a contravariant functor between the category of complex manifolds and holomorphic maps and the category of unitary bigraded rings that are graded commutative (with respect to the first degree) and associative. Moreover there exists a functorial map

c: Pic(X) = H^{1}(X, O_{X}^{∗}) ⟶ H_{D^{an}}^{2}(X, ℝ(1))

and, for each closed immersion of complex manifolds i: Y ⟶ X of codimension p, there exists a morphism

i_{∗}: H_{D^{an}}^{∗}(Y, ℝ(∗)) ⟶ H_{D^{an}}^{∗+2p}(X, ℝ(∗ + p))

satisfying the properties

A1 Let X be a complex manifold and let E be a holomorphic vector bundle of rank r. Let ℙ(E) be the associated projective bundle and let O(−1) the tautological line bundle. The map

π^{∗}: H_{D^{an}}^{∗}(X, ℝ(∗)) ⟶ H_{D^{an}}^{∗}(ℙ(E), ℝ(∗))

induced by the projection π: ℙ(E) ⟶ X gives to the second ring a structure of left module over the first. Then the elements c(cl(O(−1)))^{i}, i = 0, …, r − 1 form a basis of this module.

A2 If X is a complex manifold, L a line bundle, s a holomorphic section of L that is transverse to the zero section, Y is the zero locus of s and i: Y ⟶ X the inclusion, then

c(cl(L)) = i_{∗}(1_{Y}).

A3 If j: Z ⟶ Y and i: Y ⟶ X are closed immersions of complex manifolds then (ij)_{∗} = i_{∗}j_{∗}.

A4 If i: Y ⟶ X is a closed immersion of complex manifolds then, for every a ∈ H_{D^{an}}^{∗}(X, ℝ(∗)) and b ∈ H_{D^{an}}^{∗}(Y, ℝ(∗))

i_{∗}(bi^{∗}a) = (i_{∗}b)a.

Proof. The functoriality is clear. The product structure is described, for instance, in [15]. The morphism c is defined by the morphism in the derived category

O_{X}^{∗}[1] ⟵^{≅} s(\underline{ℤ}(1) → O_{X}) ⟶ s(\underline{ℝ}(1) → O_{X}) = ℝ_{D}(1).

The morphism i_{∗} can be constructed by resolving the sheaves ℝ_{D}(p) by means of currents (see [26] for a related construction). Properties A3 and A4 follow easily from this construction. By abuse of notation, we will denote by c_{1}(O(−1)) the first Chern class of O(−1) with the algebro-geometric twist, in any of the groups H^{2}(ℙ(E), \underline{ℝ}(1)), H^{2}(ℙ(E), \underline{ℂ}), H^{1}(ℙ(E), Ω_{ℙ(E)}^{1}). Then, we have sheaf isomorphisms (see for instance [22] for a related result),

⨁_{i=0}^{r−1} \underline{ℝ}_{X}(p − i)[−2i] ⟶ Rπ_{∗}\underline{ℝ}_{ℙ(E)}(p)

⨁_{i=0}^{r−1} Ω_{X}^{∗}[−2i] ⟶ Rπ_{∗}Ω_{ℙ(E)}^{∗}

⨁_{i=0}^{r−1} F^{p−i}Ω_{X}^{∗}[−2i] ⟶ Rπ_{∗}F^{p}Ω_{ℙ(E)}^{∗}

given, all of them, by (a_{0}, …, a_{r−1}) ⟼ ∑ a_{i}c_{1}(O(−1))^{i}. Hence we obtain a sheaf isomorphism

⨁_{i=0}^{r−1} ℝ_{X,D}(p − i)[−2i] ⟶ Rπ_{∗}ℝ_{ℙ(E),D}(p)

from which property A1 follows. Finally property A2 in this context is given by the Poincare-Lelong formula (see [13] proposition 5.64).

Notation 1.3. For the convenience of the reader, we gather here together several notations and conventions regarding the differential forms, currents and Deligne cohomology that will be used through the paper. Throughout this paper we will use consistently the algebro-geometric twist. In particular the Chern classes c_{i}, i = 0, … in Betti cohomology will live in c_{i} ∈ H^{2i}(X, ℝ(i)); hence our normalizations differ from the ones in [18] where real forms and currents are used. Moreover we will use the following notations. We will denote by ℰ_{X}^{∗} the sheaf of Dolbeault algebras of differential forms on X and by 𝒟_{X}^{∗} the sheaf of Dolbeault complexes of currents on X (see [13] §5.4 for the structure of Dolbeault complex of 𝒟_{X}^{∗}). We will denote by E^{∗}(X) and by D^{∗}(X) the complexes of global sections of ℰ_{X}^{∗} and 𝒟_{X}^{∗} respectively. Following [9] and [13] definition 5.10, we denote by (D^{∗}( , ∗), d_{D}) the functor that associates to a Dolbeault complex its corresponding Deligne complex. For shorthand, we will denote

D^{∗}(X, p) = D^{∗}(E^{∗}(X), p),
D_{D}^{∗}(X, p) = D^{∗}(D^{∗}(X), p).

To keep track of the algebro-geometric twist we will use the conventions of [13] §5.4 regarding the current associated to a locally integrable differential form

[ω](η) = \frac{1}{(2πi)^{dim X}} ∫_{X} η ∧ ω

and the current associated with a subvariety Y

δ_{Y} (η) = \frac{1}{(2πi)^{dim Y}} ∫_{Y} η.

With these conventions, we have a bigraded morphism D^{∗}(X, ∗) → D_{D}^{∗}(X, ∗) and, if Y has codimension p, the current δ_{Y} belongs to D_{D}^{2p}(X, p). Then D^{∗}(X, p) and D_{D}^{∗}(X, p) are the complex of global sections of an acyclic resolution of ℝ_{X,D}(p). Therefore

H_{D^{an}}^{∗}(X, ℝ(p)) = H^{∗}(D(X, p)) = H^{∗}(D_{D}(X, p)).

If f : X → Y is a proper smooth morphism of complex manifolds of relative dimension e, then the integral along the fibre morphism

f_{∗} : D^{k}(X, p) ⟶ D^{k−2e}(X, p − e)

is given by

f_{∗}ω = \frac{1}{(2πi)^{e}} ∫_{f} ω.  (1.4)

If (D^{∗}(∗), d_{D}) is a Deligne complex associated to a Dolbeault complex, we will write

D̃^{k}(X, p) := D^{k}(X, p)/ d_{D} D^{k−1}(X, p).

Finally, following [13] 5.14 we denote by • the product in the Deligne complex that induces the usual product in Deligne cohomology. Note that, if ω ∈ ⨁_{p} D^{2p}(X, p), then for any η ∈ D^{∗}(X, ∗) we have ω • η = η • ω = η ∧ ω. Sometimes, in this case we will just write ηω := η • ω.

We denote by ∗ the complex manifold consisting on one single point. Then

H_{D^{an}}^{n}(∗, p) = \begin{cases}ℝ(p) := (2πi)^{p}ℝ, & if n = 0, p ≤ 0, \\ ℝ(p − 1) := (2πi)^{p−1}ℝ, & if n = 1, p > 0. \\ \{0\}, & otherwise. \end{cases}

The product structure in this case is the bigraded product that is given by complex number multiplication when the degrees allow the product to be non zero. We will denote by 𝔻 this ring. This is the base ring for analytic Deligne cohomology. Note that, in particular, H_{D^{an}}^{1}(∗, 1) = ℝ = ℂ/ℝ(1). We will denote by 1_{1} the image of 1 in H_{D^{an}}^{1}(∗, 1). Following [23], theorem 1.2 implies the existence of a theory of Chern classes for holomorphic vector bundles in analytic Deligne cohomology. That is, to every vector bundle E, we can associate a collection of Chern classes c_{i}(E) ∈ H_{D^{an}}^{2i}(X, ℝ(i)), i ≥ 1 in a functorial way. We want to see that all possible characteristic classes in analytic Deligne cohomology can be derived from the Chern classes.

Definition 1.5. Let n ≥ 1 be an integer and let r_{1} ≥ 1, …, r_{n} ≥ 1 be a collection of integers. A theory of characteristic classes for n-tuples of vector bundles of rank r_{1}, …, r_{n} is an assignment that, to each n-tuple of isomorphism classes of vector bundles (E_{1}, …, E_{n}) over a complex manifold X, with rk(E_{i}) = r_{i}, assigns a class

cl(E_{1}, …, E_{n}) ∈ ⨁_{k,p} H_{D^{an}}^{k}(X, ℝ(p))

in a functorial way. That is, for every morphism f: X ⟶ Y of complex manifolds, the equality

f^{∗}(cl(E_{1}, …, E_{n})) = cl(f^{∗}E_{1}, …, f^{∗}E_{n})

holds

The first consequence of the functoriality and certain homotopy property of analytic Deligne cohomology classes is the following.

Proposition 1.6. Let cl be a theory of characteristic classes for n-tuples of vector bundles of rank r_{1}, …, r_{n}. Let X be a complex manifold and let (E_{1}, …, E_{n}) be a n-tuple of vector bundles over X with rk(E_{i}) = r_{i} for all i. Let 1 ≤ j ≤ n and let

0 ⟶ E′_{j} ⟶ E_{j} ⟶ E′′_{j} ⟶ 0,

be a short exact sequence. Then the equality

cl(E_{1}, …, E_{j}, …, E_{n}) = cl(E_{1}, …, E′_{j} ⊕ E′′_{j}, …, E_{n})

holds.

Proof. Let ι_{0}, ι_{∞}: X ⟶ X × ℙ^{1} be the inclusion as the fiber over 0 and the fiber over ∞ respectively. Then there exists a vector bundle Ẽ_{j} on X × ℙ^{1} (see for instance [19] (1.2.3.1) or definition 2.5 below) such that ι_{0}^{∗}Ẽ_{j} ≅ E_{j} and ι_{∞}^{∗}Ẽ_{j} ≅ E′_{j} ⊕ E′′_{j}. Let p_{1} : X × ℙ^{1} ⟶ X be the first projection. Let ω ∈ ⨁_{k,p} D^{k}(X, p) be any d_{D}-closed form that represents cl(p_{1}^{∗}E_{1}, …, Ẽ_{j}, …, p_{1}^{∗}E_{n}). Then, by functoriality we know that ι_{0}^{∗}ω represents cl(E_{1}, …, E_{j}, …, E_{n}) and ι_{∞}^{∗}ω represents cl(E_{1}, …, E′_{j} ⊕ E′′_{j}, …, E_{n}). We write

β = \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log tt̄• ω,

where t is the absolute coordinate of ℙ^{1}. Then

d_{D} β = ι_{∞}^{∗}ω − ι_{0}^{∗}ω

which implies the result.

A standard method to produce characteristic classes for vector bundles is to choose hermitian metrics on the vector bundles and to construct closed differential forms out of them. The following result shows that functoriality implies that the cohomology classes represented by these forms are independent from the hermitian metrics and therefore are characteristic classes. When working with hermitian vector bundles we will use the convention that, if E denotes the vector bundle, then \overline{E} = (E, h) will denote the vector bundle together with the hermitian metric.

Proposition 1.7. Let n ≥ 1 be an integer and let r_{1} ≥ 1, …, r_{n} ≥ 1 be a collection of integers. Let cl be an assignment that, to each n-tuple (\overline{E}_{1}, …, \overline{E}_{n}) = ((E_{1}, h_{1}), …, (E_{n}, h_{n})) of isometry classes of hermitian vector bundles of rank r_{1}, …, r_{n} over a complex manifold X, associates a cohomology class

cl(\overline{E}_{1}, …, \overline{E}_{n}) ∈ ⨁_{k,p} H_{D}^{k}(X, ℝ(p))

such that, for each morphism f : Y → X,

cl(f^{∗}\overline{E}_{1}, …, f^{∗}\overline{E}_{n}) = f^{∗} cl(\overline{E}_{1}, …, \overline{E}_{n}).

Then the cohomology class cl(\overline{E}_{1}, …, \overline{E}_{n}) is independent from the hermitian metrics. Therefore it is a well defined characteristic class.

Proof. Let 1 ≤ j ≤ n be an integer and let \overline{E}′_{j} = (E_{j}, h′_{j}) be the vector bundle underlying \overline{E}_{j} with a different choice of metric. Let ι_{0}, ι_{∞} and p_{1} be as in the proof of proposition 1.6. Then we can choose a hermitian metric h on p_{1}^{∗}E_{j}, such that ι_{0}^{∗}(p_{1}^{∗}E_{j}, h) = \overline{E}_{j} and ι_{∞}^{∗}(p_{1}^{∗}E_{j}, h) = \overline{E}′_{j}. Let ω be any smooth closed differential form on X × ℙ^{1} that represents cl(p_{1}^{∗}\overline{E}_{1}, …, (p_{1}^{∗}E_{1}, h), …, p_{1}^{∗}\overline{E}_{n}). Then,

β = \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log tt̄• ω

satisfies

d_{D} β = ι_{∞}^{∗}ω − ι_{0}^{∗}ω

which implies the result.

We are interested in vector bundles that can be extended to a projective variety. Therefore we will restrict ourselves to the algebraic category. So, by a complex algebraic manifold we will mean the complex manifold associated to a smooth quasi-projective variety over ℂ. When working with an algebraic manifold, by a vector bundle we will mean the holomorphic vector bundle associated to an algebraic vector bundle. We will denote by 𝔻[[x_{1}, …, x_{r}]] the ring of commutative formal power series. That is, the unknowns x_{1}, …, x_{r} commute with each other and with 𝔻. We turn it into a commutative bigraded ring by declaring that the unknowns x_{i} have bidegree (2, 1). The symmetric group in r elements, 𝔖_{r} acts on 𝔻[[x_{1}, …, x_{r}]]. The subalgebra of invariant elements is generated over 𝔻 by the elementary symmetric functions. The main result of this section is the following

Theorem 1.8. Let cl be a theory of characteristic classes for n-tuples of vector bundles of rank r_{1}, …, r_{n}. Then, there is a power series φ ∈ 𝔻[[x_{1}, …, x_{r}]] in r = r_{1} + ⋯ + r_{n} variables with coefficients in the ring 𝔻, such that, for each complex algebraic manifold X and each n-tuple of algebraic vector bundles (E_{1}, …, E_{n}) over X with rk(E_{i}) = r_{i} this equality holds:

cl(E_{1}, …, E_{n}) = φ(c_{1}(E_{1}), …, c_{r_{1}}(E_{1}), …, c_{1}(E_{n}), …, c_{r_{n}}(E_{n})). (1.9)

Conversely, any power series φ as before determines a theory of characteristic classes for n-tuples of vector bundles of rank r_{1}, …, r_{n}, by equation (1.9).

Proof. The second statement is obvious from the properties of Chern classes. Since we are assuming X quasi-projective, given n algebraic vector bundles E_{1}, …, E_{n} on X, there is a smooth projective compactification X̃ and vector bundles Ẽ_{1}, …, Ẽ_{n} on X̃, such that E_{i} = Ẽ_{i}|_{X} (see for instance [14] proposition 2.2), we are reduced to the case when X is projective. In this case, analytic Deligne cohomology agrees with ordinary Deligne cohomology. Let us assume first that r_{1} = ⋯ = r_{n} = 1 and that we have a characteristic class cl for n line bundles. Then, for each n-tuple of positive integers m_{1}, …, m_{n} we consider the space ℙ^{m_{1},...,m_{n}} = ℙ_{ℂ}^{m_{1}} × ⋯ × ℙ_{ℂ}^{m_{n}} and we denote by p_{i} the projection over the i-th factor. Then

⨁_{k,p} H_{D}^{k}(ℙ^{m_{1},...,m_{n}}, ℝ(p)) = 𝔻[x_{1}, …, x_{n}]/ (x_{1}^{m_{1}}, …, x_{n}^{m_{n}})

is a quotient of the polynomial ring generated by the classes x_{i} = c_{1}(p_{i}^{∗}O(1)) with coefficients in the ring 𝔻. Therefore, there is a polynomial φ_{m_{1},...,m_{n}} in n variables such that

cl(p_{1}^{∗}O(1), …, p_{1}^{∗}O(1)) = φ_{m_{1},...,m_{n}}(x_{1}, …, x_{n}).

If m_{1} ≤ m′_{1}, …, m_{n} ≤ m′_{n} then, by functoriality, the polynomial φ_{m_{1},...,m_{n}} is the truncation of the polynomial φ_{m′_{1},...,m′_{n}}. Therefore there is a power series in n variables, φ such that φ_{m_{1},...,m_{n}} is the truncation of φ in the appropriate quotient of the polynomial ring. Let L_{1}, …, L_{n} be line bundles on a projective algebraic manifold that are generated by global sections. Then they determine a morphism f: X ⟶ ℙ^{m_{1},...,m_{n}} such that L_{i} = f^{∗}p_{i}^{∗}O(1). Therefore, again by functoriality, we obtain

cl(L_{1}, …, L_{n}) = φ(c_{1}(L_{1}), …, c_{1}(L_{n})).

From the class cl we can define a new characteristic class for n + 1 line bundles by the formula

cl′(L_{1}, …, L_{n}, M) = cl(L_{1} ⊗ M^{∨}, …, L_{n} ⊗ M^{∨}).

When L_{1}, …, L_{n} and M are generated by global sections we have that there is a power series ψ such that

cl′(L_{1}, …, L_{n}, M) = ψ(c_{1}(L_{1}), …, c_{1}(L_{n}), c_{1}(M)).

Moreover, when the line bundles L_{i} ⊗ M^{∨} are also generated by global sections the following holds

ψ(c_{1}(L_{1}), …, c_{1}(L_{n}), c_{1}(M)) = φ(c_{1}(L_{1} ⊗ M^{∨}), …, c_{1}(L_{n} ⊗ M^{∨}))
= φ(c_{1}(L_{1}) − c_{1}(M), …, c_{1}(L_{n}) − c_{1}(M)).

Considering the system of spaces ℙ^{m_{1},...,m_{n},m_{n+1}} with line bundles

L_{i} = p_{i}^{∗}O(1) ⊗ p_{n+1}^{∗}O(1), i = 1, …, n, M = p_{n+1}^{∗}O(1),

we see that there is an identity of power series

φ(x_{1} − y, …, x_{n} − y) = ψ(x_{1}, …, x_{n}, y).

Now let X be a projective complex manifold and let L_{1}, …, L_{n} be arbitrary line bundles. Then there is a line bundle M such that M and L′_{i} = L_{i} ⊗ M, i = 1, …, n are generated by global sections. Then we have

cl(L_{1}, …, L_{n}) = cl(L′_{1} ⊗ M^{∨}, …, L′_{n} ⊗ M^{∨})
= cl′(L′_{1}, …, L′_{n}, M)
= ψ(c_{1}(L′_{1}), …, c_{1}(L′_{n}), c_{1}(M))
= φ((c_{1}(L′_{1}) − c_{1}(M), …, c_{1}(L′_{n}) − c_{1}(M)))
= φ(c_{1}(L_{1}), …, c_{1}(L_{n})).

The case of arbitrary rank vector bundles follows from the case of rank one vector bundles by proposition 1.6 and the splitting principle. We next recall the argument. Given a projective complex manifold X and vector bundles E_{1}, …, E_{n} of rank r_{1}, …, r_{n}, we can find a proper morphism π: X̃ ⟶ X, with X̃ a complex projective manifold, and such that the induced morphism

π^{∗}: H_{D}^{∗}(X, ℝ(∗)) ⟶ H_{D}^{∗}(X̃, ℝ(∗)) is injective and every bundle π^{∗}(E_{i}) admits a holomorphic filtration

0 = K_{i,0} ⊂ K_{i,1} ⊂ ⋯ ⊂ K_{i,r_{i}−1} ⊂ K_{i,r_{i}} = π^{∗}(E_{i}),

with L_{i,j} = K_{i,j}/K_{i,j−1} a line bundle. If cl is a characteristic class for ntuples of vector bundles of rank r_{1}, …, r_{n}, we define a characteristic class for r_{1} + ⋯ + r_{n}-tuples of line bundles by the formula

cl′(L_{1,1}, …, L_{1,r_{1}}, …, L_{n,1}, …, L_{n,r_{n}}) = cl(L_{1,1} ⊕ ⋯ ⊕ L_{1,r_{1}}, …, L_{n,1} ⊕ ⋯ ⊕, L_{n,r_{n}}).

By the case of line bundles we know that there is a power series in r_{1} + ⋯ + r_{n} variables ψ such that

cl′(L_{1,1}, …, L_{1,r_{1}}, …, L_{n,1}, …, L_{n,r_{n}}) = ψ(c_{1}(L_{1,1}), …, c_{1}(L_{n,r_{n}})).

Since the class cl′ is symmetric under the group 𝔖_{r_{1}} × ⋯ × 𝔖_{r_{n}}, the same is true for the power series ψ. Therefore ψ can be written in terms of symmetric elementary functions. That is, there is another power series in r_{1} + ⋯ + r_{n} variables φ, such that

ψ(x_{1,1}, …, x_{n,r_{n}}) = φ(s_{1}(x_{1,1}, …, x_{1,r_{1}}), …, s_{r_{1}}(x_{1,1}, …, x_{1,r_{1}}), … …, s_{1}(x_{n,1}, …, x_{n,r_{n}}), …, s_{r_{n}}(x_{n,1}, …, x_{n,r_{n}})),

where s_{i} is the i-th elementary symmetric function of the appropriate number of variables. Then

π^{∗}(cl(E_{1}, …, E_{n})) = cl(π^{∗}E_{1}, …, π^{∗}E_{n}))
= cl′(L_{1,1}, …, L_{n,r_{n}})
= ψ(c_{1}(L_{1,1}), …, c_{1}(L_{n,r_{n}}))

= φ(c_{1}(π^{∗}E_{1}), …, c_{r_{1}}(π^{∗}E_{1}), …, c_{1}(π^{∗}E_{n}), …, c_{r_{n}}(π^{∗}E_{n}))

= π^{∗}φ(c_{1}(E_{1}), …, c_{r_{1}}(E_{1}), …, c_{1}(E_{n}), …, c_{r_{n}}(E_{n})).

Therefore, the result follows from the injectivity of π^{∗}.

Remark 1.10. It would be interesting to know if the functoriality of a characteristic class in enough to imply that it is a power series in the Chern classes for arbitrary complex manifolds and holomorphic vector bundles.

2 Bott-Chern classes

The aim of this section is to recall the theory of Bott-Chern classes. For more details we refer the reader to [7], [4], [19], [31], [14], [10] and [12]. Note however that the theory we present here is equivalent, although not identical, to the different versions that appear in the literature.

Let X be a complex manifold and let \overline{E} = (E, h) be a rank r holomorphic vector bundle provided with a hermitian metric. Let ϕ ∈ 𝔻[[x_{1}, …, x_{r}]] be a formal power series in r variables that is symmetric under the action of 𝔖_{r}. Let s_{i}, i = 1, …, r be the elementary symmetric functions in r variables. Then ϕ(x_{1}, …, x_{r}) = φ(s_{1}, …, s_{r}) for certain power series φ. By Chern-Weil theory we can obtain a representative of the class

ϕ(E) := φ(c_{1}(E), …, c_{r}(E)) ∈ ⨁_{k,p} H_{D^{an}}^{k}(X, ℝ(p))

as follows. We denote also by ϕ the invariant power series in r × r matrices defined by ϕ. Let K be the curvature matrix of the hermitian holomorphic connection of (E, h). The entries of K in a particular trivialization of E are local sections of D^{2}(X, 1). Then we write

ϕ(E, h) = ϕ(−K) ∈ ⨁_{k,p} D^{k}(X, p).

The form ϕ(E, h) is well defined, closed, and it represents the class ϕ(E). Now let

\overline{E}_{∗} = (… ⟶^{f_{n+1}} \overline{E}_{n} ⟶^{f_{n}} \overline{E}_{n−1} ⟶^{f_{n−1}} …)

be a bounded acyclic complex of hermitian vector bundles; by this we mean a bounded acyclic complex of vector bundles, where each vector bundle is equipped with an arbitrarily chosen hermitian metric. Write

r = ∑_{i even} rk(E_{i}) = ∑_{i odd} rk(E_{i}).

and let ϕ be a symmetric power series in r variables. As before, we can define the Chern forms

ϕ( ⨁_{i even}(E_{i}, h_{i})) and ϕ( ⨁_{i odd}(E_{i}, h_{i})),

that represent the Chern classes ϕ(⨁_{i even} E_{i}) and ϕ(⨁_{i odd} E_{i}). The Chern classes are compatible with respect to exact sequences, that is,

ϕ( ⨁_{i even} E_{i}) = ϕ( ⨁_{i odd} E_{i}).

But, in general, this is not true for the Chern forms. This lack of compatibility with exact sequences on the level of Chern forms is measured by the Bott-Chern classes.

Definition 2.1. Let

\overline{E}_{∗} = (… ⟶^{f_{n+1}} \overline{E}_{n} ⟶^{f_{n}} \overline{E}_{n−1} ⟶^{f_{n−1}} …)

be an acyclic complex of hermitian vector bundles, we will say that \overline{E}_{∗} is an orthogonally split complex of vector bundles if, for any integer n, the exact sequence

0 ⟶ Ker f_{n} ⟶ \overline{E}_{n} ⟶ Ker f_{n−1} ⟶ 0

is split, there is a splitting section s_{n}: Ker f_{n−1} → E_{n} such that \overline{E}_{n} is the orthogonal direct sum of Ker f_{n} and Im s_{n} and the metrics induced in the subbundle Ker f_{n−1} by the inclusion Ker f_{n−1} ⊂ \overline{E}_{n−1} and by the section s_{n} agree.

Notation 2.2. Let (x : y) be homogeneous coordinates of ℙ^{1} and let t = x/y be the absolute coordinate. In order to make certain choices of metrics in a functorial way, we fix once and for all a partition of unity {σ_{0}, σ_{∞}}, over ℙ^{1} subordinated to the open cover of ℙ^{1} given by the open subsets {{|y| > 1/2|x|}, {|x| > 1/2|y|}}. As usual we will write ∞ = (1 : 0), 0 = (0 : 1).

The fundamental result of the theory of Bott-Chern classes is the following theorem (see [7], [4], [19]).

Theorem 2.3. There is a unique way to attach to each bounded exact complex \overline{E}_{∗} as above, a class ϕ̃(\overline{E}_{∗}) in

⨁_{k} D̃^{2k−1}(X, k) = ⨁_{k} D^{2k−1}(X, k)/ Im(d_{D})

satisfying the following properties

(i) (Differential equation)

d_{D} ϕ̃(\overline{E}_{∗}) = ϕ( ⨁_{i even}(E_{i}, h_{i})) − ϕ( ⨁_{i odd}(E_{i}, h_{i})).  (2.4)

(ii) (Functoriality) f^{∗}ϕ̃(\overline{E}_{∗}) = ϕ̃(f^{∗}\overline{E}_{∗}), for every holomorphic map f: X′ ⟶ X.

(iii) (Normalization) If \overline{E}_{∗} is orthogonally split, then ϕ̃(\overline{E}_{∗}) = 0.

Proof. We first recall how to prove the uniqueness. Let \overline{K}_{i} = (K_{i}, g_{i}), where K_{i} = Ker f_{i} and g_{i} is the metric induced by the inclusion K_{i} ⊂ E_{i}. Consider the complex manifold X × ℙ^{1} with projections p_{1} and p_{2}. For every vector bundle F on X we will denote F (i) = p_{1}^{∗}F ⊗p_{2}^{∗}O_{ℙ^{1}}(i). Let C̃_{∗} = C̃(E_{∗})_{∗} be the complex of vector bundles on X × ℙ^{1} given by C̃_{i} = E_{i}(i) ⊕ E_{i−1}(i − 1) with differential d(s, t) = (t, 0). Let D̃_{∗} = D̃(E_{∗})_{∗} be the complex of vector bundles with D̃_{i} = E_{i−1}(i) ⊕ E_{i−2}(i − 1) and differential d(s, t) = (t, 0). Using notation 2.2 we define the map ψ: C̃(E_{∗})_{i} ⟶ D̃(E_{∗})_{i} given by ψ(s, t) = (f_{i}(s) − t ⊗ y, f_{i−1}(t)). It is a morphism of complexes.

Definition 2.5. The first transgression exact sequence of E_{∗} is given by

tr_{1}(E_{∗})_{∗} = Ker ψ.

On X × 𝔸^{1}, the map p_{1}^{∗}E_{i} ⟶ C̃(E_{∗})_{i} given by s ⟼ (s ⊗ y^{i}, f_{i}(s) ⊗ y^{i−1}) induces an isomorphism of complexes

p_{1}^{∗}E_{∗} ⟶ tr_{1}(E_{∗})_{∗}|_{X×𝔸^{1}},  (2.6)

and in particular isomorphisms

tr_{1}(E_{∗})_{i}|_{X×{0}} ≅ E_{i}.  (2.7)

Moreover, we have isomorphisms

tr_{1}(E_{∗})_{i}|_{X×{∞}} ≅ K_{i} ⊕ K_{i−1}.  (2.8)

Definition 2.9. We will denote by tr_{1}(\overline{E}_{∗})_{∗} the complex tr_{1}(E_{∗})_{∗} provided with any hermitian metric such that the isomorphisms (2.7) and (2.8) are isometries. If we need a functorial choice of metric, we proceed as follows. On X × (ℙ^{1} \ {0}) we consider the metric induced by C̃ on tr_{1}(E_{∗})_{∗}. On X × (ℙ^{1} \ {∞}) we consider the metric induced by the isomorphism (2.6). We glue both metrics by means of the partition of unity of notation 2.2.

In particular, we have that tr_{1}(\overline{E}_{∗})|_{X×\{∞\}} is orthogonally split. We assume that there exists a theory of Bott-Chern classes satisfying the above properties. Thus, there exists a class of differential forms ϕ̃(tr_{1}(\overline{E}_{∗})_{∗}) with the following properties. By (i) this class satisfies

d_{D} ϕ̃(tr_{1}(\overline{E}_{∗})_{∗}) = ϕ( ⨁_{i even} tr_{1}(\overline{E}_{∗})_{i})) − ϕ( ⨁_{i odd} tr_{1}(\overline{E}_{∗})_{i}).

By (ii), it satisfies

ϕ̃(tr_{1}(\overline{E}_{∗})_{∗}) |_{X×\{0\}}= ϕ̃(tr_{1}(\overline{E}_{∗})_{∗} |_{X×\{0\}}) = ϕ̃(\overline{E}_{∗}).

Finally, by (ii) and (iii) it satisfies

ϕ̃(tr_{1}(\overline{E}_{∗})_{∗}) |_{X×\{∞\}}= ϕ̃(tr_{1}(\overline{E}_{∗})_{∗} |_{X×\{∞\}}) = 0.

Let ϕ(tr_{1}(\overline{E}_{∗})_{∗}) be any representative of the class ϕ̃(tr_{1}(\overline{E}_{∗})_{∗}). Then, in the group ⨁_{k} D̃^{2k−1}(X, k), we have

0 = d_{D} \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log(tt̄) • ϕ(tr_{1}(\overline{E}_{∗})_{∗})

= \frac{1}{2πi} ∫_{ℙ^{1}} (d_{D} \frac{−1}{2} log(tt̄) • ϕ(tr_{1}(\overline{E}_{∗})_{∗}) − \frac{−1}{2} log(tt̄) • d_{D} ϕ(tr_{1}(\overline{E}_{∗})_{∗}))

= ϕ̃(tr_{1}(\overline{E}_{∗})_{∗})|_{X×\{∞\}} − ϕ̃(tr_{1}(\overline{E}_{∗})_{∗})|_{X×\{0\}}

− \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log(tt̄) • (ϕ( ⨁_{i even} tr_{1}(\overline{E}_{∗})_{i}) − ϕ( ⨁_{i odd} tr_{1}(\overline{E}_{∗})_{i}))

= −ϕ̃(\overline{E}_{∗}) − \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log(tt̄) • (ϕ( ⨁_{i even} tr_{1}(\overline{E}_{∗})_{i}) − ϕ( ⨁_{i odd} tr_{1}(\overline{E}_{∗})_{i})).

Hence, if such a theory exists, it should satisfy the formula

ϕ̃(\overline{E}_{∗}) = \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log(tt̄) • (ϕ( ⨁_{i odd} tr_{1}(\overline{E}_{∗})_{i}) − ϕ( ⨁_{i even} tr_{1}(\overline{E}_{∗})_{i})). (2.10)

Therefore ϕ̃(\overline{E}_{∗}) is determined by properties (i), (ii) and (iii). In order to prove the existence of a theory of functorial Bott-Chern forms, we have to see that the right hand side of equation (2.10) is independent from the choice of the metric on tr_{1}(\overline{E}_{∗})_{∗} and that it satisfies the properties (i), (ii) and (iii). For this the reader can follow the proof of [4] theorem 1.29.

In view of the proof of theorem 2.3, we can define the Bott-Chern classes as follows.

Definition 2.11. Let

\overline{E}_{∗}: 0 ⟶ (E_{n}, h_{n}) ⟶ … ⟶ (E_{1}, h_{1}) ⟶ (E_{0}, h_{0}) ⟶ 0

be a bounded acyclic complex of hermitian vector bundles. Let

r = ∑_{i even} rk(E_{i}) = ∑_{i odd} rk(E_{i}).

Let ϕ ∈ 𝔻[[x_{1}, …, x_{r}]]^{𝔖_{r}} be a symmetric power series in r variables. Then the Bott-Chern class associated to ϕ and \overline{E}_{∗} is the element of ⨁_{k,p} D̃^{k}(E_{X}, p) given by

ϕ̃(\overline{E_{∗}}) = \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log(tt̄) • (ϕ( ⨁_{i odd} tr_{1}(\overline{E}_{∗})_{i}) − ϕ( ⨁_{i even} tr_{1}(\overline{E}_{∗})_{i})).

The following property is obvious from the definition.

Lemma 2.12. Let \overline{E}_{∗} be an acyclic complex of hermitian vector bundles. Then, for any integer k,

ϕ̃(\overline{E}_{∗}[k]) = (−1)^{k}ϕ̃(\overline{E}_{∗}).
□

Particular cases of Bott-Chern classes are obtained when we consider a single vector bundle with two different hermitian metrics or a short exact sequence of vector bundles. Note however that, in order to fix the sign of the Bott-Chern classes on these cases, one has to choose the degree of the vector bundles involved, for instance as in the next definition.

Definition 2.13. Let E be a holomorphic vector bundle of rank r, let h_{0} and h_{1} be two hermitian metrics and let ϕ be an invariant power series of r variables. We will denote by ϕ̃(E, h_{0}, h_{1}) the Bott-Chern class associated to the complex

\overline{ξ}: 0 ⟶ (E, h_{1}) ⟶ (E, h_{0}) ⟶ 0,

where (E, h_{0}) sits in degree zero.

Therefore, this class satisfies

d_{D} ϕ̃(E, h_{0}, h_{1}) = ϕ(E, h_{0}) − ϕ(E, h_{1}).

In fact we can characterize ϕ̃(E, h_{0}, h_{1}) axiomatically as follows.

Proposition 2.14. Given ϕ, a symmetric power series in r variables, there is a unique way to attach, to each rank r vector bundle E on a complex manifold X and metrics h_{0} and h_{1}, a class ϕ̃(E, h_{0}, h_{1}) satisfying

(i) d_{D} ϕ̃(E, h_{0}, h_{1}) = ϕ(E, h_{0}) − ϕ(E, h_{1}).

(ii) f^{∗}ϕ̃(E, h_{0}, h_{1}) = ϕ̃(f^{∗}(E, h_{0}, h_{1})) for every holomorphic map f: Y ⟶ X.

(iii) ϕ̃(E, h, h) = 0.

Moreover, if we denote Ẽ := tr_{1}(\overline{ξ})_{1}, then it satisfies

Ẽ|_{X×{∞}} ≅ (E, h_{0}), Ẽ|_{X×{0}} ≅ (E, h_{1})

and

ϕ̃(E, h_{0}, h_{1}) = \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log(tt̄) • ϕ(Ẽ).  (2.15)

Proof. The axiomatic characterization is proved as in theorem 2.3. In order to prove equation (2.15), if we follow the notations of the proof of theorem 2.3 we have K_{0} = (E, h_{0}) and K_{1} = 0. Therefore tr_{1}(\overline{ξ})_{0} = p_{1}^{∗}(E, h_{0}), while Ẽ := tr_{1}(\overline{ξ})_{1} satisfies Ẽ|_{X×\{0\}} = (E, h_{1}) and Ẽ|_{X×\{∞\}} = (E, h_{0}). Using the antisymmetry of log tt̄ under the involution t ↦ 1/t we obtain

ϕ̃(E, h_{0}, h_{1}) = ϕ̃(\overline{ξ}) = \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log(tt̄) • ϕ(Ẽ).

We can also treat the case of short exact sequences. If

\overline{ε}: 0 ⟶ \overline{E}_{2} ⟶ \overline{E}_{1} ⟶ \overline{E}_{0} ⟶ 0

is a short exact sequence of hermitian vector bundles, by convention, we will assume that \overline{E}_{0} sits in degree zero. This fixs the sign of ϕ̃(\overline{ε}).

Proposition 2.16. Given ϕ, a symmetric power series in r variables, there is a unique way to attach, to each short exact sequence of hermitian vector bundles on a complex manifold X

\overline{ε}: 0 ⟶ \overline{E}_{2} ⟶ \overline{E}_{1} ⟶ \overline{E}_{0} ⟶ 0,

where \overline{E}_{1} has rank r, a class ϕ̃(\overline{ε}) satisfying (i) d_{D} ϕ̃(\overline{ε}) = ϕ(\overline{E}_{0} ⊕ \overline{E}_{2}) − ϕ(\overline{E}_{1}).

(ii) f^{∗}ϕ̃(\overline{ε}) = ϕ̃(f^{∗}(\overline{ϵ})) for every holomorphic map f: Y ⟶ X.

(iii) ϕ̃(\overline{ε}) = 0 whenever \overline{ε} is orthogonally split.

□

The following additivity result of Bott-Chern classes will be useful later.

Lemma 2.17. Let \overline{A}_{∗,∗} be a bounded exact sequence of bounded exact sequences of hermitian vector bundles. Let

r = ∑_{i,j even} rk(A_{i,j}) = ∑_{i,j odd} rk(A_{i,j}) = ∑_{\begin{matrix}i odd \\ j even \end{matrix}} rk(A_{i,j}) = ∑_{\begin{matrix}i even \\ j odd \end{matrix}} rk(A_{i,j}).

Let ϕ be a symmetric power series in r variables. Then

ϕ̃( ⨁_{k even} \overline{A}_{k,∗}) − ϕ̃( ⨁_{k odd} \overline{A}_{k,∗}) = ϕ̃( ⨁_{k even} \overline{A}_{∗,k}) − ϕ̃( ⨁_{k odd} \overline{A}_{∗,k}).

Proof. The proof is analogous to the proof of proposition 6.13 and is left to the reader.

Corollary 2.18. Let \overline{A}_{∗,∗} be a bounded double complex of hermitian vector bundles with exact rows, let

r = ∑_{i+j even} rk(A_{i,j}) = ∑_{i+j odd} rk(A_{i,j})

and let ϕ be a symmetric power series in r variables. Then

ϕ̃(Tot \overline{A}_{∗,∗}) = ϕ̃(⨁_{k} \overline{A}_{∗,k}[−k]).

Proof. Let k_{0} be an integer such that \overline{A}_{k,l} = 0 for k < k_{0}. For any integer n we denote by Tot_{n} = Tot((\overline{A}_{k,l})_{k≥n}) the total complex of the exact complex formed by the rows with index greater or equal than n. Then Tot_{k_{0}} = Tot(\overline{A}_{∗,∗}). For each k there is an exact sequence of complexes

0 ⟶ Tot_{k+1} ⟶ Tot_{k} ⊕ ⨁_{l<k} \overline{A}_{l,∗}[−l] ⟶ ⨁_{l≤k} \overline{A}_{l,∗}[−l] ⟶ 0,

which is orthogonally split in each degree. Therefore by lemma 2.17 we obtain

ϕ̃(Tot_{k} ⊕ ⨁_{l<k} \overline{A}_{l,∗}[−l]) = ϕ̃(Tot_{k−1} ⊕ ⨁_{l≤k} \overline{A}_{l,∗}[−l]).

Hence the result follows by induction.

A particularly important characteristic class is the Chern character. This class is additive for exact sequences. Specializing lemma 2.17 and corollary 2.18 to the Chern character we obtain

Corollary 2.19. With the hypothesis of lemma 2.17, the following equality holds:

∑_{k}(−1)^{k}c̃h(\overline{A}_{k,∗}) = ∑_{k}(−1)^{k}c̃h(\overline{A}_{∗,k}) = c̃h(Tot \overline{A}_{∗,∗}).

□

Our next aim is to extend the Bott-Chern classes associated to the Chern character to metrized coherent sheaves. This extension is due to Zha [32], although it is still unpublished.

Definition 2.20. A metrized coherent sheaf \overline{F} on X is a pair (F, \overline{E}_{∗} → F) where F is a coherent sheaf on X and

0 → \overline{E}_{n} → \overline{E}_{n−1} → ⋯ → \overline{E}_{0} → F → 0

is a finite resolution by hermitian vector bundles of the coherent sheaf F. This resolution is also called the metric of \overline{F}. If \overline{E} is a hermitian vector bundle, we will also denote by \overline{E} the metrized coherent

sheaf (E, \overline{E} ⟶^{id} E).

Note that the coherent sheaf 0 may have non trivial metrics. In fact, any exact sequence of hermitian vector bundles

0 → \overline{A}_{n} → ⋯ → \overline{A}_{0} → 0 → 0

can be seen as a metric on 0. It will be denoted \overline{0}_{A_{∗}}. A metric on 0 is said to be orthogonally split if the exact sequence is orthogonally split. A morphism of metrized coherent sheaves \overline{F}_{1} → \overline{F}_{2} is just a morphism of sheaves F_{1} → F_{2}. A sequence of metrized coherent sheaves

\overline{ε}: … ⟶ \overline{F}_{n+1} ⟶ \overline{F}_{n} ⟶ \overline{F}_{n−1} ⟶ …

is said to be exact if it is exact as a sequence of coherent sheaves.

Definition 2.21. Let \overline{F} = (F, \overline{E}_{∗} → F) be a metrized coherent sheaf. Then the Chern character form associated to \overline{F} is given by

ch(\overline{F}) = ∑_{i}(−1)^{i} ch(\overline{E}_{i}).

Definition 2.22. An exact sequence of metrized coherent sheaves with compatible metrics is a commutative diagram

\begin{aligned}& & ⋮ & & ⋮ & & ⋮ \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & \overline{E}_{n,1} & → & … & → & \overline{E}_{0,1} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & \overline{E}_{n,0} & → & … & → & \overline{E}_{0,0} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & F_{n} & → & … & → & F_{0} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0 \end{aligned}  (2.23)

where all the rows and columns are exact. The columns of this diagram are the individual metrics of each coherent sheaf. We will say that an exact sequence with compatible metrics is orthogonally split if each row of vector bundles is an orthogonally split exact sequence of hermitian vector bundles.

As in the case of exact sequences of hermitian vector bundles, the Chern character form is not compatible with exact sequences of metrized coherent sheaves and we can define a secondary Bott-Chern character which measures the lack of compatibility between the metrics.

Theorem 2.24. 1) There is a unique way to attach to every finite exact sequence of metrized coherent sheaves with compatible metrics

\overline{ε}: 0 → \overline{F}_{n} → ⋯ → \overline{F}_{0} → 0

on a complex manifold X a Bott-Chern secondary character

c̃h(\overline{ε}) ∈ ⨁_{p} D̃^{2p−1}(X, p)

such that the following axioms are satisfied:

(i) (Differential equation)

d_{D} c̃h(\overline{ε}) = ∑_{k}(−1)^{k} ch(\overline{F_{k}}).

(ii) (Functoriality) If f: X′ ⟶ X is a morphism of complex manifolds, that is tor-independent from the coherent sheaves F_{k}, then

f^{∗}(c̃h)(\overline{ε}) = c̃h(f^{∗}\overline{ε}),

where the exact sequence f^{∗}\overline{ε} exists thanks to the tor-independence. (iii) (Horizontal normalization) If \overline{ε} is orthogonally split then

c̃h(\overline{ε}) = 0.

2) There is a unique way to attach to every finite exact sequence of metrized coherent sheaves

\overline{ε}: 0 → \overline{F}_{n} → ⋯ → \overline{F}_{0} → 0

on a complex manifold X a Bott-Chern secondary character

c̃h(\overline{ε}) ∈ ⨁_{p} D̃^{2p−1}(X, p)

such that the axioms (i), (ii) and (iii) above and the axiom (iv) below are satisfied:

(iv) (Vertical normalization) For every bounded complex of hermitian

vector bundles

⋯ → \overline{A}_{k} → ⋯ → \overline{A}_{0} → 0

that is orthogonally split, and every bounded complex of metrized coherent sheaves

\overline{ε}: 0 → \overline{F}_{n} → ⋯ → \overline{F}_{0} → 0

where the metrics are given by \overline{E}_{i,∗} → F_{i}, if, for some i_{0} we denote

\overline{F}′_{i_{0}} = (F_{i_{0}}, \overline{E}_{i_{0},∗} ⊕ \overline{A}_{∗} → F_{i_{0}})

and

\overline{ε}′: 0 → \overline{F}_{n} → ⋯ → \overline{F}′_{i_{0}} → ⋯ → \overline{F}_{0} → 0,

then c̃h(\overline{ε}′) = c̃h(\overline{ε}).

Proof. 1) The uniqueness is proved using the standard deformation argument. By definition, the metrics of the coherent sheaves form a diagram like (2.23). On X × ℙ^{1}, for each j ≥ 0 we consider the exact sequences Ẽ_{∗,j} = tr_{1}(E_{∗,j}) associated to the rows of the diagram with the hermitian metrics of definition 2.9. Then, for each i, j there are maps d: Ẽ_{i,j} → Ẽ_{i−1,j}, and δ: Ẽ_{i,j} → Ẽ_{i,j−1}. We denote

F̃_{i} = Coker(δ: Ẽ_{i,1} → Ẽ_{i,0}).

Using the definition of tr_{1} and diagram chasing one can prove that there is a commutative diagram

\begin{aligned}& & ⋮ & & ⋮ & & ⋮ \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & Ẽ_{n,1} & → & … & → & Ẽ_{0,1} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & Ẽ_{n,0} & → & … & → & Ẽ_{0,0} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & F̃_{n} & → & … & → & F̃_{0} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0 \end{aligned}  (2.25)

where all the rows and columns are exact. In particular this implies that the inclusions i_{0} : X → X × {0} → X × ℙ^{1} and i_{∞}: X → X × {∞} → X × ℙ^{1} are tor-independent from the sheaves F̃_{i}. But i_{0}^{∗}F̃_{∗} is isometric with \overline{F}_{∗} and i_{∞}^{∗}F̃_{∗} is orthogonally split. Hence, by the standard argument, axioms (i), (ii) and (iii) imply that

c̃h(\overline{ε}) = ∑_{j}(−1)^{j}c̃h(\overline{E}_{∗,j}).  (2.26)

To prove the existence we use equation (2.26) as definition. Then the properties of the Bott-Chern classes of exact sequences of hermitian vector bundles imply that axioms (i), (ii) and (iii) are satisfied. Proof of 2). We first assume that such theory exists. Let

⋯ → \overline{A}_{k} → ⋯ → \overline{A}_{0} → 0

be a bounded complex of hermitian vector bundles, non necessarily orthogonally split, and

\overline{ε}: 0 → \overline{F}_{n} → ⋯ → \overline{F}_{0} → 0

a bounded complex of metrized coherent sheaves where the metrics are given by \overline{E}_{i,∗} → F_{i}. As in axiom (iv), for some i_{0} we denote

\overline{F}′_{i_{0}} = (F_{i_{0}}, \overline{E}_{i,∗} ⊕ \overline{A}_{∗} → F_{i_{0}})

and

\overline{ε}′: 0 → \overline{F}_{n} → ⋯ → \overline{F}′_{i_{0}} → ⋯ → \overline{F}_{0} → 0.

By axioms (i), (ii) and (iv), the class (−1)^{i_{0}}(c̃h(\overline{ε}′) − c̃h(\overline{ε})) satisfies the properties that characterize c̃h(A_{∗}). Therefore c̃h(\overline{ε}′) = c̃h(\overline{ε}) + (−1)^{i_{0}} c̃h(A_{∗}). Fix again a number i_{0} and assume that there is an exact sequence of resolutions

[figure: 0; \overline{A}_{•}; \overline{E}′_{i_{0},∗}; \overline{E}_{i_{0},∗}; 0; 0; F_{i_{0}}; F_{i_{0}}]  (2.27)

Let now \overline{ε}′ denote the exact sequence \overline{ε} but with the metric \overline{E}′_{i_{0},∗} in the position i_{0}. Let \overline{η}_{j} denote the j-th row of the diagram (2.27). Again using a deformation argument one sees that

c̃h(\overline{ε}′) − c̃h(\overline{ε}) = (−1)^{i_{0}} (c̃h(\overline{A}_{∗}) − ∑_{j}(−1)^{j} c̃h(η_{j})) .  (2.28)

Choose now a compatible system of metrics

\begin{aligned}& & ⋮ & & ⋮ & & ⋮ \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & \overline{D}_{n,1} & → & … & → & \overline{D}_{0,1} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & \overline{D}_{n,0} & → & … & → & \overline{D}_{0,0} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & F_{n} & → & … & → & F_{0} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0 \end{aligned}  (2.29)

we denote by \overline{λ}_{j} each row of the above diagram. For each i, choose a resolution \overline{E}′_{i,∗} ⟶ F_{i} such that there exist exact sequences of resolutions

[figure: 0; \overline{A}_{i,∗}; \overline{E}′_{i,∗}; \overline{E}_{i,∗}; 0; 0; F_{i}; F_{i}]  (2.30)

and

[figure: 0; \overline{B}_{i,∗}; \overline{E}′_{i,∗}; \overline{D}_{i,∗}; 0; 0; F_{i}; F_{i}]  (2.31)

We denote by \overline{η}_{i,j} each row of the diagram (2.30) and by \overline{μ}_{i,j} each row of the diagram (2.31). Then, by (2.28) and (2.26), we have

c̃h(\overline{ε}) = ∑_{j}(−1)^{j} c̃h(\overline{λ}_{j}) + ∑_{i}(−1)^{i}(c̃h(\overline{B}_{i,∗}) − c̃h(\overline{A}_{i,∗}))

+ ∑_{i,j}(−1)^{i+j}(c̃h(\overline{η}_{i,j}) − c̃h(\overline{μ}_{i,j})) (2.32)

Thus, c̃h(\overline{ε}) is uniquely determined by axioms (i) to (iv). To prove the existence we use equation (2.32) as definition. We have to show that this definition is independent of the choices of the new resolutions. This independence follows from corollary 2.19. Once we know that the Bott-Chern classes are well defined, it is clear that they satisfy axioms (i), (ii), (iii) and (iv).

Proposition 2.33. (Compatibility with exact squares) If

⋮  ⋮  ⋮
↓  ↓  ↓
… → \overline{F}_{n+1,m+1} → \overline{F}_{n+1,m} → \overline{F}_{n+1,m−1} → …
↓ ↓ ↓
… → \overline{F}_{n,m+1} → \overline{F}_{n,m} → \overline{F}_{n,m−1} → …
↓ ↓ ↓
… → \overline{F}_{n−1,m+1} → \overline{F}_{n−1,m} → \overline{F}_{n−1,m−1} → …
↓  ↓  ↓
⋮  ⋮  ⋮

is a bounded commutative diagram of metrized coherent sheaves, where all the rows … (\overline{ε}_{n−1}), (\overline{ε}_{n}), (\overline{ε}_{n+1}), … and all the columns (\overline{η}_{m−1}), (\overline{η}_{m}), (\overline{η}_{m+1}) are exact, then

∑_{n}(−1)^{n}c̃h(\overline{ε}_{n}) = ∑_{m}(−1)^{m}c̃h(\overline{η}_{m}).

Proof. This follows from equation (2.32) and corollary 2.19.

We will use the notation of definition 2.13 also in the case of metrized coherent sheaves. It is easy to verify the following result.

Proposition 2.34. Let

(\overline{ε}) … ⟶ \overline{E}_{n+1} ⟶ \overline{E}_{n} ⟶ \overline{E}_{n−1} ⟶ …

be a finite exact sequence of hermitian vector bundles. Then the Bott-Chern classes obtained by theorem 2.24 and by theorem 2.3 agree. □

Proposition 2.35. Let \overline{F} = (F, \overline{E}_{∗} → F) be a metrized coherent sheaf. We consider the exact sequence of metrized coherent sheaves

\overline{ε}: 0 ⟶ \overline{E}_{n} → ⋯ → \overline{E}_{0} → \overline{F} → 0,

where, by abuse of notation, \overline{E}_{i} = (E_{i}, \overline{E}_{i} →^{=} E_{i}). Then c̃h(\overline{ε}) = 0.

Proof. Define K_{i} = Ker(E_{i} → E_{i−1}), i = 1, …, n and K_{0} = Ker(E_{0} → F). Write

\overline{K_{i}} = (K_{i}, 0 → \overline{E}_{n} → ⋯ → \overline{E}_{i+1} → K_{i}), i = 0, …, n,

and \overline{K}_{−1} = \overline{F}. If we prove that

c̃h(0 → \overline{K_{i}} → \overline{E}_{i} → \overline{K}_{i−1} → 0) = 0,  (2.36)

then we obtain the result by induction using proposition 2.33. In order to prove equation (2.36) we apply equation (2.32). To this end consider resolutions

\overline{D}_{0,∗} ⟶ K_{i−1},  \overline{D}_{0,k} = \overline{E}_{k+i}
\overline{D}_{1,∗} ⟶ E_{i},  \overline{D}_{1,k} = \overline{E}_{k+i+1} ⊕ \overline{E}_{k+i}
\overline{D}_{2,∗} ⟶ K_{i},  \overline{D}_{2,k} = \overline{E}_{k+i+1}

with the map D_{2,k} →^{Δ} D_{1,k} given by s ↦ (s, d s) and the map D_{1,k} →^{∇} D_{0,k} given by (s, t) ↦ t − d s. The differential of the complex D_{1,k} is given by (s, t) ↦ (t, 0). Using equations (2.32) and (2.26) we write the left hand side of equation (2.36) in terms of Bott-Chern classes of vector bundles. All the exact sequences involved are orthogonally split except maybe the sequences

\overline{λ}_{k}: 0 → \overline{D}_{2,k} → \overline{D}_{1,k} → \overline{D}_{0,k} → 0.

But now we consider the diagrams

\xymatrix{\overline{E}_{k+i+1} \ar[r]^{i_{1}} \ar[d]^{id} & \overline{E}_{k+i+1} ⊕ \overline{E}_{k+i} \ar[r]^{p_{2}} \ar[d]^{f} & \overline{E}_{k+i} \ar[d]^{id} \\ _{\overline{E}_{k+i+1}} \ar[r]^{Δ} & \overline{E}_{k+i+1} ⊕ \overline{E}_{k+i} \ar[r]^{∇} & _{\overline{E}_{k+i}}}

and

\xymatrix{\overline{E}_{k+i} \ar[r]^{i_{2}} \ar[d]^{id} & \overline{E}_{k+i+1} ⊕ \overline{E}_{k+i} \ar[r]^{p_{1}} \ar[d]^{f} & \overline{E}_{k+i+1} \ar[d]^{id} \\ _{\overline{E}_{k+i}} \ar[r]^{i_{2}} & \overline{E}_{k+i+1} ⊕ \overline{E}_{k+i} \ar[r]^{p_{1}} & _{\overline{E}_{k+i+1}}} ,

where i_{i}, i_{2} are the natural inclusions, p_{1} and p_{2} are the projections and f(s, t) = (s, t + f(s)). These diagrams and corollary 2.19 imply that c̃h(\overline{λ}_{k}) = 0.

Remark 2.37. In [32], Zha shows that the Bott-Chern classes associated to exact sequences of metrized coherent sheaves are characterized by proposition 2.34, proposition 2.35 and proposition 2.33. We prefer the characterization in terms of the differential equation, the functoriality and the normalization, because it relies on natural extensions of the corresponding axioms that define the Bott-Chern classes for exact sequences of hermitian vector bundles. Moreover, this approach will be used in a subsequent paper where we will study singular Bott-Chern classes associated to arbitrary proper morphisms.

The following generalization of proposition 2.35 will be useful later. Let

ε: 0 → G_{n} → G_{n−1} → ⋯ → G_{0} → F → 0 be a finite resolution of a coherent sheaf by coherent sheaves. Assume that we have a commutative diagram

⋮  ⋮  ⋮
↓  ↓  ↓
\overline{E}_{1,n} → … → \overline{E}_{1,0}
↓ ↓ ↓
\overline{E}_{0,n} → … → \overline{E}_{0,0}
↓ ↓ ↓
0 → \overline{G}_{n} → … → \overline{G}_{0} → F → 0
↓  ↓  ↓
0  0  0

where the columns are exact, the rows are complexes and the \overline{E}_{i,j} are hermitian vector bundles. The columns of this diagram define metrized coherent sheaves \overline{G}_{i}. Let \overline{F} be the metrized coherent sheaf defined by the resolution Tot(\overline{E}_{∗,∗}) ⟶ F.

Proposition 2.38. With the notations above, let \overline{ε} be the exact sequence of metrized coherent sheaves

\overline{ε}: 0 → \overline{G}_{n} → \overline{G}_{n−1} → ⋯ → \overline{G}_{0} → \overline{F} → 0

Then c̃h(\overline{ε}) = 0.

Proof. For each k, let Tot_{k} = Tot((E_{∗,j})_{j≥k}). There are inclusions Tot_{k} ⟶ Tot_{k−1}. Let \overline{D}_{∗,j} = s(Tot_{j+1} → Tot_{j}) with the hermitian metric induced by \overline{E}_{∗,∗}. There are exact sequences of complexes

0 ⟶ \overline{E}_{∗,j} ⟶ \overline{D}_{∗,j} ⟶ s(Tot_{j+1} → Tot_{j+1}) ⟶ 0  (2.39)

that are orthogonally split at each degree. The third complex is orthogonally split. Therefore, if we denote by h_{E} and h_{D} the metric structures of G_{j} induced respectively by the first and second column of diagram (2.39), then

c̃h(G_{j}, h_{E}, h_{D}) = 0.  (2.40)

There is a commutative diagram of resolutions

⋮  ⋮  ⋮  ⋮
↓  ↓  ↓  ↓
0 → \overline{D}_{1,n} → … → \overline{D}_{1,0} → (Tot_{0})_{1} → 0
↓ ↓ ↓ ↓
0 → \overline{D}_{0,n} → … → \overline{D}_{0,0} → (Tot_{0})_{0} → 0
↓  ↓  ↓  ↓
0 → G_{n} → … → G_{0} →  F  → 0
↓  ↓  ↓  ↓
0  0  0  0

where the rows of degree greater or equal than zero are orthogonally split. Hence the result follows from equation (2.26), equation (2.40) and proposition 2.33.

Remark 2.41. We have only defined the Bott-Chern classes associated to the Chern character. Everything applies without change to any additive characteristic class. The reader will find no difficulty to adapt the previous results to any multiplicative characteristic class like the Todd genus or the total Chern class.

3 Direct images of Bott-Chern classes

The aim of this section is to show that certain direct images of Bott-Chern classes are closed. This result is a generalization of results of Bismut, Gillet and Soulé [6] page 325 and of Mourougane [29] proposition 6. The fact that these direct images of Bott-Chern classes are closed implies that certain relations between characteristic classes are true at the level of differential forms (see corollary 3.7 and corollary 3.8). In the first part of this section we deal with differential geometry. Thus all the varieties will be differentiable manifolds. Let G_{1} be a Lie group and let π: N_{2} ⟶ M_{2} be a principal bundle with structure group G_{2} and connection ω_{2}. Assume that there is a left action of G_{1} over N_{2} that commutes with the right action of G_{2} and such that the connection ω_{2} is G_{1}-invariant. Let 𝔤_{1} and 𝔤_{2} be the Lie algebras of G_{1} and G_{2}. Every element γ ∈ 𝔤_{1} defines a tangent vector field γ^{∗} over N_{2} given by

γ_{p}^{∗} = \frac{d}{dt}|_{t=0} exp(tγ)p.

Let (γ^{∗})^{V} be the vertical component of γ^{∗} with respect to the connection ω_{2}. For every point p ∈ N_{2}, we denote by φ(γ, p) ∈ 𝔤_{2} the element characterized by (γ^{∗})_{p}^{V} = φ(γ, p)_{p}^{∗}, where φ(γ, p)^{∗} is the fundamental vector field associated to φ(γ, p). The commutativity of the actions of G_{1} and G_{2} and the invariance of the connection ω_{2} implies that, for g ∈ G_{1} and γ ∈ 𝔤_{1}, the following equalities hold

L_{g∗}(γ^{∗}) = (ad(g)γ^{∗}),  (3.1)
L_{g∗}(γ^{∗})^{V} = (ad(g)γ^{∗})^{V} ,  (3.2)
φ(ad(g)γ, p) = φ(γ, g^{−1}p).  (3.3)

Let G_{2} be the vector bundle over M_{2} associated to N_{2} and the adjoint representation of G_{2}. That is,

G_{2} = N_{2} × 𝔤_{2} /⟨(pg, v) ∼ (p, ad(g)v)⟩ .

Thus, we can identify smooth sections of G_{2} with 𝔤_{2}-valued functions on N_{2} that are invariant under the action of G_{2}. In this way, φ(γ, p) determines a section

φ(γ) ∈ C^{∞}(N_{2}, 𝔤_{2})^{G_{2}} = C^{∞}(M_{2}, G_{2}).

Equation (3.3) implies that, for g ∈ G_{1} and γ ∈ 𝔤_{1},

φ(ad(g)γ) = L_{g^{−1}}^{∗}φ(γ).

We denote by Ω^{ω_{2}} the curvature of the connection ω_{2}. Let P be an invariant function on 𝔤_{2}, then P (Ω^{ω_{2}} + φ(γ)) is a well defined differential form on M_{2}.

Proposition 3.4. Let P be an invariant function on 𝔤_{2} and let μ be a current on M_{2} invariant under the action of G_{1}. Then μ(P (Ω^{ω_{2}}+φ(γ))) is an invariant function on 𝔤_{1}.

Proof. Let g ∈ G_{1}. Then,

μ(P (Ω^{ω_{2}} + φ(ad(g)γ))) = μ(P (Ω^{ω_{2}} + L_{g^{−1}}^{∗}φ(γ)))
= μ(P (L_{g^{−1}}^{∗}Ω^{ω_{2}} + L_{g^{−1}}^{∗}φ(γ)))
= L_{g^{−1}∗}(μ)(P (Ω^{ω_{2}} + φ(γ)))
= μ(P (Ω^{ω_{2}} + φ(γ)))

Let now N_{1} ⟶ M_{1} be a principal bundle with structure group G_{1} and provided with a connection ω_{1}. Then we can form the diagram

N_{1} × N_{2} ⟶^{π_{1}} N_{1} ×_{G_{1}} N_{2}
↓_{π′}  ↓^{π}
N_{1} × M_{2} ⟶^{π_{2}} N_{1} ×_{G_{1}} M_{2}
↓^{q}
M_{1}

Then π is a principal bundle with structure group G_{2}. The connections ω_{1} and ω_{2} induce a connection on the principal bundle π. The subbundle of horizontal vectors with respect to this connection is given by π_{1∗}(T^{H}N_{1}⊕T^{H}N_{2}). We will denote this connection by ω_{1,2}. We are interested in computing the curvature ω_{1,2}. In fact, all the maps in the above diagram are fiber bundles provided with a connection. When applicable, given a vector field U in any of these spaces, we will denote by U^{H,1} the horizontal lifting to N_{1} × N_{2}, by U^{H,2} the horizontal lifting to N_{1} ×_{G_{1}} N_{2} and by U^{H,3} the horizontal lifting to N_{1} ×_{G_{1}} M_{2}.

The tangent space T (N_{1}×N_{2}) can be decomposed as direct sum in the following ways

T (N_{1} × N_{2}) = T^{H}N_{1} ⊕ T^{V} N_{1} ⊕ T^{H}N_{2} ⊕ T^{V} N_{2}
= T^{H}N_{1} ⊕ T^{V} N_{1} ⊕ T^{H}N_{2} ⊕ Ker π_{1∗},  (3.5)

For every point (x, y) ∈ N_{1} × N_{2} we have that (Ker π_{1∗})_{(x,y)} ⊂ T_{x}^{V} N_{1} ⊕ T_{y}N_{2}. Moreover, there is an isomorphism 𝔤_{1} ⟶ (Ker π_{1∗})_{(x,y)} that sends an element γ ∈ 𝔤_{1} to the element (γ_{x}^{∗}, −γ_{y}^{∗}) ∈ T_{x}^{V} N_{1} ⊕ T_{y}N_{2}. The tangent space to N_{1} ×_{G_{1}} M_{2} can be decomposed as the sum of the subbundle

of vertical vectors with respect to q and the subbundle of horizontal vectors defined by the connection ω_{1}. The horizontal lifting to N_{1} × N_{2} of a vertical vector lies in T^{H}N_{2} and the horizontal lifting of a horizontal vector lies in T^{H}N_{1}. Let U, V be two vector fields on M_{1} and let U^{H,3}, V^{H,3} be the horizontal liftings to N_{1} ×_{G_{1}} M_{2}. Then

Ω^{ω_{1,2}}(U^{H,3},V^{H,3}) = [U^{H,3}, V^{H,3}]^{H,2} − [U^{H,2}, V^{H,2}]
= π_{1∗}([U^{H,3}, V^{H,3}]^{H,1} − [U^{H,1}, V^{H,1}])

= π_{1∗}([U^{H,3}, V^{H,3}]^{H,1} − [U, V]^{H,1} + [U, V]^{H,1} − [U^{H,1}, V^{H,1}])

= π_{1,∗}([U^{H,3}, V^{H,3}]^{H,1} − [U, V]^{H,1} + Ω^{ω_{1}}(U, V)).

But, we have

Ω^{ω_{1,2}}(U^{H,3}, V^{H,3}) ∈ T^{V} N_{2},
Ω^{ω_{1}}(U, V) ∈ T^{V} N_{1},
[U^{H,3}, V^{H,3}]^{H,1} − [U, V]^{H,1} ∈ T^{H}N_{2}.

Therefore, by the direct sum decomposition (3.5) we obtain that

Ω^{ω_{1,2}}(U^{H,3}, V^{H,3}) = ((π_{1∗}Ω^{ω_{1}}(U, V)))^{V} ,

where the vertical part is taken with respect to the fib re bundle π. If U is a horizontal vector field over N_{1} ×_{G_{1}} M_{2} and V is a vertical vector field,

a similar argument shows that Ω^{ω_{1,2}}(U, V) = 0. Finally, if U and V are vector fields on M_{2}, they determine vertical vector fields on N_{1} ×_{G_{1}} M_{2}. Then the

horizontal liftings U^{H,1} and V^{H,1} are induced by horizontal liftings of U and V to N_{2}. Therefore, reasoning as before we see that

Ω^{ω_{1,2}}(U, V) = Ω^{ω_{2}}(U, V).

Proposition 3.6. Let G_{1} and G_{2} be Lie groups, with Lie algebras 𝔤_{1} and 𝔤_{2}. For i = 1, 2, let N_{i} ⟶ M_{i} be a principal bundle with structure group G_{i}, provided with a connection ω_{i}. Assume that there is a left action of G_{1} over N_{2} that commutes with the right action of G_{2} and that the connection ω_{2} is invariant under the G_{1}-action. We form the G_{2}-principal bundle π: N_{1} ×_{G_{1}}

N_{2} ⟶ N_{1} ×_{G_{1}} M_{2} with the induced connection ω_{1,2} and curvature Ω^{ω_{1,2}}. Let

P be any invariant function on 𝔤_{2}. Thus P (Ω^{ω_{1,2}}) is a well defined closed differential form on N_{1} ×_{G_{1}} M_{2}. Let μ be a current on M_{2} invariant under the

G_{1}-action. Being G_{1} invariant, the current μ induces a current on N_{1} ×_{G_{1}} M_{2},

that we denote also by μ. Let q: N_{1} ×_{G_{1}} M_{2} ⟶ M_{1} be the projection. Then

q_{∗}(P (Ω^{ω_{1,2}}) ∧ μ) is a closed differential form on M_{1}.

Proof. Let U ⊂ M_{1} be a trivializing open subset for N_{1} and choose a trivialization of N_{1} |_{U} ≅ U ×G_{1}. With this trivialization, we can identify Ω^{ω_{1}} |_{U} with a 2-form on U with values in 𝔤_{1}. For γ ∈ 𝔤_{1}, we denote by

ψ_{μ}(γ) = μ(P (Ω^{ω_{2}} + φ(γ)))

the invariant function provided by proposition 3.4. Then

q_{∗}(P (Ω^{ω_{1,2}}) ∧ μ) = ψ_{μ}(Ω^{ω_{1}}).

Therefore, the result follows from the usual Chern-Weil theory.

We go back now to complex geometry and analytic real Deligne cohomology and to the notations 1.3, in particular (1.4).

Corollary 3.7. Let X be a complex manifold and let \overline{E} = (E, h^{E}) be a rank r hermitian holomorphic vector bundle on X. Let π: ℙ(E) ⟶ X be the associated projective bundle. On ℙ(E) we consider the tautological exact sequence

\overline{ξ}: 0 ⟶ \overline{O(−1)} ⟶ π^{∗}\overline{E} ⟶ \overline{Q} ⟶ 0

where all the vector bundles have the induced metric. Let P_{1}, P_{2} and P_{3} be invariant power series in 1, r − 1 and r variables respectively with coefficients in 𝔻. Let P_{1}(\overline{O(−1)}) and P_{2}(\overline{Q}) be the associated Chern forms and let P̃_{3}(\overline{ξ}) the associated Bott-Chern class. Then

π_{∗}(P_{1}(\overline{O(−1)}) • P_{2}(\overline{Q}) • P̃_{3}(\overline{ξ})) ∈ ⨁_{k} D̃^{2k−1}(X, k)

is closed. Hence it defines a class in analytic real Deligne cohomology. This class does not depend on the hermitian metric of E.

Proof. We consider ℂ^{r} with the standard hermitian metric. On the space ℙ(ℂ^{r}) we have the tautological exact sequence

0 ⟶ O_{ℙ(ℂ^{r})}(−1) ⟶^{f} ℂ^{r} ⟶ Q ⟶ 0.

Let (x : y) be homogeneous coordinates on ℙ^{1} and let t = x/y be the absolute coordinate. Let p_{1} and p_{2} be the two projections of M_{2} = ℙ(ℂ^{r}) × ℙ^{1}. Let Ẽ be the cokernel of the map

\begin{matrix}p_{1}^{∗}O_{ℙ(ℂ^{r})}(−1) & ⟶ & p_{1}^{∗}O_{ℙ(ℂ^{r})}(−1) ⊗ p_{2}^{∗}O_{ℙ^{1}}(1) ⊕ p_{1}^{∗}ℂ^{r} ⊗ p_{2}^{∗}O_{ℙ^{1}}(1) \\ s & ⟼ & s ⊗ y + f(s) ⊗ x \end{matrix}

with the metric induced by the standard metric of ℂ^{r} and the Fubini-Study metric of O_{ℙ(1)}(1). Let N_{2} be the principal bundle over M_{2} formed by the triples (e_{1}, e_{2}, e_{3}), where e_{1}, e_{2} and e_{3} are unitary frames of p_{1}^{∗}O_{ℙ(ℂ^{r})}(−1), p_{1}^{∗}Q and Ẽ respectively. The structure group of this principal bundle is G_{2} = U(1) × U(r − 1) × U(r). Let ω_{2} be the connection induced by the hermitian holomorphic connections on the vector bundles p_{1}^{∗}O_{ℙ(ℂ^{r})}(−1), p_{1}^{∗}Q and Ẽ. Now we denote M_{1} = X, and let N_{1} be the bundle of unitary frames of \overline{E}. This is a principal bundle over M_{1} with structure group G_{1} = U(r). The group G_{1} acts on the left on N_{2}. This action commutes with the right action of G_{2} and the connection ω_{2} is invariant under this action. Let μ = [− log(|t|)] be the current on M_{2} associated to the locally integrable function − log(|t|). This current is invariant under the action of G_{1} because this group acts trivially on the factor ℙ^{1}. The invariant power series P_{1}, P_{2} and P_{3} determine an invariant function P on 𝔤_{2}, the Lie algebra of G_{2}. Let ω_{1} be the connection induced in N_{1} by the holomorphic hermitian connection on \overline{E}. As before let ω_{1,2} be the connection on N_{1} ×_{G_{1}} N_{2} induced

by ω_{1} and ω_{2} and let q: N_{1} ×_{G_{1}} M_{2} ⟶ M_{1} be the projection. Observe that

N_{1} ×_{G_{1}} M_{2} = ℙ(E) × ℙ^{1} and q = π ∘ p_{1}.

By the projection formula and the definition of Bott-Chern classes we have

π_{∗}(P_{1}(\overline{O(−1)}) ∧ P_{2}(\overline{Q}) ∧ P̃_{3}(\overline{ξ})) = q_{∗}(μ • P (Ω^{ω_{1,2}})),

Therefore the fact that it is closed follows from 3.6. Since, for fixed P_{1}, P_{2} and P_{3}, the construction is functorial on (X.\overline{E}), the fact that the class in analytic real Deligne cohomology does not depend on the choice of the hermitian metric follows from proposition 1.7.

Corollary 3.8. Let \overline{E} = (E, h^{E}) be a hermitian holomorphic vector bundle on a complex manifold X. We consider the projective bundle π: ℙ(E ⊕ ℂ) ⟶ X. Let \overline{Q} be the universal quotient bundle on the space ℙ(E ⊕ ℂ) with the induced metric. Then the following equality of differential forms holds

π_{∗} ∑_{i}(−1)^{i} ch(⋀^{i} \overline{Q}^{∨}) = π_{∗}(c_{r}(\overline{Q}) Td^{−1}(\overline{Q})) = Td^{−1}(\overline{E}).

Proof. Let \overline{ξ} be the tautological exact sequence with induced metrics. We first prove that

π_{∗}(c_{r}(\overline{Q}) Td(\overline{O(−1)})) = 1.

We can write Td(\overline{O(−1)}) = 1 + c_{1}(\overline{O(−1)})ϕ(\overline{O(−1)}) for certain power series ϕ. Since c_{r+1}(\overline{E} ⊕ ℂ) = 0 we have

c_{r}(\overline{Q})c_{1}(\overline{O(−1)}) = d_{D} c̃_{r+1}(\overline{ξ}).

Therefore, by corollary 3.7, we have

π_{∗}(c_{r}(\overline{Q}) Td(\overline{O(−1)})) = π_{∗}(c_{r}(\overline{Q})) + π_{∗}(c_{r}(\overline{Q})c_{1}(\overline{O(−1)})ϕ(\overline{O(−1)}))
= 1 + d_{D} π_{∗}(c̃_{r+1}(\overline{ξ})ϕ(\overline{O(−1)}))
= 1.

Then the corollary follows from corollary 3.7 by using the identity

π_{∗}(c_{r}(\overline{Q}) Td^{−1}(\overline{Q})) = π_{∗}(c_{r}(\overline{Q}) Td(\overline{O(−1)})π^{∗} Td^{−1}(\overline{E}))

+ d_{D} π_{∗}(c_{r}(\overline{Q}) Td(\overline{O(−1)})\begin{matrix} \\ Td & −1 \end{matrix}(\overline{ξ})).

The following generalization of corollary 3.7 provides many relations between integrals of Bott-Chern classes and is left to the reader.

Corollary 3.9. Let X be a complex manifold and let \overline{E} = (E, h^{E}) be a rank r hermitian holomorphic vector bundle on X. Let π: ℙ(E) ⟶ X be the associated projective bundle. On ℙ(E) we consider the tautological exact sequence

\overline{ξ}: 0 ⟶ \overline{O(−1)} ⟶ π^{∗}\overline{E} ⟶ \overline{Q} ⟶ 0

where all the vector bundles have the induced metric. Let P_{1} and P_{2} be invariant power series in 1 and r − 1 variables respectively with coefficients in 𝔻 and let P_{3}, …, P_{k} be invariant power series in r variables with coefficients in 𝔻. Let P_{1}(\overline{O(−1)}) and P_{2}(\overline{Q}) be the associated Chern forms and let P̃_{3}(\overline{ξ}), …, P̃_{k}(\overline{ξ}) be the associated Bott-Chern classes. Then

π_{∗}(P_{1}(\overline{O(−1)}) • P_{2}(\overline{Q}) • P̃_{3}(\overline{ξ}) • ⋯ • P̃_{k}(\overline{ξ}))

is a closed differential form on X for any choice of the ordering in computing the non associative product under the integral.

4 Cohomology of currents and wave front sets

The aim of this section is to prove the Poincaré lemma for the complex of currents with fixed wave front set. This implies in particular a certain ∂ ¯∂- lemma (corollary 4.7) that will allow us to control the singularities of singular Bott-Chern classes.

Let X be a complex manifold of dimension n. Following notation 1.3 recall that there is a canonical isomorphism

H_{D^{an}}^{∗}(X, ℝ(p)) ≅ H^{∗}(D_{D}^{∗}(X, p)).

A current η can be viewed as a generalized section of a vector bundle and, as such, has a wave front set that is denoted by WF(η). The theory of wave front sets of distributions is developed in [25] chap. VIII. For the theory of wave front sets of generalized sections, the reader can consult [24] chap. VI. Although we will work with currents and hence with generalized sections of vector bundles, we will follow [25]. The wave front set of η is a closed conical subset of the cotangent bundle of X minus the zero section T^{∗}X_{0} = T^{∗}X \ {0}. This set describes the points and directions of the singularities of η and it allows us to define certain products and inverse images of currents. Let S ⊂ T^{∗}X_{0} be a closed conical subset, we will denote by 𝒟_{X,S}^{∗} the subsheaf of currents whose wave front set is contained in S. We will denote by D^{∗}(X, S) its complex of global sections. For every open set U ⊂ X there is an appropriate notion of convergence in 𝒟_{X,S}^{∗}(U) (see [25] VIII Definition 8.2.2). All references to continuity below are with respect to this notion of convergence. We next summarize the basic properties of wave front sets.

Proposition 4.1. Let u be a generalized section of a vector bundle and let P be a differential operator with smooth coefficients. Then

WF(P u) ⊆ WF(u).

Proof. This is [25] VIII (8.1.11).

Corollary 4.2. The sheaf 𝒟_{X,S}^{∗} is closed under ∂ and ¯∂. Therefore it is a sheaf of Dolbeault complexes.

Let f: X ⟶ Y be a morphism of complex manifolds. The set of normal directions of f is

N_{f} = {(f(x), v) ∈ T^{∗}Y | df(x)^{t}v = 0}.

This set measures the singularities of f. For instance, if f is a smooth map then N_{f} = 0 whereas, if f is a closed immersion, N_{f} is the conormal bundle of f(X). Let S ⊂ T^{∗}Y_{0} be a closed conical subset. We will say that f is transverse to S if N_{f} ∩ S = ∅. We will denote

f^{∗}S = {(x, df(x)^{t}v) ∈ T^{∗}X_{0} | (f(x), v) ∈ S}.

Theorem 4.3. Let f: X ⟶ Y be a morphism of complex manifolds that is transverse to S. Then there exists one and only one extension of the pull-back morphism f^{∗}: ℰ_{Y}^{∗} ⟶ ℰ_{X}^{∗} to a continuous morphism

f^{∗}: 𝒟_{Y,S}^{∗} ⟶ 𝒟_{X,f^{∗}S}^{∗}.

In particular there is a continuous morphism of complexes

D^{∗}(Y, S) ⟶ D^{∗}(X, f^{∗}S).

Proof. This follows from [25] theorem 8.2.4.

We now recall the effect of correspondences on the wave front sets. Let K ∈ D^{∗}(X × Y), and let S be a conical subset of T^{∗}Y_{0}. We will write

WF(K)_{X} = {(x, ξ) ∈ T^{∗}X_{0} | ∃y ∈ Y, (x, y, ξ, 0) ∈ WF(K)}
WF′(K)_{Y} = {(y, η) ∈ T^{∗}Y_{0} | ∃x ∈ X, (x, y, 0, −η) ∈ WF(K)}
WF′(K) ∘ S = {(x, ξ) ∈ T^{∗}X_{0} | ∃(y, η) ∈ S, (x, y, ξ, −η) ∈ WF(K)}.

Theorem 4.4. The image of the correspondence map

\begin{matrix}E_{c}^{∗}(Y) & ⟶ & D^{∗}(X) \\ η & ⟼ & p_{1∗}(K ∧ p_{2}^{∗}(η)) \end{matrix}

is contained in D^{∗}(X, W F (K)_{X}). Moreover, if S ∩ WF′(K)_{Y} = ∅, then there exists one and only one extension to a continuous map

D_{c}^{∗}(Y, S) ⟶ D^{∗}(X, S′),

where S′ = WF(K)_{X} ∪ WF′(K) ∘ S.

Proof. This is [25] theorem 8.2.13.

We are now in a position to state and prove the Poincaré lemma for currents with fixed wave front set. As usual, we will denote by F the Hodge filtration of any Dolbeault complex.

Theorem 4.5 (Poincaré lemma). Let S be any conical subset of T^{∗}X_{0}. Then the natural morphism

ι: (E^{∗}(X), F) ⟶ (D^{∗}(X, S), F)

is a filtered quasi-isomorphism.

Proof. Let K be the Bochner-Martinelli integral operator on ℂ^{n} ×ℂ^{n}. It is the operator

\begin{matrix}E_{c}^{p,q}(ℂ^{n}) & ⟶ & E^{p,q−1}(ℂ^{n}) \\ φ & ⟼ & ∫_{w∈ℂ^{n}} k(z, w) ∧ φ(w), \end{matrix}

where k is the Bochner-Martinelli kernel ([21] pag. 383). Thus k is a differential form on ℂ^{n} × ℂ^{n} with singularities only along the diagonal. Using the explicit description of k in [21], it can be seen that W F (k) = N^{∗}Δ_{0}, the conormal bundle of the diagonal. By theorem 4.4, the operator K defines a continuous linear map from Γ_{c}(ℂ^{n}, 𝒟_{ℂ^{n},S}^{∗}) to Γ(ℂ^{n}, 𝒟_{ℂ^{n},S}^{∗}). This is the key fact that allows us to adapt the proof of the Poincaré Lemma for arbitrary currents to the case of currents with fixed wave front set. We will prove that the sheaf inclusion

(ℰ_{X}, F) ⟶ (𝒟_{X,S}, F)

is a filtered quasi-isomorphism. Then the theorem will follow from the fact that both are fine sheaves. The previous statement is equivalent to the fact that, for any integer p ≥ 0, the inclusion

ι: ℰ_{X}^{p,∗} ⟶ 𝒟_{X,S}^{p,∗}

is a quasi-isomorphism. Let x ∈ X, since exactness can be checked at the level of stalks, we need to show that

ι_{x}: ℰ_{X,x}^{p,∗} ⟶ 𝒟_{X,S,x}^{p,∗}

is a quasi-isomorphism. let U be a coordinate neighborhood around x and let x ∈ V ⊂ U be a relatively compact open subset. Let ρ ∈ C_{c}^{∞}(U) be a function with compact support such that ρ |_{V} = 1. We define an operator

Kρ: 𝒟_{X,S}^{p,q}(U) ⟶ 𝒟_{X,S}^{p,q−1}(V).

If T ∈ 𝒟_{X,S}^{p,q}(U) and φ ∈ E_{c}^{∗}(V) is a test form, then

Kρ(T)(φ) = (−1)^{p+q}T (ρK(φ)).

Hence, using that ¯∂K(φ) + K(¯∂φ) = φ, and that φ = ρφ, we have

(¯∂KρT + Kρ¯∂T + T)(φ) = −T (¯∂(ρ) ∧ K(φ)).

Observe that, even if the support of φ is contained in V, the support of K(φ) can be ℂ^{n}; therefore the right hand side of the above equation may be non zero. We compute

T (¯∂(ρ) ∧ K(φ)) = T (¯∂(ρ) ∧ ∫_{w∈ℂ^{n}} k(w, z) ∧ φ(w))
= T (∫_{w∈ℂ^{n}} ¯∂(ρ) ∧ k(w, z) ∧ φ(w)) .

Since supp(φ) ⊂ V and ¯∂(ρ)|_{V} ≡ 0, we can find a number ϵ > 0 such that, if ∥z − w∥ < ϵ, then ¯∂(ρ) ∧ k(w, z) ∧ φ(w) = 0. Since the singularities of k(w, z) are concentrated on the diagonal, it follows that the differential form ¯∂(ρ) ∧ k(w, z) ∧ φ(w) is smooth. Therefore, the current in V given by

φ ⟼ T (∫_{w∈ℂ^{n}} ¯∂(ρ) ∧ k(w, z) ∧ φ(w)) , is the current associated to the smooth differential form T_{z} (¯∂(ρ) ∧ k(w, z)), where the subindex z means that T only acts on the z variable, being w ∈ V a parameter. This smooth form will be denoted by Ψ(T). Summing up, we have shown that, for any current T ∈ 𝒟_{X,S}^{p,q}(U) there exists a smooth differential form Ψ(T) ∈ ℰ_{X}^{p,q}(V) such that

T |_{V} = −¯∂KρT − Kρ¯∂T − Ψ(T).

Observe that we can not say that Ψ is a quasi-inverse of ι_{x} because it depends on the choice of ρ and it is not possible to choose a single ρ that can be applied to all T. Hence it is not a well defined operator at the level of stalks. Let now T ∈ 𝒟_{X,S,x}^{p,∗} be closed. It is defined in some neighborhood of x, say U′. Applying the above procedure we find a smooth differential form Ψ(T) defined on a relatively compact subset of U′, say V′, that is cohomologous to T. Hence the map induced by ι_{x} in cohomology is surjective. Let ω ∈ ℰ_{X,x}^{p,∗} be closed and such that ι_{x}ω = ¯∂T for some T ∈ 𝒟_{X,S,x}^{p,∗−1}. We may assume that ω and T are defined is some neighborhood U′′ of x. Then, on some relatively compact subset V′′ ⊂ U′′, we have

ω |_{V′′}= ¯∂T |_{V′′}= −¯∂Kρω − ¯∂Ψ(T).

Since Kρω and Ψ(T) are smooth differential forms we conclude that the map induced by ι_{x} in cohomology is injective.

We will denote by D_{D}^{∗}(X, S, p) the Deligne complex associated to D^{∗}(X, S). The following two results are direct consequences of theorem 4.5.

Corollary 4.6. The inclusion D_{D}^{∗}(X, S, p) ⟶ D_{D}^{∗}(X, p) induces an isomorphism

H^{∗}(D_{D}^{∗}(X, S, p)) ≅ H_{D^{an}}^{∗}(X, ℝ(p)).

Corollary 4.7. (i) Let η ∈ D_{D}^{n}(X, p) be a current such that

d_{D} η ∈ D_{D}^{n+1}(X, S, p),

then there is a current a ∈ D_{D}^{n−1}(X, p) such that η + d_{D} a ∈ D_{D}^{n}(X, S, p).

(ii) Let η ∈ D_{D}^{n}(X, S, p) be a current such that there is a current a ∈ D_{D}^{n−1}(X, p) with η = d_{D} a, then there is a current b ∈ D_{D}^{n−1}(X, S, p) such that η = d_{D} b.

□

5 Deformation of resolutions

In this section we will recall the deformation of resolutions based on the Grassmannian graph construction of [1]. We will also recall the Koszul resolution associated to a section of a vector bundle.

The main theme is that given a bounded complex E_{∗} of locally free sheaves (with some properties) on a complex manifold X, one can construct a bounded complex tr_{1}(E_{∗})_{∗} over a certain manifold W. This new manifold has a birational map π: W ⟶ X × ℙ^{1}, that is an isomorphism over X × ℙ^{1} \ {∞}. The complex tr_{1}(E_{∗})_{∗} agrees with the original complex over X ×{0} and is particularly simple over π^{−1}(X ×{∞}). Thus tr_{1}(E_{∗})_{∗} is a deformation of the original complex to a simpler one. The two examples we are interested in are: first, when the original complex is exact, then W agrees with X × ℙ^{1} and tr_{1}(E_{∗})_{∗} was defined in 2.5. Its restriction to π^{−1}(X × {∞}) is split; second, when i: Y ⟶ X is a closed immersion of complex manifolds, and E_{∗} is a bounded resolution of i_{∗}O_{Y}, then W agrees with the deformation to the normal cone of Y and the restriction of tr_{1}(E_{∗})_{∗} to π^{−1}(X × {∞}) is an extension of a Koszul resolution by a split complex. Note that, if we allow singularities, then the Grassmannian graph construction is much more general. The deformation of resolutions is based on the Grassmannian graph construction of [1], and, in the form that we present here, has been developed in [6] and [20]. In order to fix notations we first recall the deformation to the normal cone and the Koszul resolution associated to the zero section of a vector bundle. Let Y ↪ X be a closed immersion of complex manifolds, with Y of pure codimension n. In the sequel we will use notation 2.2. Let W = W_{Y/X} be the blow-up of X × ℙ^{1} along Y × {∞}. Since Y and X × ℙ^{1} are manifolds, W is also a manifold. The map π: W ⟶ X × ℙ^{1} is an isomorphism away from Y × {∞}; we will write P for the exceptional divisor of the blow-up. Then

P = ℙ(N_{Y/X} ⊗ N_{∞/ℙ^{1}}^{−1} ⊕ ℂ).

Thus P can be seen as the projective completion of the vector bundle N_{Y/X} ⊗ N_{∞/ℙ^{1}}^{−1}. Note that N_{∞/ℙ^{1}} is trivial although not canonically trivial. Nevertheless we can choose to trivialize it by means of the section y ∈ O_{ℙ^{1}}(1). Sometimes we will tacitly assume this trivialization and omit N_{∞/ℙ^{1}} from the formulae. The map q_{W} : W ⟶ ℙ^{1}, obtained by composing π with the projection q: X × ℙ^{1} ⟶ ℙ^{1}, is flat and, for t ∈ ℙ^{1}, we have

q_{W}^{−1}(t) ≅ \begin{cases}X × \{t\}, & if t ≠ ∞, \\ P ∪ X̃, & if t = ∞, \end{cases}

where X̃ is the blow-up of X along Y, and P ∩ X̃ is, at the same time, the divisor at ∞ of P and the exceptional divisor of X̃. Following [6] we will use the following notations

\xymatrix{P \ar[r]^{f} \ar[d]_{π_{P}} & W \ar[d]^{π} \\ Y × \{∞\} \ar[r]^{i_{∞}} & X × ℙ^{1}}

i: Y ⟶ X,

W_{∞} = π^{−1}(∞) = P ∪ X̃,
q: X × ℙ^{1} ⟶ ℙ^{1},  the projection,
p: X × ℙ^{1} ⟶ X,  the projection,

q_{W} = q ∘ π p_{W} = p ∘ π

q_{Y} : Y × ℙ^{1} ⟶ ℙ^{1},  the projection,
p_{Y} : Y × ℙ^{1} ⟶ Y,  the projection,
j: Y × ℙ^{1} ⟶ W  the induced map,

j_{∞}: Y × {∞} ⟶ P.

Given any map g: Z ⟶ X × ℙ^{1}, we will denote p_{Z} = p ∘ g and q_{Z} = q ∘ g. For instance p_{P} = p ∘ π ∘ f = p_{W} ∘ f = i ∘ π_{P}, where, in the last equality, we are identifying Y with Y × {∞}. We next recall the construction of the Koszul resolution. Let Y be a complex manifold and let N be a rank n vector bundle. Let P = ℙ(N ⊕ ℂ) be the projective bundle of lines in N ⊕ ℂ. It is obtained by completing N with the divisor at infinity. Let π_{P} : P ⟶ Y be the projection and let s: Y ⟶ P be the zero section. On P there is a tautological short exact sequence

0 ⟶ O(−1) ⟶ π_{P}^{∗} (N ⊕ ℂ) ⟶ Q ⟶ 0.  (5.1)

The above exact sequence and the inclusion ℂ ⟶ π_{P}^{∗} (N ⊕ ℂ) induce a section σ: O_{P} ⟶ Q that vanishes along the zero section s(Y). By duality we obtain a morphism Q^{∨} ⟶ O_{P} that induces a long exact sequence

0 ⟶ ⋀^{n} Q^{∨} ⟶ … ⟶ ⋀^{1} Q^{∨} ⟶ O_{P} ⟶ s_{∗}O_{Y} ⟶ 0.

If F is another vector bundle over Y, we obtain an exact sequence,

0 ⟶ ⋀^{n} Q^{∨} ⊗ π_{P}^{∗} F ⟶ … ⟶ ⋀^{1} Q^{∨} ⊗ π_{P}^{∗} F ⟶ π_{P}^{∗} F ⟶ s_{∗}F ⟶ 0. (5.2)

Definition 5.3. The Koszul resolution of s_{∗}(F) is the resolution (5.2). The complex

0 ⟶ ⋀^{n} Q^{∨} ⊗ π_{P}^{∗} F ⟶ … ⟶ ⋀^{1} Q^{∨} ⊗ π_{P}^{∗} F ⟶ π_{P}^{∗} F ⟶ 0

will be denoted by K(F, N). When \overline{N} is a hermitian vector bundle, the exact sequence (5.1) induces a hermitian metric on Q. If, moreover, \overline{F} is also a hermitian vector bundle, all the vector bundles that appear in the Koszul resolution have an induced hermitian metric. We will denote by K(\overline{F} , \overline{N}) the corresponding complex of hermitian vector bundles.

In particular, we shall write K(\overline{O_{Y}} , \overline{N}) if F = O_{Y} is endowed with the trivial metric ∥1∥ = 1, unless expressly stated otherwise.

We finish this section by recalling the results about deformation of resolutions that will be used in the sequel. For more details see [1] II.1, [6] Section 4 (c) and [20] Section 1.

Theorem 5.4. Let i : Y ↪ X be a closed immersion of complex manifolds, where Y may be empty. Let U = X \ Y. Let F be a vector bundle over Y and E_{∗} ⟶ i_{∗}F ⟶ 0 be a resolution of i_{∗}F. Then there exists a complex manifold W = W (E_{∗}), called the Grassmannian graph construction, with a birational map π: W ⟶ X ×ℙ^{1} and a complex of vector bundles, tr_{1}(E_{∗})_{∗}, over W such that

(i) The map π is an isomorphism away from Y × {∞}. The restriction of tr_{1}(E_{∗})_{∗} to X × (ℙ^{1} \ {∞}) is isomorphic to p_{W}^{∗} E_{∗} restricted to X × (ℙ^{1} \ {∞}). Moreover, If X̃ is the Zariski closure of U × {∞} inside W, the restriction of tr_{1}(E_{∗})_{∗} to X̃ is split acyclic. In particular, if Y is empty or F is the zero vector bundle, hence E_{∗} is acyclic in the whole X, then W = X × ℙ^{1} and tr_{1}(E_{∗})_{∗} is the first transgression exact sequence introduced in 2.5.

(ii) When Y is non-empty and F is a non-zero vector bundle over Y, then W (E_{∗}) agrees with W_{Y/X}, the deformation to the normal cone of Y. Moreover, there is an exact sequence of resolutions on P

\xymatrix{0 \ar[r] & A_{∗} \ar[r] \ar[d] & tr_{1}(E_{∗})_{∗} |_{P} \ar[r] \ar[d] & K(F, N_{Y/X} ⊗ N_{∞/ℙ^{1}}^{−1}) \ar[r] \ar[d] & 0 \\ & 0 \ar[r] & (j_{∞})_{∗}F \ar[r]^{=} & (j_{∞})_{∗}F},

where A_{∗} is split acyclic and K(F, N_{Y/X} ⊗ N_{∞/ℙ^{1}}^{−1}) is the Koszul resolution.

(iii) Let f: X′ ⟶ X be a morphism of complex manifolds and assume that we are in one of the following cases:

(a) The map f is smooth.

(b) The map f is arbitrary and E_{∗} is acyclic.

(c) f is transverse to Y.

Then E′_{∗} := f^{∗}(E_{∗}) is exact over f^{−1}(U),

W′ := W (E′_{∗}) = W ×_{X} X′,

with f_{W} : W′ ⟶ W the induced map, and we have f_{W}^{∗} (tr_{1}(E_{∗})_{∗}) = tr_{1}(f^{∗}(E_{∗}))_{∗}.

(iv) If the vector bundles E_{i} are provided with hermitian metrics, then one can choose a hermitian metric on tr_{1}(E_{∗})_{∗} such that its restriction to X × {0} is isometric to E_{∗} and the restriction to U × {∞} is orthogonally split. We will denote by tr_{1}(\overline{E}_{∗})_{∗} the complex tr_{1}(E_{∗})_{∗} with such a choice of hermitian metrics. Moreover, this choice of metrics can be made functorial. That is, if f is a map as in item (iii), then

f_{W}^{∗} (tr_{1}(\overline{E}_{∗})_{∗}) = tr_{1}(f^{∗}(\overline{E}_{∗}))_{∗}

Proof. The case when E_{∗} is acyclic has already been treated. For the case when Y is non-empty and F is non zero, we first recall the construction of the Grassmannian graph of an arbitrary complex from [20], which is more general than what we need here. If E is a vector bundle over X we will denote by E(i) the vector bundle over X × ℙ^{1} given by E(i) = p^{∗}E ⊗ q^{∗}O(i). Let C̃_{∗} be the complex of locally free sheaves given by C̃_{i} = E_{i}(i) ⊕ E_{i−1}(i − 1) with differential given by d(a, b) = (b, 0). On X × (ℙ^{1} \ {∞}) we consider, for each i, the inclusion of vector bundles γ_{i}: E_{i} ↪ C̃_{i} given by s ⟼ (s ⊗ y^{i}, d s ⊗ y^{i−1}). Let G be the product of the Grassmann bundles Gr(n_{i}, C̃_{i}) that parametrize rank n_{i} = rk E_{i} subbundles of C̃_{i} over X × ℙ^{1}. The inclusion γ_{∗}: ⨁ E_{i} ⟶ ⨁ C̃_{i} induces a section s of G over X × 𝔸^{1}. Then W (E_{∗}) is defined to be the closure of s(X ×𝔸^{1}) in G. Since the projection from G to X × ℙ^{1} is proper, the same is true for the induced map π: W ⟶ X × ℙ^{1}. For each i, the induced map W ⟶ Gr(n_{i}, C̃_{i}) defines a subbundle tr_{1}(E_{∗})_{i} of π^{∗}C̃_{i}. This subbundle agrees with E_{i} over X × 𝔸^{1}. The differential of C̃_{∗} induces a differential on tr_{1}(E_{∗})_{∗}. Assume now that the bundles E_{i} are provided with hermitian metrics. Using the Fubini-Study metric of O(1) we obtain induced metrics on C̃_{i}. Over π^{−1}(X × (ℙ^{1} \ {∞})) we induce a metric on tr_{1}(E_{∗})_{i} by means of the identification with E_{i}. Over π^{−1}(X × (ℙ^{1} \ {0})) we consider on tr_{1}(E_{∗})_{i} the metric induced by C̃_{i}. We glue together both metrics with the partition of unity {σ_{0}, σ_{∞}} of notation 2.2. In the case we are interested there is a more explicit description of tr_{1}(E_{∗})_{∗} given in [6] Section 4 (c). Namely, tr_{1}(E_{∗})_{i} is the kernel of the morphism

ϕ: p_{W}^{∗} C̃_{i} = p_{W}^{∗} E_{i}(i) ⊕ p_{W}^{∗} E_{i−1}(i − 1) ⟶ p_{W}^{∗} E_{i−1}(i) ⊕ p_{W}^{∗} E_{i−2}(i − 1) (5.5)

given by ϕ(s, t) = (d s − t ⊗ y, d t). The only statements that are not explicitly proved in [6] Section 4 (c) or [20] Section 1 are the functoriality when f is not smooth and the properties of the explicit choice of metrics. If the complex E_{∗} is acyclic, then the same is true for E′_{∗} = f^{∗}E_{∗}. In this case W = X × ℙ^{1} and W′ = X′ × ℙ^{1}. Then the functoriality follows from the definition of tr_{1}(E_{∗})_{∗}.

Assume now that we are in case (iii)c. We can form the Cartesian square

\xymatrix{Y′ \ar[r]^{i′} \ar[d]_{g} & X′ \ar[d]^{f} \\ Y \ar[r]^{i} & X}

where i′ is also a closed immersion of complex manifolds. Then we have that E′_{∗} is a resolution of i′_{∗}g^{∗}F. Hence W′ = W (E′_{∗}) is the deformation to the normal cone of Y′ and therefore W′ = W ×_{X} X′. Again the functoriality of

tr_{1}(E_{∗})_{∗} can be checked using the explicit construction of [20] Section 1 that we have recalled above.

Remark 5.6. (i) The definition of tr_{1}(E_{∗}) can be extended to any bounded chain complex over a integral scheme (see [20]).

(ii) There is a sign difference in the definition of the inclusion γ used in [20] and the one used in [6]. We have followed the signs of the first reference.

6 Singular Bott-Chern classes

Throughout this section we will use notation 1.3. In particular we will write

D̃_{D}^{n}(X, p) = D_{D}^{n}(X, p)/ d_{D} D_{D}^{n−1}(X, p),
D̃_{D}^{n}(X, S, p) = D_{D}^{n}(X, S, p)/ d_{D} D_{D}^{n−1}(X, S, p).

A particularly important current is W_{1} ∈ D_{D}^{1}(ℙ^{1}, 1) given by

W_{1} = [\frac{−1}{2} log ∥t∥^{2}].  (6.1)

With the above convention, this means that

W_{1}(η) = \frac{1}{2πi} ∫_{ℙ^{1}} \frac{−1}{2} log ∥t∥^{2} • η.  (6.2)

By the Poincaré-Lelong equation

d_{D} W_{1} = δ_{∞} − δ_{0}.  (6.3)

Note that the current W_{1} was used in the construction of Bott-Chern classes (definition 2.11) and will also have a role in the definition of singular Bott-Chern classes. Before defining singular Bott-Chern classes we need to define the objects that give rise to them.

Definition 6.4. Let i: Y ⟶ X be a closed immersion of complex manifolds. Let N be the normal bundle of Y and let h_{N} be a hermitian metric on N. We denote \overline{N} = (N, h_{N}). Let r_{N} be the rank of N, that agrees with the codimension of Y in X. Let \overline{F} = (F, h_{F}) be a hermitian vector bundle on Y of rank r_{F}. Let \overline{E}_{∗} → i_{∗}F be a metric on the coherent sheaf i_{∗}F. The four-tuple

\overline{ξ} = (i, \overline{N}, \overline{F} , \overline{E}_{∗}).  (6.5)

is called a hermitian embedded vector bundle. The number r_{F} will be called the rank of \overline{ξ} and the number r_{N} will be called the codimension of \overline{ξ}. By convention, any exact complex of hermitian vector bundles on X will be considered a hermitian embedded vector bundle of any rank and codimension.

Obviously, to any hermitian embedded vector bundle we can associate the metrized coherent sheaf (i_{∗}F, \overline{E}_{∗} → i_{∗}F).

Definition 6.6. A singular Bott-Chern class for a hermitian embedded vector bundle \overline{ξ} is a class η̃ ∈ ⨁_{p} D̃_{D}^{2p−1}(X, p) such that

d_{D} η = ∑_{i=0}^{n}(−1)^{i}[ch(\overline{E}_{i})] − i_{∗}([Td^{−1}(\overline{N}) ch(\overline{F})])  (6.7)

for any current η ∈ η̃.

The existence of this class is guaranteed by the Grothendieck-Riemann-Roch theorem, which implies that the two currents in the right hand side of equation (6.7) are cohomologous. Even if we have defined singular Bott-Chern classes as classes of currents with arbitrary singularities, it is an important observation that in each singular Bott-Chern class we can find representatives with controlled singularities. Let N_{Y,0}^{∗} be the conormal bundle of Y with the zero section deleted. It is a closed conical subset of T_{0}^{∗}(X). Since the current

∑_{i=0}^{n}(−1)^{i}[ch(\overline{E}_{i})] − i_{∗}([Td^{−1}(\overline{N}) ch(\overline{F})])

= ∑_{i=0}^{n}(−1)^{i}[ch(\overline{E}_{i})] − Td^{−1}(\overline{N}) ch(\overline{F})δ_{Y}

belongs to D_{D}^{∗}(X, N_{Y,0}^{∗}, p), by corollary 4.7, we obtain

Proposition 6.8. Let \overline{ξ} = (i, \overline{N}, \overline{F} , \overline{E}_{∗}) be a hermitian embedded vector bundle as before. Then any singular Bott-Chern class for \overline{ξ} belongs to the subset

⨁_{p} D̃_{D}^{2p−1}(X, N_{Y,0}^{∗}, p) ⊂ ⨁_{p} D̃_{D}^{2p−1}(X, p).

□

This result will allow us to define inverse images of singular Bott-Chern classes for certain maps. Let f: X′ ⟶ X be a morphism of complex manifolds that is transverse to Y. We form the Cartesian square

\xymatrix{Y′ \ar[r]^{i′} \ar[d]^{g} & X′ \ar[d]^{f} \\ Y \ar[r]^{i} & X}.

Observe that, by the transversality hypothesis, the normal bundle to Y′ on X′ is the inverse image of the normal bundle to Y on X and f^{∗}E_{∗} is a resolution of i′_{∗}g^{∗}F. Thus we write f^{∗}\overline{ξ} = (i′, f^{∗}\overline{N}, g^{∗}\overline{F} , f^{∗}\overline{E}_{∗}), which is a hermitian embedded vector bundle. By proposition 6.8, given any singular Bott-Chern class η̃ for ξ, we can find a representative η ∈ ⨁_{p} D_{D}^{2p−1}(X, N_{Y,0}^{∗}, p). By theorem 4.3, there is a well defined current f^{∗}η and it is a singular Bott-Chern current for f^{∗}ξ. Therefore we can define f^{∗}(η̃) = { \atop f^{∗}(η)}. Again by theorem 4.3, this class does not depend on the choice of the representative η. Our next objective is to study the possible definitions of functorial singular Bott-Chern classes.

Definition 6.9. Let r_{F} and r_{N} be two integers. A theory of singular Bott-Chern classes of rank r_{F} and codimension r_{N} is an assignment which, to each hermitian embedded vector bundle \overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗}) of rank r_{F} and codimension r_{N}, assigns a class of currents

T (\overline{ξ}) ∈ ⨁_{p} D̃_{D}^{2p−1}(X, p)

satisfying the following properties

(i) (Differential equation) The following equality holds

d_{D} T (\overline{ξ}) = ∑_{i}(−1)^{i}[ch(\overline{E}_{i})] − i_{∗}([Td^{−1}(\overline{N}) ch(\overline{F})]).  (6.10)

(ii) (Functoriality) For every morphism f: X′ ⟶ X of complex manifolds that is transverse to Y, then

f^{∗}T (\overline{ξ}) = T (f^{∗}\overline{ξ}).

(iii) (Normalization) Let \overline{A} = (A_{∗}, g_{∗}) be a non-negatively graded orthogonally split complex of vector bundles. Write \overline{ξ} ⊕ \overline{A} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗} ⊕ \overline{A}_{∗}). Then T (\overline{ξ}) = T (\overline{ξ} ⊕ \overline{A}). Moreover, if X = Spec ℂ is one point, Y = ∅ and \overline{E}_{∗} = 0, then T (\overline{ξ}) = 0.

A theory of singular Bott-Chern classes is an assignment as before, for all positive integers r_{F} and r_{M}. When the inclusion i and the bundles F and N are clear from the context, we will denote T (\overline{ξ}) by T (\overline{E}_{∗}). Sometimes we will have to restrict ourselves to complex algebraic manifolds and algebraic vector bundles. In this case we will talk of theory of singular Bott-Chern classes for algebraic vector bundles.

Remark 6.11. (i) Recall that the case when Y = ∅ and \overline{E}_{∗} is any bounded exact sequence of hermitian vector bundles is considered a hermitian embedded vector bundle of arbitrary rank. In this case, the properties above imply that

T (\overline{ξ}) = [c̃h(\overline{E}_{∗})],

where c̃h is the Bott-Chern class associated to the Chern character. That is, for acyclic complexes, any theory of singular Bott-Chern classes agrees with the Bott-Chern classes associated to the Chern character.

(ii) If the map f is transverse to Y, then either f^{−1}(Y) is empty or it has the same codimension as Y. Moreover, it is clear that f^{∗}F has the same rank as F. Therefore, the properties of singular Bott-Chern classes do not mix rank or codimension. This is why we have defined singular Bott-Chern classes for a particular rank and codimension.

(iii) By contrast with the case of Bott-Chern classes, the properties above are not enough to characterize singular Bott-Chern classes.

For the rest of this section we will assume the existence of a theory of singular Bott-Chern classes and we will obtain some consequences of the definition. We start with the compatibility of singular Bott-Chern classes with exact sequences and Bott-Chern classes. Let

\overline{χ}: 0 ⟶ \overline{F}_{n} ⟶ … ⟶ \overline{F}_{1} ⟶ \overline{F}_{0} ⟶ 0  (6.12)

be a bounded exact sequence of hermitian vector bundles on Y. For j = 0, …, n, let \overline{E}_{j,∗} ⟶ i_{∗}F_{j} be a resolution, and assume that they fit in a commutative diagram

\xymatrix{0 \ar[r] & \overline{E}_{n,∗} \ar[r] \ar[d] & … \ar[r] & \overline{E}_{1,∗} \ar[r] \ar[d] & \overline{E}_{0,∗} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & i_{∗}F_{n} \ar[r] & … \ar[r] & i_{∗}F_{1} \ar[r] & i_{∗}F_{0} \ar[r] & 0},

with exact rows. We write \overline{ξ_{j}} = (i: Y ⟶ X, \overline{N}, \overline{F}_{j}, \overline{E}_{j,∗}). For each k, we denote by \overline{η}_{k} the exact sequence

0 ⟶ \overline{E}_{n,k} ⟶ … ⟶ \overline{E}_{1,k} ⟶ \overline{E}_{0,k} ⟶ 0.

Proposition 6.13. With the above notations, the following equation holds:

T ( ⨁_{j even} \overline{ξ}_{j}) − T ( ⨁_{j odd} \overline{ξ}_{j}) = ∑_{k}(−1)^{k}[c̃h(\overline{η}_{k})] − i_{∗}([Td^{−1}(\overline{N})c̃h(\overline{χ})]).

Here the direct sum of hermitian embedded vector bundles, involving the same embedding and the same hermitian normal bundle, is defined in the obvious manner.

Proof. We consider the construction of theorem 5.4 for each of the exact sequences \overline{η}_{k} and the exact sequence \overline{χ}. For each k, we have W_{X} := W (\overline{η}_{k}) = X × ℙ^{1} and we denote W_{Y} := W (\overline{χ}) = Y × ℙ^{1}. On W_{Y} we consider the transgression exact sequence tr_{1}(\overline{χ})_{∗} and on W_{X} we consider the transgression exact sequences tr_{1}(\overline{η_{k}})_{∗}. We denote by j: W_{Y} ⟶ W_{X} the induced morphism. Then there is an exact sequence (of exact sequences)

… ⟶ tr_{1}(\overline{η}_{1})_{∗} ⟶ tr_{1}(\overline{η}_{0})_{∗} ⟶ j_{∗} tr_{1}(\overline{χ})_{∗} ⟶ 0.

We denote

tr_{1}(\overline{χ})_{+} = ⨁_{j even} tr_{1}(\overline{χ})_{j}, tr_{1}(\overline{χ})_{−} = ⨁_{j odd} tr_{1}(\overline{χ})_{j},
tr_{1}(\overline{η}_{k})_{+} = ⨁_{j even} tr_{1}(\overline{η}_{k})_{j}, tr_{1}(\overline{η}_{k})_{−} = ⨁_{j odd} tr_{1}(\overline{η}_{k})_{j},

and

tr_{1}(\overline{ξ})_{+} = (j: W_{Y} ⟶ W_{X}, p_{Y}^{∗} \overline{N}, tr_{1}(\overline{χ})_{+}, tr_{1}(\overline{η}_{∗})_{+}),
tr_{1}(\overline{ξ})_{−} = (j: W_{Y} ⟶ W_{X}, p_{Y}^{∗} \overline{N}, tr_{1}(\overline{χ})_{−}, tr_{1}(\overline{η}_{∗})_{−}),

where here p_{Y} : W_{Y} ⟶ Y denotes the projection. We consider the current on X × ℙ^{1} given by W_{1} • (T (tr_{1}(\overline{ξ})_{+}) − T (tr_{1}(\overline{ξ})_{−})). This current is well defined because the wave front set of W_{1} is the conormal bundle of (X × \{0\}) ∪ (X × \{∞\}), whereas the wave front set of T (tr_{1}(\overline{ξ})_{±}) is the conormal bundle of Y × ℙ^{1}. By the functoriality of the transgression exact sequences, we obtain that

tr_{1}(\overline{ξ})_{+} |_{X×\{0\}}= ⨁_{j even} \overline{ξ}_{j}, tr_{1}(\overline{ξ})_{−} |_{X×\{0\}}= ⨁_{j odd} \overline{ξ}_{j}.

Moreover, using the fact that, for any bounded acyclic complex of hermitian vector bundles \overline{E}_{∗}, the exact sequence tr_{1}(\overline{E}_{∗}) |_{X×\{∞\}} is orthogonally split, we have an isometry

tr_{1}(\overline{ξ})_{+} |_{X×\{∞\}}≅ tr_{1}(\overline{ξ})_{−} |_{X×\{∞\}} .

We now denote by p_{X}: W_{X} ⟶ X the projection. Using the properties that define a theory of singular Bott-Chern classes, in the group ⨁_{p} D̃_{D}^{2p−1}(X, N_{Y,0}^{∗}, p), the following holds

0 = d_{D}(p_{X})_{∗} (W_{1} • T (tr_{1}(\overline{ξ})_{+}) − W_{1} • T (tr_{1}(\overline{ξ})_{−}))

= (T (tr_{1}(\overline{ξ})_{+}) − T (tr_{1}(\overline{ξ})_{−})) |_{X×\{∞\}} − (T (tr_{1}(\overline{ξ})_{+}) − T (tr_{1}(\overline{ξ})_{−})) |_{X×\{0\}}

− (p_{X})_{∗} ∑_{k}(−1)^{k}W_{1} • (ch(tr_{1}(\overline{η}_{k})_{+}) − ch(tr_{1}(\overline{η}_{k})_{−}))

+ (p_{X})_{∗} (W_{1} • j_{∗} [Td^{−1}(p_{Y}^{∗} \overline{N}) ch(tr_{1}(\overline{χ})_{+}) − Td^{−1}(p_{Y}^{∗} \overline{N}) ch(tr_{1}(\overline{χ})_{−})])

= −T ( ⨁_{j even} \overline{ξ}_{j}) + T ( ⨁_{j odd} \overline{ξ}_{j}) + ∑(−1)^{k}[c̃h(\overline{η}_{k})] − i_{∗}[Td^{−1}(\overline{N}) • c̃h(\overline{χ})],

which implies the proposition.

The following result is a consequence of proposition 6.13 and theorem 2.24.

Corollary 6.14. Let Y ⟶ X be a closed immersion of complex manifolds. Let \overline{χ} be an exact sequence of hermitian vector bundles on Y as (6.12). For each j, let ξ_{j} = (i: Y ⟶ X, \overline{N}, \overline{F}_{j}, \overline{E}_{j,∗}) be a hermitian embedded vector bundle. We denote by \overline{ε} the induced exact sequence of metrized coherent sheaves. Then

T ( ⨁_{j even} \overline{ξ}_{j}) − T ( ⨁_{j odd} \overline{ξ}_{j}) = [c̃h(\overline{ε})] − i_{∗}([Td^{−1}(\overline{N})c̃h(\overline{χ})]).

□

We now study the effect of changing the metric of the normal bundle N.

Proposition 6.15. Let \overline{ξ}_{0} = (i, \overline{N}_{0}, \overline{F} , \overline{E}_{∗}) be a hermitian embedded vector bundle, where \overline{N}_{0} = (N, h_{0}). Let h_{1} be another metric in the vector bundle N and write \overline{N}_{1} = (N, h_{1}), \overline{ξ}_{1} = (i, \overline{N}_{1}, \overline{F} , \overline{E}_{∗}). Then

T (\overline{ξ}_{0}) − T (\overline{ξ}_{1}) = −i_{∗}[\begin{matrix} \\ Td & −1 \end{matrix}(N, h_{0}, h_{1}) ch(\overline{F})].

Proof. The proof is completely analogous to the proof of proposition 6.13.

We now study the case when Y is the zero section of a completed vector bundle. Let \overline{F} and \overline{N} be hermitian vector bundles over Y. We denote P = ℙ(N ⊕ ℂ), the projective bundle of lines in N ⊕ O_{Y}. Let s: Y ⟶ P denote the zero section and let π_{P} : P ⟶ Y denote the projection. Let K(\overline{F} , \overline{N}) be the Koszul resolution of definition 5.3. We will use the notations before this definition. The following result is due to Bismut, Gillet and Soulé for the particular choice of singular Bott-Chern classes defined in [6].

Theorem 6.16. Let T be a theory of singular Bott-Chern classes of rank r_{F} and codimension r_{N}. Let Y be a complex manifold and let \overline{F} and \overline{N} be hermitian vector bundles of rank r_{F} and r_{N} respectively. Then the current (π_{P})_{∗}(T (K(\overline{F} , \overline{N}))) is closed. Moreover the cohomology class that it represents does not depend on the metric of N and F and determines a characteristic class for pairs of vector bundles of rank r_{F} and r_{N}. We denote this class by C_{T} (F, N).

Proof. We have that

d_{D}(π_{P})_{∗}(T (K(\overline{F} , \overline{N})))

= (π_{P})_{∗}(d_{D} T (K(\overline{F} , \overline{N})))

= (π_{P})_{∗} (∑_{k=0}^{r}(−1)^{k}[ch(⋀^{k} \overline{Q}^{∨})π_{P}^{∗} ch(\overline{F})] − s_{∗}[Td^{−1}(\overline{N}) ch(\overline{F})])

= ((π_{P})_{∗}[c_{r}(\overline{Q}) Td^{−1}(\overline{Q})] − [Td^{−1}(\overline{N})]) ch(\overline{F})).

Therefore, the fact that the current (π_{P})_{∗}(T (K(\overline{F} , \overline{N}))) is closed follows from corollary 3.8. The fact that this class is functorial on (Y, \overline{N}, \overline{F}) is clear from the construction Thus, the fact that it does not depend on the hermitian metrics of N and F follows from proposition 1.7.

Remark 6.17. By theorem 1.8 we know that, if we restrict ourselves to the algebraic category, C_{T} (F, N) is given by a power series on the Chern classes with coefficients in 𝔻. By degree reasons

C_{T} (F, N) ∈ ⨁_{p} H_{D^{an}}^{2p−1}(Y, ℝ(p)).

Let 1_{1} ∈ H_{D}^{1}(∗, ℝ(1)) be the element determined by the constant function with value 1 in D^{1}(∗, 1). Then C_{T} (F, N)/1_{1} is a power series in the Chern classes of N and F with real coefficients.

7 Classification of theories of singular Bott-Chern classes

The aim of this section is to give a complete classification of the possible theories of singular Bott-Chern classes. This classification is given in terms of the characteristic class C_{T} introduced in the previous section.

Theorem 7.1. Let r_{F} and r_{N} be two positive integers. Let C be a characteristic class for pairs of vector bundles of rank r_{F} and r_{N}. Then there exists a unique theory T_{C} of singular Bott-Chern classes of rank r_{F} and codimension r_{N} such that C_{T_{C}} = C.

Proof. We first prove the uniqueness. Assume that T is a theory of singular Bott-Chern classes such that C_{T} = C. Let \overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗}) be a hermitian embedded vector bundle as in section 6. Let W be the deformation to the normal cone of Y. We will use all the notations of section 5. In particular, we will denote by pX̃: X̃ ⟶ X and p_{P} : P ⟶ X the morphisms induced by restricting p_{W}. Recall that p_{P} can be factored as

P ⟶^{π_{P}} Y ⟶^{i} X.

The normal vector bundle to the inclusion j: Y × ℙ^{1} ⟶ W is isomorphic to p_{Y}^{∗} N ⊗ q_{Y}^{∗} O(−1). We provide it with the hermitian metric induced by the metric of N and the Fubini-Study metric of O(−1) and we denote it by \overline{N}′.

By theorem 5.4 we have a complex of hermitian vector bundles, tr_{1}(E_{∗})_{∗} such that the restriction tr_{1}(E_{∗})_{∗}|_{X×{0}} is isometric to E_{∗}, the restriction tr_{1}(E_{∗})_{∗}|X̃ is orthogonally split and there is an exact sequence on P

0 ⟶ A_{∗} ⟶ tr_{1}(E_{∗})_{∗}|_{P} ⟶ K(F, N) ⟶ 0,

where A_{∗} is split acyclic and K(F, N) is the Koszul resolution. Recall that we have trivialized N_{∞/ℙ^{1}}^{−1} by means of the section y of O_{ℙ^{1}}(1). We choose a hermitian metric in every bundle of A_{∗} such that it becomes orthogonally split. For each k we will denote by \overline{η}_{k} the exact sequence of hermitian vector bundles

0 ⟶ \overline{A}_{k} ⟶ tr_{1}(\overline{E}_{∗})_{k}|_{P} ⟶ K(\overline{F} , \overline{N})_{k} ⟶ 0.  (7.2)

Observe that the current W_{1} is defined as the current associated to a locally integrable differential form. The pull-back of this form to W is also locally integrable. Therefore it defines a current on W that we also denote by W_{1}. Moreover, since the wave front sets of W_{1} and of T (tr_{1}(\overline{E}_{∗})_{∗}) are disjoint, there is a well defined current W_{1} • T (tr_{1}(\overline{E}_{∗})_{∗}). Then, using the properties of singular Bott-Chern classes in definition 6.9, the equality

0 = d_{D}(p_{W})_{∗} (W_{1} • T (tr_{1}(\overline{E}_{∗})_{∗}))

= (pX̃)_{∗}(T (tr_{1}(\overline{E}_{∗})_{∗})|X̃) + (p_{P})_{∗}(T (tr_{1}(\overline{E}_{∗})_{∗})|_{P}) − T (\overline{ξ})

− (p_{W})_{∗} (W_{1} • (∑_{k}(−1)^{k} ch(tr_{1}(\overline{E}_{∗})_{∗}) − (j_{∗}(ch(p_{Y}^{∗} \overline{F}) Td^{−1}(\overline{N}′))))

holds in the group ⨁_{k} D̃^{2k−1}(X, k). By properties 6.9(ii) and 6.9(iii), T (tr_{1}(\overline{E}_{∗})_{∗})|X̃ = T (tr_{1}(\overline{E}_{∗})_{∗}|X̃) = 0. By proposition 6.13 we have

T (tr_{1}(\overline{E}_{∗})_{∗}|_{P}) = T (K(\overline{F} , \overline{N})) − ∑_{k}(−1)^{k}[c̃h(\overline{η}_{k})].

Moreover, we have

(p_{P})_{∗}(T (K(\overline{F} , \overline{N}))) = i_{∗}(π_{P})_{∗}(T (K(\overline{F} , \overline{N}))) = i_{∗}C_{T} (F, N).

By the definition of N′ and the choice of its metric, there are two differential forms a, b on Y, such that

ch(p_{Y}^{∗} \overline{F}) Td^{−1}(\overline{N}′) = p_{Y}^{∗} (a) + p_{Y}^{∗} (b) ∧ q_{Y}^{∗} (c_{1}(O(−1))).

We denote ω = −c_{1}(O(−1)). By the properties of the Fubini-Study metric, ω is invariant under the involution of ℙ^{1} that sends t to 1/t. Then

(p_{W})_{∗} (W_{1} • (j_{∗}(ch(p_{Y}^{∗} \overline{F}) Td^{−1}(\overline{N}′))) = i_{∗}(p_{Y})_{∗}(W_{1} • (p_{Y}^{∗} a + p_{Y}^{∗} bω)) = 0 because the current W_{1} changes sign under the involution t ⟼ 1/t. Summing up, we have obtained the equation

T (\overline{ξ}) = −(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(\overline{E}_{∗})_{k}))

− ∑_{k}(−1)^{k}(p_{P})_{∗}[c̃h(\overline{η}_{k})] + i_{∗}C_{T} (F, N). (7.3)

Hence the singular Bott-Chern class is characterized by the properties of definition 6.9 and the characteristic class C_{T}. In order to prove the existence of a theory of singular Bott-Chern classes, we use equation (7.3) to define a class T_{C}(ξ) as follows.

Definition 7.4. Let C be a characteristic class for pairs of vector bundles of rank r_{F} and r_{N} as in theorem 7.1. Let \overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗}) be as in definition 6.9. Let \overline{A}_{∗}, tr_{1}(\overline{E}_{∗})_{∗} and \overline{η}_{∗} be as in (7.2). Then we define

T_{C}(\overline{ξ}) = −(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(\overline{E}_{∗})_{k}))

− ∑_{k}(−1)^{k}(p_{P})_{∗}[c̃h(\overline{η}_{k})] + i_{∗}C(F, N). (7.5)

We have to prove that this definition does not depend on the choice of the metric of tr_{1}(\overline{E}_{∗})_{∗} or the metric of \overline{A}_{∗}, that T_{C} satisfies the properties of definition 6.9 and that the characteristic class C_{T_{C}} agrees with C. First we prove the independence from the metrics. We denote by h_{k} the hermitian metric on tr_{1}(\overline{E}_{∗})_{k} and by g_{k} the hermitian metric on A_{k}. Let h′_{k} and g′_{k} be another choice of metrics satisfying also that (A_{∗}, g′_{∗}) is orthogonally split, that (tr_{1}(E_{∗})_{k}, h′_{k})|_{X×\{0\}} is isometric to \overline{E}_{k} and that (tr_{1}(E_{∗})_{k}, h′_{k})|X̃ is orthogonally split. We denote by \overline{η}′_{k} the exact sequence η_{k} provided with the metrics g′ and h′. Then, in the group ⨁_{p} D̃^{2p−1}(X, p), we have

∑_{k}(−1)^{k}(p_{P})_{∗}[c̃h(\overline{η}_{k})] − ∑_{k}(−1)^{k}(p_{P})_{∗}[c̃h(\overline{η}′_{k})] =

∑_{k}(−1)^{k}(p_{P})_{∗} [c̃h(A_{k}, g_{k}, g′_{k})] − ∑_{k}(−1)^{k}(p_{P})_{∗} [c̃h(tr_{1}(E_{∗})_{k}|_{P} , h_{k}, h′_{k})] . (7.6)

Observe that the first term of the right hand side vanishes due to the hypothesis of A_{∗} being orthogonally split for both metrics.

Moreover, we also have,

(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(E_{∗})_{k}, h_{k})) −

(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(E_{∗})_{k}, h′_{k})) =

(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • d_{D} c̃h(tr_{1}(E_{∗})_{k}, h_{k}, h′_{k})) . (7.7)

But, in the group ⨁_{p} D̃^{2p−1}(X, p),

(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • d_{D} c̃h(tr_{1}(E_{∗})_{k}, h_{k}, h′_{k})) =

∑_{k}(−1)^{k}(pX̃)_{∗}[c̃h(tr_{1}(E_{∗})_{k}, h_{k}, h′_{k})]|X̃

+ ∑_{k}(−1)^{k}(p_{P})_{∗}[c̃h(tr_{1}(E_{∗})_{k}, h_{k}, h′_{k})]|_{P})

− ∑_{k}(−1)^{k}[c̃h(tr_{1}(E_{∗})_{k}, h_{k}, h′_{k})]|_{X×{0}}. (7.8)

The last term of the right hand side vanishes because the metrics h_{k} and h′_{k} agree when restricted to X × {0} and the first term vanishes by the hypothesis that tr_{1}(E_{∗})_{∗}|X̃ is orthogonally split with both metrics. Combining equations (7.6), (7.7) and (7.8) we obtain that the right hand side of equation (7.5) does not depend on the choice of metrics. We next prove the property (i) of definition 6.9. We compute

d_{D} T_{C}(\overline{ξ}) = − ∑_{k}(−1)^{k} ((pX̃)_{∗} ch(tr_{1}(\overline{E}_{∗})_{k}|X̃) + (p_{P})_{∗} ch(tr_{1}(\overline{E}_{∗})_{k}|_{P}))

+ ∑_{k}(−1)^{k} ch(tr_{1}(\overline{E}_{∗})_{k}|_{X×\{0\}})

− ∑_{k}(−1)^{k}(p_{P})_{∗} (ch(\overline{A}_{k}) + ch(K(\overline{F} , \overline{N})_{k}) − ch(tr_{1}(\overline{E}_{∗})_{k}|_{P})) .

Using that \overline{A}_{∗} and that tr_{1}(\overline{E}_{∗})_{∗}|X̃ are orthogonally split and corollary 3.8 we obtain

d_{D} T_{C}(\overline{ξ}) = ∑_{k}(−1)^{k} ch(\overline{E}_{k}) − ∑_{k}(−1)^{k}(p_{P})_{∗} ch(K(\overline{F} , \overline{N})_{k})

= ∑_{k}(−1)^{k}[ch(\overline{E}_{k})] − (p_{P})_{∗}[c_{r}(\overline{Q}) Td^{−1}(\overline{Q})]

= ∑_{k}(−1)^{k}[ch(\overline{E}_{k})] − i_{∗}[ch(\overline{F}) Td^{−1}(\overline{N})].

We now prove the normalization property. We consider first the case when Y = ∅ and \overline{E}_{∗} is a non-negatively graded orthogonally split complex. We denote by

\overline{K}_{i} = Ker(d_{i}: E_{i} ⟶ E_{i−1})

with the induced metric. By hypothesis there are isometries

\overline{E}_{i} = \overline{K}_{i} ⊕ \overline{K}_{i−1}.

Under these isometries, the differential is d(s, t) = (t, 0). Following the explicit construction of tr_{1}(E_{∗}) given in [20], recalled in definition 2.5, we see that

tr_{1}(E_{∗})_{i} = p^{∗}K_{i} ⊗ q^{∗}O(i) ⊕ p^{∗}K_{i−1} ⊗ q^{∗}O(i − 1) = K_{i}(i) ⊕ K_{i−1}(i − 1).

Moreover, we can induce a metric on tr_{1}(E_{∗})_{∗} satisfying the hypothesis of definition 2.9 by means of the metric of the bundles K_{i} and the Fubini-Study metric on the bundles O(i). It is clear that the second and third terms of the right hand side of equation (7.3) are zero. For the first term we have

∑_{k}(−1)^{k}(p_{W})_{∗}W_{1} • (ch(tr_{1}(\overline{E}_{∗})_{k}))

= (p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(\overline{K}_{k}(k) ⊕^{⊥} \overline{K}_{k−1}(k − 1)))

= (p_{W})_{∗} (W_{1} • (a + b ∧ ω)) ,

where ω is the Fubini-Study (1, 1)-form on ℙ^{1} and a, b are inverse images of differential forms on X. Therefore we obtain that T_{C}(\overline{E}_{∗}) = 0. Now let \overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗}) and let \overline{B}_{∗} be a non-negatively graded orthogonally split complex of vector bundles. By [20] section 1.1, we have that W (E_{∗} ⊕ B_{∗}) = W (E_{∗}) and that

tr_{1}(E_{∗} ⊕ B_{∗}) = tr_{1}(E_{∗}) ⊕ π^{∗} tr_{1}(B_{∗}).

In order to compute T_{C}(\overline{ξ}), we have to consider the exact sequences of hermitian vector bundles over P

\overline{η}_{k}: 0 ⟶ \overline{A}_{k} ⟶ tr_{1}(\overline{E}_{∗})_{k}|_{P} ⟶ K(\overline{F} , \overline{N})_{k} ⟶ 0,

whereas, in order to compute T_{C}(\overline{ξ} ⊕ \overline{B}_{∗}), we consider the sequences

\overline{η}′_{k}:

0 ⟶ \overline{A}_{k}⊕π^{∗}(tr_{1}(\overline{B})_{k})|_{P} ⟶ tr_{1}(\overline{E}_{∗})_{k}⊕π^{∗}(tr_{1}(\overline{B})_{k})|_{P} ⟶ K(\overline{F} , \overline{N})_{k} ⟶ 0.

By the additivity of Bott-Chern classes, we have that c̃h(\overline{η}_{k}) = c̃h(\overline{η}′_{k}). Therefore

T_{C}(\overline{ξ} ⊕ B̄_{∗}) − T_{C}(\overline{ξ}) = −(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(\overline{E}_{∗} ⊕ \overline{B}_{∗})_{k}))

+ (p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(\overline{E}_{∗})_{k}))

= −(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(\overline{B}_{∗})_{k}))

= 0.

The proof of the functoriality is left to the reader. Finally we prove that C_{T_{C}} = C. Let Y be a complex manifold and let \overline{F} and \overline{N} be two hermitian vector bundles. We write X = ℙ(N ⊕ ℂ). Let i: Y ⟶ X be the inclusion given by the zero section and let π_{X}: X ⟶ Y be the projection. On X we have the tautological exact sequence

0 ⟶ O(−1) ⟶ π_{X}^{∗}(N ⊕ ℂ) ⟶ Q ⟶ 0

and the Koszul resolution, denoted K(\overline{F} , \overline{N}). We denote

\overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , K(\overline{F} , \overline{N})).

Using the definition of T_{C}, that is, equation (7.5), and the fact that T_{C} satisfies the properties of definition 6.9, hence equation (7.3) is satisfied, we obtain that

i_{∗}C(F, N) = i_{∗}C_{T_{C}} (F, N)

Applying (π_{X})_{∗} we obtain that C(F, N) = C_{T_{C}} (F, N) which finishes the proof of theorem 7.1.

8 Transitivity and projection formula

We now investigate how different properties of the characteristic class C_{T} are reflected in the corresponding theory of singular Bott-Chern classes.

Proposition 8.1. Let i: Y ↪ X be a closed immersion of complex manifolds. Let \overline{F} be a hermitian vector bundle on Y and \overline{G} a hermitian vector bundle on X. Let \overline{N} denote the normal bundle to Y provided with a hermitian metric. Let \overline{E}_{∗} be a finite resolution of i_{∗}F by hermitian vector bundles. We denote \overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗}) and \overline{ξ} ⊗ \overline{G} = (i: Y ⟶ X, \overline{N}, \overline{F} ⊗ i^{∗}\overline{G}, \overline{E}_{∗} ⊗ \overline{G}). Then

T (\overline{ξ} ⊗ \overline{G}) − T (\overline{ξ}) • ch(\overline{G}) = i_{∗}(C_{T} (F ⊗ i^{∗}G, N)) − i_{∗}(C_{T} (F, N)) • ch(\overline{G}).

Proof. Since the construction of tr_{1}(E_{∗})_{∗} is local on X and Y and compatible with finite sums, we have that

W (E_{∗}) = W (E_{∗} ⊗ G), tr_{1}(\overline{E}_{∗} ⊗ \overline{G})_{∗} = tr_{1}(\overline{E}_{∗})_{∗} ⊗ p_{W}^{∗} \overline{G}.

We first compute

(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(\overline{E}_{∗} ⊗ \overline{G})_{∗}))

= (p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(\overline{E}_{∗})_{∗})p_{W}^{∗} ch(\overline{G}))

= (p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(\overline{E}_{∗})_{∗})) ch(\overline{G}). (8.2)

The Koszul resolution of i_{∗}(F ⊗ i^{∗}G) is given by

K(F ⊗ i^{∗}G, N) = K(F, N) ⊗ p_{P}^{∗} G.

For each k ≥ 0, we will denote by \overline{η}_{k} ⊗ p_{P}^{∗} \overline{G} the exact sequence

0 ⟶ \overline{A}_{k} ⊗ p_{P}^{∗} \overline{G} ⟶ tr_{1}(\overline{E}_{∗} ⊗ \overline{G})_{k}|_{P} ⟶ K(\overline{F} , \overline{N})_{k} ⊗ p_{P}^{∗} \overline{G} ⟶ 0.

Then, we have

(p_{P})_{∗}[c̃h(\overline{η}_{k} ⊗ p_{P}^{∗} \overline{G})] = (p_{P})_{∗}[c̃h(\overline{η}_{k}) • p_{P}^{∗} ch(\overline{G})] = (p_{P})_{∗}[c̃h(\overline{η}_{k})] • ch(\overline{G}) (8.3)

Thus the proposition follows from equation (8.2), equation (8.3) and formula (7.3).

Definition 8.4. We will say that a theory of singular Bott-Chern classes is compatible with the projection formula if, whenever we are in the situation of proposition 8.1, the following equality holds:

T (\overline{ξ} ⊗ \overline{G}) = T (\overline{ξ}) • ch(\overline{G}).

We will say that a characteristic class C (of pairs of vector bundles) is compatible with the projection formula if it satisfies

C(F, N) = C(O_{Y} , N) • ch(F).

Corollary 8.5. A theory of singular Bott-Chern classes T is compatible with the projection formula if and only if it is the case for the associated characteristic class C_{T}.

Proof. Assume that C_{T} is compatible with the projection formula and that we are in the situation of proposition 8.1. Then

i_{∗}C_{T} (F ⊗ i^{∗}G, N)) = i_{∗}(C_{T} (O_{Y} , N) • ch(F ⊗ i^{∗}G))
= i_{∗}(C_{T} (O_{Y} , N) • ch(F)i^{∗} ch(G))
= i_{∗}(C_{T} (O_{Y} , N) • ch(F)) ch(G)
= i_{∗}(C_{T} (F, N)) • ch(G).

Thus, by proposition 8.1, T is compatible with the projection formula. Assume that T is compatible with the projection formula. Let \overline{F} and \overline{N} be hermitian vector bundles over a complex manifold Y. Let s: Y ↪ P := ℙ(N ⊕ ℂ) be the zero section and let π: P ⟶ Y be the projection. Then

C_{T} (F, N) = π_{∗}(T (K(\overline{F} , \overline{N})))
= π_{∗}(T (K(\overline{O}_{Y} , \overline{N}) ⊗ π^{∗}\overline{F}))
= π_{∗}(T (K(\overline{O}_{Y} , \overline{N})) • π^{∗} ch(F))
= π_{∗}(T (K(\overline{O}_{Y} , \overline{N}))) • ch(F)
= C_{T} (O_{Y} , N) • ch(F).

We will next investigate the relationship between singular Bott-Chern classes and compositions of closed immersions. Thus, let

\xymatrix{Y \ar@{^{(}->}[r]^{i_{Y /X}} \ar@{^{(}->}[rr]_{i_{Y /M}} & X \ar@{^{(}->}[r]^{i_{X/M}} & M}

be a composition of closed immersions. Assume that the normal bundles N_{Y/X}, N_{X/M} and N_{Y/M} are provided with hermitian metrics. We will denote by \overline{ε} the exact sequence

\overline{ε}: 0 → \overline{N}_{Y/X} → \overline{N}_{Y/M} → i_{Y/X}^{∗} \overline{N}_{X/M} → 0.  (8.6)

Let P_{X/M} = ℙ(N_{X/M} ⊕ ℂ) be the projective completion of the normal cone to X in M. Then there is an isomorphism

N_{Y/P_{X/M}} ≅ N_{Y/X} ⊕ i_{Y/X}^{∗}N_{X/M}.  (8.7)

We denote by \overline{N}_{Y/P_{X/M}} the vector bundle on the left hand side with the hermitian metric induced by the isomorphism (8.7). Let \overline{F} be a hermitian vector bundle over Y, let \overline{E}_{∗} ⟶ (i_{Y/X})_{∗}F be a resolution by hermitian vector bundles. Let \overline{E}′_{∗,∗} be a complex of complexes of vector bundles over M, such that, for each k ≥ 0, \overline{E}′_{k,∗} ⟶ (i_{X/M})_{∗}E_{k} is a resolution, and there is a commutative diagram of resolutions

\xymatrix{… \ar[r] & E′_{k+1,∗} \ar[r] \ar[d] & E′_{k,∗} \ar[r] \ar[d] & E′_{k−1,∗} \ar[r] \ar[d] & … \\ … \ar[r] & (i_{X/M})_{∗}E_{k+1} \ar[r] & (i_{X/M})_{∗}E_{k} \ar[r] & (i_{X/M})_{∗}E_{k−1} \ar[r] & …} .

It follows that we have a resolution Tot(\overline{E}′_{∗,∗}) ⟶ (i_{Y/M})_{∗}F of (i_{Y/M})_{∗}F by hermitian vector bundles.

Notation 8.8. We will denote

\overline{ξ}_{Y ↪X} = (i_{Y/X}, \overline{N}_{Y/X}, \overline{F} , \overline{E}_{∗}),
\overline{ξ}_{Y ↪M} = (i_{Y/M}, \overline{N}_{Y/M}, \overline{F} , Tot(\overline{E}′_{∗,∗})),
\overline{ξ}_{X↪M,k} = (i_{X/M}, \overline{N}_{X/M}, \overline{E}_{k}, \overline{E}′_{k,∗}).

We will also denote by \overline{ξ}_{Y ↪P_{X/M}} the hermitian embedded vector bundle

(Y ↪ P_{X/M}, \overline{N}_{Y/P_{X/M}} , \overline{F} , Tot(π_{P_{X/M}}^{∗} \overline{E}_{∗} ⊗ K(O_{X}, \overline{N}_{X/M}))) .

Let T be a theory of singular Bott-Chern classes, and let C_{T} be its associated characteristic class. Our aim now is to relate T (\overline{ξ}_{Y ↪X}), T (\overline{ξ}_{Y ↪M}) and T (\overline{ξ}_{X↪M,k}). Let W_{X} be the deformation to the normal cone of X in M. As before we denote by j_{X}: X × ℙ^{1} ⟶ W_{X} the inclusion. We denote by W the deformation to the normal cone of j_{X}(Y × ℙ^{1}) in W_{X}. This double deformation is represented in figure 1. There is a proper map q_{W} : W ⟶ ℙ^{1} × ℙ^{1}. The fibers of q_{W} over the corners of ℙ^{1} × ℙ^{1} are as follows:

q_{W}^{−1}(0, 0) = M,
q_{W}^{−1}(∞, 0) = M̃_{X} × {0} ∪ P_{X/M},
q_{W}^{−1}(0, ∞) = M̃_{Y} ∪ P_{Y/M},
q_{W}^{−1}(∞, ∞) = M̃_{X} × {∞} ∪ P̃_{X/M} ∪ P_{Y/P_{X/M}} ,

where M̃_{X} and M̃_{Y} are the blow-up of M along X and Y respectively, P_{Y/M} = ℙ(N_{Y/M} ⊕ ℂ) is the projective completion of the normal cone to Y in M, P_{Y/P_{X/M}} of the normal cone to Y in P_{X/M} and P̃_{X/M} is the blow-up of P_{X/M}

M  W_{Y} P_{Y/M}  W̃_{X} P_{Y ×ℙ^{1}}  M̃_{X} × ℙ^{1} W_{Y/P} W_{X}  M̃_{X} × {∞} P̃_{X/M} P_{Y/P_{X/M}}  W ℙ^{1} × ℙ^{1} (0, ∞) (0, 0)  (∞, 0)  (∞, ∞) P_{X/M} M̃_{X} × {0} M̃_{Y}

Figure 1: Double deformation

along Y. The preimages by π of the different faces of ℙ^{1} × ℙ^{1} are as follows:

q_{W}^{−1}(ℙ^{1} × {0}) = W_{X},
q_{W}^{−1}({0} × ℙ^{1}) = W_{Y} ,
q_{W}^{−1}(ℙ^{1} × {∞}) = W̃_{X} ∪ P_{Y ×ℙ^{1}},
q_{W}^{−1}({∞} × ℙ^{1}) = M̃_{X} × ℙ^{1} ∪ W_{Y/P} ,

where W_{Y} is the deformation to the normal cone of Y in M, the component W̃_{X} is the blow-up of W_{X} along j_{X}(Y × ℙ^{1}), while P_{Y ×ℙ^{1}} = ℙ(N_{Y ×ℙ^{1}/W_{X}} ⊕ ℂ) is the projective completion of the normal cone to j_{X}(Y × ℙ^{1}) in W_{X} and W_{Y/P} is the deformation to the normal cone of Y inside P_{X/M}. All the above subvarieties will be called boundary components of W.

We will use the following notations for the different maps.

p_{X}: X × ℙ^{1} ⟶ X  p_{Y} : Y × ℙ^{1} ⟶ Y
p_{Y ×ℙ^{1}} : Y × ℙ^{1} × ℙ^{1} ⟶ Y × ℙ^{1}  p_{M̃_{X}×ℙ^{1}} : M̃_{X} × ℙ^{1} ⟶ M
p_{W_{Y /P}} : W_{Y/P} ⟶ M  p_{W_{Y}} : W_{Y} ⟶ M
p_{W_{X}}: W_{X} ⟶ M  p_{P_{Y ×ℙ^{1}}} : P_{Y ×ℙ^{1}} ⟶ M
pW̃_{X}: W̃_{X} ⟶ M  p_{P_{Y /P_{X/M}}} : P_{Y/P_{X/M}} ⟶ M
p_{P_{X/M}} : P_{X/M} ⟶ M  pP̃_{X/M} : P̃_{X/M} ⟶ M
p_{P_{Y /M}} : P_{Y/M} ⟶ M  p_{W} : W ⟶ M
j_{Y} : Y × ℙ^{1} ⟶ W_{Y}  j′_{Y} : Y × ℙ^{1} ⟶ W_{X}
j_{Y ×ℙ^{1}} : Y × ℙ^{1} × ℙ^{1} ⟶ W  i_{Y/P_{X/M}} : Y ⟶ P_{X/M}
π_{P_{X/M}} : P_{X/M} ⟶ X  π_{P_{Y /M}} : P_{Y/M} ⟶ Y
π_{P_{Y /P}} : P_{Y/P_{X/M}} ⟶ Y  π_{P_{Y ×ℙ^{1}}} : P_{Y ×ℙ^{1}} ⟶ Y × ℙ^{1}
πM̃_{X}: M̃_{X} ⟶ M  πM̃_{Y} : M̃_{Y} ⟶ M

Note that the map p_{M̃_{X}×ℙ^{1}} factors through the blow-up M̃_{X} ⟶ M and the

map pW̃_{X} factors through the blow-up M̃_{Y} ⟶ M, whereas the maps p_{W_{Y /P}}, p_{P_{X/M}} and pP̃_{X/M} factor through the inclusion X ↪ M and the maps p_{P_{Y ×ℙ^{1}}} , p_{P_{Y /M}} and p_{P_{Y /P_{X/M}}} factor through the inclusion Y ↪ M. The normal bundle to X ×ℙ^{1} in W_{X} is isomorphic to p_{X}^{∗}N_{X/M} ⊗q_{X}^{∗}O(−1) and we consider on it the metric induced by the metric on \overline{N}_{X/M} and the Fubini-Study metric on O(−1). We denote it by \overline{N}_{X×ℙ^{1}/W_{X}}. The normal bundle to Y × ℙ^{1} in W_{X} satisfies

N_{Y ×ℙ^{1}/W_{X}} |_{Y ×{0}} ≅ N_{Y/M}
N_{Y ×ℙ^{1}/W_{X}} |_{Y ×{∞}} ≅ N_{Y/X} ⊕ i_{Y/X}^{∗}N_{X/M}.

On N_{Y ×ℙ^{1}/W_{X}} we choose a hermitian metric such that the above isomorphisms are isometries. Finally, on the normal bundle to Y × ℙ^{1} × ℙ^{1} in W, we define a metric using the same procedure as the definition of the metric of \overline{N}_{X×ℙ^{1}/W_{X}}. On W_{X} we obtain a sequence of resolutions tr_{1}(\overline{E}′)_{n,∗} ⟶ (j_{X})_{∗}p_{X}^{∗}E_{n}. They form a complex of complexes tr_{1}(\overline{E}′)_{∗,∗} and the associated total complex Tot(tr_{1}(\overline{E}′)_{∗,∗}) provides us with a resolution

Tot(tr_{1}(\overline{E}′)_{∗,∗})_{∗} ⟶ (j′_{Y})_{∗}p_{Y}^{∗} F.  (8.9)

The restriction of Tot(tr_{1}(\overline{E}′)_{∗,∗}) to M is Tot(\overline{E}′_{∗,∗}). The restriction of each complex tr_{1}(\overline{E}′)_{n,∗} to M̃_{X} × \{0\} is orthogonally split. Therefore the restriction of Tot(tr_{1}(\overline{E}′)) to M̃_{X} × \{0\} is the total complex of a complex of orthogonally split complexes. So it is acyclic although not necessarily orthogonally split. The restriction of each complex tr_{1}(\overline{E}′)_{n,∗} to P_{X/M} fits in an exact sequence

0 ⟶ \overline{A}_{n,∗} ⟶ tr_{1}(\overline{E}′)_{n,∗}|_{P_{X/M}} ⟶ π_{P_{X/M}}^{∗} \overline{E}_{n} ⊗ K(\overline{O}_{X}, \overline{N}_{X/M})_{∗} ⟶ 0.

These exact sequences glue together giving a commutative diagram

\xymatrix{Tot(\overline{A}_{∗,∗}) \ar@{^{(}->}[r] \ar[d] & Tot(tr_{1}(\overline{E}′)_{∗,∗}|_{P_{X/M}}) \ar@{->>}[r] \ar[d] & Tot(π_{P_{X/M}}^{∗} \overline{E}_{∗} ⊗ K(\overline{O}_{X}, \overline{N}_{X/M})_{∗}) \ar[d] \\ 0 \ar@{^{(}->}[r] & (i_{Y/P_{X/M}})_{∗}F \ar@{->>}[r] & (i_{Y/P_{X/M}})_{∗}F}

where the rows are short exact sequences. Even if the complexes (\overline{A}_{n})_{∗} are orthogonally split, this is not necessarily the case for Tot(\overline{A}_{∗,∗}). To ease the notation we will denote \overline{A}_{∗} = Tot(\overline{A}_{∗,∗}). Applying theorem 5.4 to the resolution (8.9), we obtain a complex of hermitian vector bundles Ẽ′_{∗} = tr_{1}(Tot(tr_{1}(\overline{E}′)_{∗,∗})) which is a resolution of the coherent sheaf (j_{Y ×ℙ^{1}})_{∗}p_{Y ×ℙ^{1}}^{∗}p_{Y}^{∗} F. We now study the restriction of Ẽ′_{∗} to each of the boundary components of W.

• The restriction of Ẽ′_{∗} to W_{X} is just Tot(tr_{1}(\overline{E}′)) which has already been described. For each k ≥ 0, we will denote by η_{k}^{1} the short exact sequence of hermitian vector bundles on P_{X/M}

\xymatrix{\overline{A}_{k} \ar@{^{(}->}[r] & Tot(tr_{1}(\overline{E}′)_{∗,∗}|_{P_{X/M}})_{k} \ar@{->>}[r] & Tot(π_{P_{X/M}}^{∗} \overline{E} ⊗ K(O_{X}, \overline{N}_{X/M}))_{k}} ,

whereas, for each n, k ≥ 0 we will denote by η_{n,k}^{1} the short exact sequence

\xymatrix{\overline{A}_{n,k} \ar@{^{(}->}[r] & tr_{1}(\overline{E}′)_{n,k}|_{P_{X/M}} \ar@{->>}[r] & π_{P_{X/M}}^{∗} \overline{E}_{n} ⊗ K(O_{X}, \overline{N}_{X/M})_{k}} .

• Its restriction to W_{Y} is tr_{1}(Tot(\overline{E}′)). It is a resolution of (j_{Y})_{∗}p_{Y}^{∗} F. Its restriction to M̃_{Y} is orthogonally split, whereas its restriction to P_{Y/M} fits in an exact sequence

0 ⟶ \overline{B}_{∗} ⟶ tr_{1}(Tot(\overline{E}′))_{∗}|_{P_{Y /M}} ⟶ π_{P_{Y /M}}^{∗} \overline{F} ⊗ K(\overline{O}_{Y} , \overline{N}_{Y/M}) ⟶ 0.

For each k ≥ 0 we will denote by η_{k}^{2} the degree k piece of the above exact sequence.

• Its restriction to M̃_{X} × ℙ^{1} is an acyclic complex, such that its further restriction to M̃_{X} × {0} is acyclic and its restriction to M̃_{X} × {∞} is orthogonally split.

• Its restriction to W_{Y/P} fits in a short exact sequence

0 → tr_{1}(\overline{A}_{∗}) → Ẽ′_{∗}|_{W_{Y /P}} → tr_{1}(Tot(π_{P_{X/M}}^{∗} \overline{E} ⊗ K(\overline{O}_{X}, \overline{N}_{X/M}))) → 0.

For each k ≥ 0, we will denote by μ_{k}^{1} the exact sequence of hermitian vector bundles over W_{Y/P} given by the piece of degree k of this exact sequence. The three terms of the above exact sequence become orthogonally split when restricted to P̃_{X/M}. By contrast, when restricted to P_{Y/P_{X/M}} they fit in a commutative diagram

\xymatrix{\overline{C}_{∗}^{1} \ar@{^{(}->}[r] \ar[d] & \overline{C}_{∗}^{2} \ar@{->>}[r] \ar[d] & \overline{C}^{3} \ar[d] \\ tr_{1}(\overline{A})_{∗}|_{P_{Y /P_{X/M}}} \ar@{^{(}->}[r] \ar[d] & Ẽ′_{∗}|_{P_{Y /P_{X/M}}} \ar@{->>}[r] \ar[d] & _{\overline{D}{2 \atop ∗}} \ar[d] \\ 0 \ar@{^{(}->}[r] & _{\overline{D}{1 \atop ∗}} \ar@{->>}[r] & _{\overline{D}{1 \atop ∗}}}

where the complexes \overline{C}_{∗}^{i} are orthogonally split, and

\overline{D}_{∗}^{1} = π_{P_{Y /P}}^{∗} \overline{F} ⊗ K(\overline{O}_{Y} , \overline{N}_{Y/P_{X/M}}),
\overline{D}_{∗}^{2} = tr_{1}(Tot(π_{P_{X/M}}^{∗} \overline{E} ⊗ K(\overline{O}_{X}, \overline{N}_{X/M})))|_{P_{Y /P_{X/M}}} .

For each k ≥ 0, we will denote by η_{k}^{3} the exact sequence corresponding to the piece of degree k of the second row of the above diagram, by η_{k}^{4} that of the second column and by η_{k}^{5} that of the third column. Notice that the map in the third row is an isometry. We assume that the metric on C_{∗}^{1} is chosen in such a way that the first column is an isometry. Since the complexes \overline{C}_{∗}^{i} are orthogonally split, by lemma 2.17 we obtain

∑_{k}(−1)^{k} (c̃h(η_{k}^{3}) − c̃h(η_{k}^{4}) + c̃h(η_{k}^{5})) = 0.  (8.10)

Note that the restriction of μ_{k}^{1} to P_{X/M} agrees with η_{k}^{1}, whereas its restriction to P_{Y/P_{X/M}} agrees with η_{k}^{3}.

• Its restriction to W̃_{X} is orthogonally split.

• Finally its restriction to P_{Y ×ℙ^{1}} fits in an exact sequence

\xymatrix{\overline{D}_{∗} \ar@{^{(}->}[r] & Ẽ′_{∗}|_{P_{Y ×ℙ^{1}}} \ar@{->>}[r] & π_{P_{Y ×ℙ^{1}}}^{∗} p_{Y ×ℙ^{1}}^{∗} \overline{F} ⊗ K(O_{Y ×ℙ^{1}}, \overline{N}_{Y ×ℙ^{1}/W_{X}})} ,

where \overline{D}_{∗} is orthogonally split. For each k ≥ 0 we will denote by μ_{k}^{2} the piece of degree k of this exact sequence. Note that the restriction of μ_{k}^{2} to P_{Y/M} agrees with η_{k}^{2} and the restriction of μ_{k}^{2} to P_{Y/P_{X/M}} agrees with η_{k}^{4}.

On ℙ^{1} × ℙ^{1} we denote the two projections by p_{1} and p_{2}. Since the currents p_{1}^{∗}W_{1} and p_{2}^{∗}W_{1} have disjoint wave front sets we can define the current W_{2} = p_{1}^{∗}W_{1} • p_{2}^{∗}W_{1} ∈ D_{D}^{2}(ℙ^{1} × ℙ^{1}, 2) which satisfies

d_{D} W_{2} = (δ_{{∞}×ℙ^{1}} − δ_{{0}×ℙ^{1}}) • p_{2}^{∗}W_{1} − p_{1}^{∗}W_{1} • (δ_{ℙ^{1}×{∞}} − δ_{ℙ^{1}×{0}}). (8.11)

The key point in order to study the compatibility of singular Bott-Chern classes and composition of closed immersions is that, in the group ⨁_{p} D̃^{2p−1}(M, p), we have

d_{D}(p_{W})_{∗} (∑_{k}(−1)^{k}W_{2} • ch(Ẽ′_{k})) = 0.

We compute this class using the equation (8.11). It can be decomposed as follows.

d_{D}(p_{W})_{∗} (∑_{k}(−1)^{k}W_{2} • ch(Ẽ′_{k})) =

(p_{M̃_{X}×ℙ^{1}})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(Ẽ′_{k}|_{M̃_{X}×ℙ^{1}}))  (a)

+ (p_{W_{Y /P}})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(Ẽ′_{k}|_{W_{Y /P}}))  (b)

− (p_{W_{Y}})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(Ẽ′_{k}|_{W_{Y}}))  (c)

− (pW̃_{X})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(Ẽ′_{k}|W̃_{X}))  (d)

− (p_{P_{Y ×ℙ^{1}}} )_{∗} (∑_{k}(−1)^{k}W_{1} • ch(Ẽ′_{k}|_{P_{Y ×ℙ^{1}}} ))  (e)

+ (p_{W_{X}})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(Ẽ′_{k}|_{W_{X}}))  (f)
=: I_{a} + I_{b} − I_{c} − I_{d} − I_{e} + I_{f}

We compute each of the above terms.

(a) Since the restriction Ẽ′|_{M̃_{X}×{∞}} is orthogonally split, we have

I_{a} = −(πM̃_{X})_{∗}c̃h(Ẽ′|_{M̃_{X}×{0}}).

But, using lemma 2.17 and the fact, for each k, the complexes tr_{1}(\overline{E}′)_{k,∗}|M̃_{X} are orthogonally split, we obtain that I_{a} = 0.

(b) We compute

I_{b} =(p_{W_{Y /P}})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(Ẽ′_{k}|_{W_{Y /P}}))

=(p_{W_{Y /P}})_{∗} (W_{1} • ∑_{k}(−1)^{k}(− d_{D} c̃h(μ_{k}^{1}) + ch(tr_{1}(\overline{A}_{∗})_{k})

+ ch(tr_{1}(Tot(π_{P_{X/M}}^{∗} \overline{E} ⊗ K(O_{X}, \overline{N}_{X/M})))_{k})))
= ∑_{k}(−1)^{k}(−(p_{P_{Y /P_{X/M}}})_{∗}c̃h(η_{k}^{3}) − (pP̃_{X/M})_{∗}c̃h(μ_{k}^{1}|P̃_{X/M}) + (p_{P_{X/M}})_{∗}c̃h(η_{k}^{1}))

− c̃h(\overline{A})

− (i_{X/M})_{∗}(π_{P_{X/M}})_{∗}T (\overline{ξ}_{Y ↪P_{X/M}}) + (i_{Y/M})_{∗}C_{T} (F, N_{Y/P_{X/M}})

− ∑_{k}(−1)^{k}(p_{P_{Y /P_{X/M}}})c̃h(η_{k}^{5}),

where ξ_{Y ↪P_{X/M}} is as in notation 8.8. By corollary 2.19 and the fact that the exact sequences \overline{A}_{k,∗} are orthogonally split, the term c̃h(\overline{A}) vanishes. Also by corollary 2.19 we can see that

∑_{k}(−1)^{k}(pP̃_{X/M})_{∗}c̃h(μ_{k}^{1}|P̃_{X/M})

vanishes. Therefore we conclude

I_{b} = ∑_{k}(−1)^{k}(−(p_{P_{Y /P_{X/M}}})_{∗}c̃h(η_{k}^{3}) + (p_{P_{X/M}})_{∗}c̃h(η_{k}^{1})) − (p_{P_{Y /P_{X/M}}})c̃h(η_{k}^{5})

− (i_{X/M})_{∗}(π_{P_{X/M}})_{∗}T (\overline{ξ}_{Y ↪P_{X/M}}) + (i_{Y/M})_{∗}C_{T} (F, N_{Y/P_{X/M}}).

(c) By the definition of singular Bott-Chern forms we have

I_{c} = −T (\overline{ξ}_{Y ↪M}) + (i_{Y/M})_{∗}C_{T} (F, N_{Y/M}) − ∑_{k}(−1)^{k}(p_{P_{Y /M}})_{∗}c̃h(η_{k}^{2}),

(d) Since the restriction of Ẽ′_{∗} to W̃_{X} is orthogonally split, we have I_{d} = 0.

(e) We compute

I_{e} =(p_{P_{Y ×ℙ^{1}}} )_{∗} (∑_{k}(−1)^{k}W_{1} • ch(Ẽ′_{k}|_{P_{Y ×ℙ^{1}}} ))

=(p_{P_{Y ×ℙ^{1}}} )_{∗} (W_{1} • ∑_{k}(−1)^{k}( − d_{D} c̃h(μ_{k}^{2}) + ch(\overline{D}_{k})

+ ch(π_{P_{Y ×ℙ^{1}}}^{∗} p_{Y}^{∗} \overline{F} ⊗ K(O_{Y ×ℙ^{1}}, \overline{N}_{Y ×ℙ^{1}/W_{X}})_{k}))) .

The term ∑(−1)^{k} ch(\overline{D}_{k}) vanishes because the complex D_{∗} is orthogonally split. We have

∑_{k}(−1)^{k}(p_{P_{Y ×ℙ^{1}}} )_{∗}(W_{1} • ch(π_{P_{Y ×ℙ^{1}}}^{∗} p_{Y}^{∗} \overline{F} ⊗ K(\overline{O}_{Y ×ℙ^{1}}, \overline{N}_{Y ×ℙ^{1}/W_{X}})_{k}))

= (i_{Y/M})_{∗} ch(\overline{F})•(p_{Y})_{∗} (W_{1} • π_{P_{Y ×ℙ^{1}}}^{∗} ∑_{k}(−1)^{k} ch(K(\overline{O}_{Y ×ℙ^{1}}, \overline{N}_{Y ×ℙ^{1}/W_{X}})_{k}))

= (i_{Y/M})_{∗} ch(\overline{F}) • (p_{Y})_{∗} (W_{1} • Td^{−1}(\overline{N}_{Y ×ℙ^{1}/W_{X}}))

= (i_{Y/M})_{∗} ch(\overline{F}) • \begin{matrix} \\ Td & −1 \end{matrix}(\overline{ε}_{N}), (8.12)

where \overline{ε}_{N} is the exact sequence (8.6). Therefore we obtain

I_{e} = − ∑_{k}(−1)^{k}(p_{P_{Y /P_{X/M}}})_{∗}c̃h(η_{k}^{4}) + ∑_{k}(−1)^{k}(p_{P_{Y /M}})_{∗}c̃h(η_{k}^{2})

+ (i_{Y/M})_{∗} ch(\overline{F}) • \begin{matrix} \\ Td & −1 \end{matrix}(\overline{ε}_{N}).

(f) Finally we have

I_{f} = − ∑_{k}(−1)^{k}T (\overline{ξ}_{X↪M,k}) + ∑_{k}(−1)^{k}(i_{X/M})_{∗}C_{T} (E_{k}, N_{X/M})

− ∑_{k,l}(−1)^{k+l}(p_{P_{X/M}})_{∗}c̃h(η_{k,l}^{1}).

By corollary 2.19 we have that

∑_{m,l}(−1)^{m+l}(p_{P_{X/M}})_{∗}c̃h(η_{m,l}^{1}) = ∑_{k}(−1)^{k}(p_{P_{X/M}})_{∗}c̃h(η_{k}^{1}).

Thus

I_{f} = − ∑_{k}(−1)^{k}T (\overline{ξ}_{X↪M,k}) + ∑_{k}(−1)^{k}(i_{X/M})_{∗}C_{T} (E_{k}, N_{X/M})

− ∑_{k}(−1)^{k}(p_{P_{X/M}})_{∗}c̃h(η_{k}^{1}).

Summing up all the terms we have computed, and taking into account equation (8.10) and the fact that

C_{T} (F, N_{Y/M}) = C_{T} (F, N_{Y/P_{X/M}})

we have obtained the following partial result.

Lemma 8.13. Let i_{Y/M} = i_{X/M} ∘ i_{Y/X} be a composition of closed immersions of complex manifolds. Let T be a theory of singular Bott-Chern classes with C_{T} its associated characteristic class. Let \overline{ξ}_{Y ↪M}, \overline{ξ}_{X↪M,k} and \overline{ξ}_{Y ↪P_{X/M}} be as

in notation 8.8, and let \overline{ε} be as in (8.6). Then, in the group ⨁_{p} D̃^{2p−1}(M, p), the equation

T (\overline{ξ}_{Y ↪M}) = ∑_{k}(−1)^{k}T (\overline{ξ}_{X↪M,k}) − ∑_{k}(−1)^{k}(i_{X/M})_{∗}C_{T} (E_{k}, N_{X/M})

+ (i_{X/M})_{∗}(π_{P_{X/M}})_{∗}T (\overline{ξ}_{Y ↪P_{X/M}}) + (i_{Y/M})_{∗} ch(\overline{F}) • \begin{matrix} \\ Td & −1 \end{matrix}(\overline{ε}_{N}) (8.14)

holds.

In order to compute the third term of the right hand side of equation (8.14) we consider the following situation

\xymatrix{Y ×_{X} P_{X/M} \ar[r]^{j} \ar[d]_{π} & P_{X/M} \ar[d]_{π} \\ Y \ar[u]_{s} \ar[r]^{i} & X \ar[u]_{s}}.

To ease the notation, we denote P_{X/M} by P, Y ×_{X}P_{X/M} by X′ and we denote by

P′ the projective completion of the normal cone to X′ in P and by π_{P′}: P′ ⟶ X′, π_{X′/Y} : X′ ⟶ Y and π_{P′/Y} : P′ ⟶ Y the projections. Observe that X and X′ intersect transversely along Y. Moreover, N_{Y/X′} = i_{Y/X}^{∗}N_{X/M}, N_{X′/P} = π_{X′/Y}^{∗} N_{Y/X} and N_{Y/P} = N_{Y/X} ⊕N_{Y/X′}. We use these identifications to define metrics on N_{Y/X′}, N_{X′/P} and N_{Y/P}. Therefore the exact sequence

0 ⟶ \overline{N}_{Y/X′} ⟶ \overline{N}_{Y/P} ⟶ i_{Y/X′}^{∗} \overline{N}_{X′/P} ⟶ 0

is orthogonally split. We apply the previous lemma to the composition of closed inclusions

Y ↪ X′ ↪ P,

the vector bundle \overline{F} over Y and the resolutions

π^{∗}\overline{F} ⊗ j^{∗}K(\overline{O}_{X}, \overline{N}_{X/M})_{∗} ⟶ s_{∗}F

π^{∗}\overline{E}_{∗} ⊗ K(\overline{O}_{X}, \overline{N}_{X/M})_{k} ⟶ j_{∗}(π^{∗}F ⊗ j^{∗}K(O_{X}, N_{X/M})_{k}).

We denote by \overline{ξ}_{Y ↪P} and \overline{ξ}_{X′↪P,k} the hermitian embedded vector bundles corresponding to the above resolutions. If i_{Y/P′}: Y ↪ P′ is the induced inclusion, we denote by \overline{ξ}_{Y ↪P′} the hermitian embedded vector bundle

(i_{Y/P′}, \overline{N}_{Y/P′}, \overline{F} , Tot(π_{P′}^{∗}j^{∗}K(\overline{O}_{X}, \overline{N}_{X/M}) ⊗ K(\overline{O}_{X′}, \overline{N}_{X′/P}) ⊗ (π_{P′/Y})^{∗}\overline{F})) .

Note that the hermitian embedded vector bundle \overline{ξ}_{Y ↪P} agrees with the hermitian embedded vector bundle denoted \overline{ξ}_{Y ↪P_{X/M}} in lemma 8.13. Moreover, we have that

\overline{ξ}_{X′↪P,k} = π^{∗}\overline{ξ}_{Y ↪X} ⊗ K(\overline{O}_{X}, \overline{N}_{X/M})_{k}.

Applying lemma 8.13, we obtain

T (\overline{ξ}_{Y ↪P_{X/M}}) = ∑_{k}(−1)^{k}T (\overline{ξ}_{X′↪P_{X/M} ,k})

− ∑_{k}(−1)^{k}j_{∗}C_{T} (π^{∗}F ⊗ j^{∗}K(O_{X}, N_{X/M})_{k}, N_{X′/P})

+ j_{∗}(π_{P′})_{∗}T (\overline{ξ}_{Y ↪P′}) (8.15)

By proposition 8.1,

∑_{k}(−1)^{k}T (\overline{ξ}_{X′↪P_{X/M} ,k}) = ∑_{k}(−1)^{k}T (π^{∗}\overline{ξ}_{Y ↪X} ⊗ K(\overline{O}_{X}, \overline{N}_{X/M})_{k})

= T (π^{∗}\overline{ξ}_{Y ↪X}) • ∑_{k}(−1)^{k} ch(K(\overline{O}_{X}, \overline{N}_{X/M})_{k})

+ ∑_{k}(−1)^{k}j_{∗}C_{T} (π^{∗}F ⊗ j^{∗}K(O_{X}, N_{X/M})_{k}, N_{X′/P})

− ∑_{k}(−1)^{k}j_{∗}C_{T} (π^{∗}F, N_{X′/P}) • ch(K(O_{X}, N_{X/M})_{k}) (8.16)

We now want to compute the term (i_{X/M})_{∗}(π_{P_{X/M}})_{∗}j_{∗}(π_{P′})_{∗}T (\overline{ξ}_{Y ↪P′}). Observe that we can identify

P′ = ℙ(i_{Y/X}^{∗}N_{X/M} ⊕ ℂ) ×_{Y} ℙ(s^{∗}N_{X′/P} ⊕ ℂ),

where s^{∗}N_{X′/P} is canonically isomorphic to N_{Y/X}. Moreover

(i_{X/M})_{∗}(π_{P_{X/M}})_{∗}j_{∗}(π_{P′})_{∗}T (\overline{ξ}_{Y ↪P′}) = (i_{Y/M})_{∗}(π_{P′/Y})_{∗}T (\overline{ξ}_{Y ↪P′}).

Definition 8.17. We denote

C_{T}^{ad}(F, N_{Y/X}, i_{Y/X}^{∗}N_{X/M}) = (π_{P′/Y})_{∗}T (\overline{ξ}_{Y ↪P′})

and we define

ρ(F, N_{Y/X}, i_{Y/X}^{∗}N_{X/M}) = C_{T} (F, N_{Y/M}) − C_{T}^{ad}(F, N_{Y/X}, i_{Y/X}^{∗}N_{X/M}). (8.18)

Lemma 8.19. The current C_{T}^{ad}(F, N_{Y/X}, i_{Y/X}^{∗}N_{X/M}) is closed and defines a characteristic class of triples of vector bundles. Therefore ρ is also a characteristic class. Moreover the class ρ does not depend on the theory of singular Bott-Chern classes T.

Proof. The fact that C_{T}^{ad}(F, N_{Y/X}, i_{Y/X}^{∗}N_{X/M}) is closed and determines a characteristic class is proved as in 6.16. The independence of ρ from to T is seen as follows. We denote by \overline{K}′_{∗} the complex

Tot(π_{P′}^{∗}j^{∗}K(\overline{O}_{X}, \overline{N}_{X/M}) ⊗ K(\overline{O}_{X′}, \overline{N}_{X′/P})) ⊗ (π_{P′/Y})^{∗}\overline{F} .

This complex is a resolution of (i_{Y/P′})_{∗}\overline{F} Let W be the blow-up of P′ × ℙ^{1} along Y × ∞, and let tr_{1}(\overline{K}′)_{∗} be the deformation of complexes on W given by theorem 5.4. Just by looking at the rank of the different vector bundles we see that the restriction of tr_{1}(\overline{K}′)_{∗} to P_{Y/P′}, the exceptional divisor of this blow-up, is isomorphic (although not necessarily isometric) to the Koszul complex K(\overline{F} , \overline{N}_{X/M})_{∗}. Then, by equation (7.3)

T (\overline{ξ}_{Y ↪P′}) − (i_{Y/P′})_{∗}C_{T} (F, N_{Y/M}) =

− (p_{W})_{∗} (W_{1} • ∑_{k}(−1)^{k} ch(tr_{1}(\overline{K}′)_{k}))

− ∑_{k}(−1)^{k}(p_{P})_{∗}c̃h(tr_{1}(\overline{K}′)_{k}|_{P_{Y /P′}} , K(\overline{F} , \overline{N}_{X/M})_{k}).

Since the right hand side of this equation does not depend on the theory T, the result is proved.

Using equations (8.15), (8.16), lemma 8.19 and the projection formula, we obtain

(π_{P_{X/M}})_{∗}T (\overline{ξ}_{Y ↪P_{X/M}}) = (T (\overline{ξ}_{Y ↪X}) − (i_{Y/X})_{∗}C_{T} (F, N_{Y/X}))

• (π_{P_{X/M}})_{∗} ∑_{k}(−1)^{k} ch(K(O_{X}, \overline{N}_{X/M})_{k})

+ (π_{P_{X/M}})_{∗}j_{∗}(π_{P′})_{∗}T (\overline{ξ}_{Y ↪P′})

= (T (\overline{ξ}_{Y ↪X}) − (i_{Y/X})_{∗}C_{T} (F, N_{Y/X})) • Td^{−1}(\overline{N}_{X/M})

+ (i_{Y/X})_{∗}C_{T}^{ad}(F, N_{Y/X}, i_{Y/X}^{∗}N_{X/M})

= (T (\overline{ξ}_{Y ↪X}) − (i_{Y/X})_{∗}C_{T} (F, N_{Y/X})) • Td^{−1}(\overline{N}_{X/M})

+ (i_{Y/X})_{∗}C_{T} (F, N_{Y/M}) − ρ(F, N_{Y/X}, i_{Y/X}^{∗}N_{X/M}). (8.20)

Joining this equation and lemma 8.13 we obtain the main relationship between singular Bott-Chern classes and composition of closed immersions.

Proposition 8.21. Let i_{Y/M} = i_{X/M} ∘ i_{Y/X} be a composition of closed immersions of complex manifolds. Let T be a theory of singular Bott-Chern classes with C_{T} its associated characteristic class. Let \overline{ξ}_{Y ↪M}, \overline{ξ}_{X↪M,k} and \overline{ξ}_{Y ↪P_{X/M}} be as in notation 8.8 and let \overline{ε} be as in (8.6). Then, in the group

⨁_{p} D̃^{2p−1}(M, p), we have the equation

T (\overline{ξ}_{Y ↪M}) = ∑_{k}(−1)^{k}T (\overline{ξ}_{X↪M,k}) + (i_{X/M})_{∗}(T (\overline{ξ}_{Y ↪X}) • Td^{−1}(\overline{N}_{X/M}))

+ (i_{Y/M})_{∗} ch(\overline{F}) • \begin{matrix} \\ Td & −1 \end{matrix}(\overline{ε}_{N})

+ (i_{Y/M})_{∗}C_{T}^{ad}(F, N_{Y/X}, i_{Y/X}^{∗}N_{X/M})
− (i_{X/M})_{∗}((i_{Y/X})_{∗}C_{T} (F, N_{Y/X}) • Td^{−1}(N_{X/M}))

− (i_{X/M})_{∗} ∑_{k}(−1)^{k}C_{T} (E_{k}, N_{X/M})

We can simplify the formula of proposition 8.21 if we assume that our theory of singular Bott-Chern classes is compatible with the projection formula.

Corollary 8.22. With the hypothesis of proposition 8.21, assume furthermore that T is compatible with the projection formula. Then

T (\overline{ξ}_{Y ↪M}) = ∑_{k}(−1)^{k}T (\overline{ξ}_{X↪M,k}) + (i_{X/M})_{∗}(T (\overline{ξ}_{Y ↪X}) • Td^{−1}(\overline{N}_{X/M}))

+ (i_{Y/M})_{∗} ch(\overline{F}) • \begin{matrix} \\ Td & −1 \end{matrix}(\overline{ε}_{N})

+ (i_{Y/M})_{∗} [C_{T}^{ad}(F, N_{Y/X}, i_{Y/X}^{∗}N_{X/M}) − C_{T} (F, N_{Y/X}) • Td^{−1}(i_{Y/X}^{∗}N_{X/M}))

−C_{T} (F, i_{Y/X}^{∗}N_{X/M}) • Td^{−1}(N_{Y/X})]

Proof. Since T is compatible with the projection formula, then C_{T} is also. Therefore, using the Grothendieck-Riemann-Roch theorem for closed immersions at the level of analytic Deligne cohomology classes, we have

∑_{k}(−1)^{k}C_{T} (E_{k},N_{X/M}) = C_{T} (O_{X}, N_{X/M}) • ∑_{k}(−1)^{k} ch(E_{k})
= C_{T} (O_{X}, N_{X/M}) • (i_{Y/X})_{∗}(ch(F) • Td^{−1}(N_{Y/X}))

= (i_{Y/X})_{∗}(i_{Y/X}^{∗}C_{T} (O_{X}, N_{X/M}) • ch(F) • Td^{−1}(N_{Y/X}))

= (i_{Y/X})_{∗}(C_{T} (F, i_{Y/X}^{∗}N_{X/M}) • Td^{−1}(N_{Y/X})),

which implies the result.

Definition 8.23. Let T be a theory of singular Bott-Chern classes. We will say that T is transitive if the equation

T (\overline{ξ}_{Y ↪M}) = ∑_{k}(−1)^{k}T (\overline{ξ}_{X↪M,k}) + (i_{X/M})_{∗}(T (\overline{ξ}_{Y ↪X}) • Td^{−1}(\overline{N}_{X/M}))

+ (i_{Y/M})_{∗} ch(\overline{F}) • \begin{matrix} \\ Td & −1 \end{matrix}(\overline{ε}_{N}) (8.24)

holds. When equation (8.24) is satisfied for a particular choice of complex immersions and resolutions, we say that the theory T is transitive with respect to this particular choice.

We now introduce an abstract version of definition 8.17.

Definition 8.25. Given any characteristic class C of pairs of vector bundles, we will denote

C^{ρ}(F, N_{1}, N_{2}) := C(F, N_{1} ⊕ N_{2}) − ρ(F, N_{1}, N_{2}),

where ρ is the characteristic class of definition 8.17.

Note that, when T is a theory of singular Bott-Chern classes we have

C_{T}^{ρ} (F, N_{1}, N_{2}) = C_{T}^{ad}(F, N_{1}, N_{2}).

Definition 8.26. We will say that a characteristic class C (of pairs of vector bundles) is ρ-Todd additive (in the second variable) if it satisfies

C(F, N_{1} ⊕ N_{2}) = C(F, N_{1}) • Td^{−1}(N_{2}) + C(F, N_{2}) • Td^{−1}(N_{1}) + ρ(F, N_{1}, N_{2})

or, equivalently,

C^{ρ}(F, N_{1}, N_{2}) = C(F, N_{1}) • Td^{−1}(N_{2}) + C(F, N_{2}) • Td^{−1}(N_{1}).

A direct consequence of corollary 8.22 is

Corollary 8.27. Let T be a theory of singular Bott-Chern classes that is compatible with the projection formula. Then it is transitive if and only if the associated characteristic class C_{T} is ρ-Todd additive.

Since we are mainly interested in singular Bott-Chern classes that are transitive and compatible with the projection formula, we will study characteristic classes that are compatible with the projection formula and ρ-Todd-additive in the second variable. Since we want to express any characteristic class in terms of a power series we will restrict ourselves to the algebraic category.

Proposition 8.28. Let C be a class that is compatible with the projection formula and ρ-Todd additive in the second variable. Then C determines a power series ϕ_{C}(x) given by

C(O_{Y} , L) = ϕ_{C}(c_{1}(L)),  (8.29)

for every complex algebraic manifold Y and algebraic line bundle Y. Conversely, given any power series in one variable ϕ(x), there exists a unique characteristic class for algebraic vector bundles that is compatible with the projection formula and ρ-Todd additive in the second variable such that equation (8.29) holds.

Proof. This result follows directly from the splitting principle and theorem 1.8.

Remark 8.30. The utility of corollary 8.27 and proposition 8.28 is limited by the fact that we do not know an explicit formula for the class ρ(O_{Y} , N_{1}, N_{2}). This class is related with the arithmetic difference between ℙ_{Y} (N_{1} ⊕ N_{2} ⊕ ℂ) and ℙ_{Y} (N_{1} ⊕ ℂ) ×_{Y} ℙ_{Y} (N_{2} ⊕ ℂ), the second space being simpler than the first.

The main ingredients needed to compute this class are the Bott-Chern classes of the tautological exact sequence. Therefore the work of Mourougane [29] might be useful for computing this class.

Recall that an additive genus is a characteristic class for algebraic vector bundles S such that

S(N_{1} ⊕ N_{2}) = S(N_{1}) + S(N_{2}).

Let ϕ(x) = ∑_{i=0}^{∞} a_{i}x^{i} be a power series in one variable. There is a one to one correspondence between additive genus and power series characterized by the condition that S(L) = ϕ(c_{1}(L)), for each line bundle L. Since the class ρ does not depend on the theory T it cancels out when considering the difference between two different theories of singular Bott-Chern classes.

Proposition 8.31. Let C_{1} and C_{2} be two characteristic classes for pairs of algebraic vector bundles that are compatible with the projection formula and ρ-Todd-additive in the second variable. Then there is a unique additive genus S_{12} such that

C_{1}(F, N) − C_{2}(F, N) = ch(F) • Td(N)^{−1} • S_{12}(N).  (8.32)

We can summarize the results of this section in the following theorem.

Theorem 8.33. There is a one to one correspondence between theories of singular Bott-Chern classes for complex algebraic manifolds that are transitive and compatible with the projection formula, and formal power series ϕ(x) ∈ ℝ[[x]]. To each theory of singular Bott-Chern classes corresponds the power series ϕ such that

C_{T} (O_{Y} , L) = 1_{1} • ϕ(c_{1}(L)),  (8.34)

for every complex algebraic manifold Y and every algebraic line bundle L. To each power series ϕ it corresponds a unique class C, compatible with the projection formula and ρ-Todd-additive in the second variable, characterized by equation (8.34) and a theory of singular Bott-Chern given by definition 7.4.

Even if we do not know the exact value of the class ρ another consequence of corollary 8.27 is that, in order to prove the transitivity of a theory of singular Bott-Chern classes it is enough to check it for a particular class of compositions.

Corollary 8.35. Let T be a theory of singular Bott-Chern classes compatible with the projection formula. Then T is transitive if and only if for any compact complex manifold Y and vector bundles N_{1}, N_{2}, the theory T is transitive with respect to the composition of inclusions

Y ↪ ℙ_{Y} (N_{1} ⊕ ℂ) ↪ ℙ_{Y} (N_{1} ⊕ ℂ) ×_{Y} ℙ_{Y} (N_{2} ⊕ ℂ)

and the Koszul resolutions. □

We can make the previous corollary a little more explicit. Let π_{1} and π_{2} be the projections from P := ℙ_{Y} (N_{1} ⊕ ℂ) ×_{Y} ℙ_{Y} (N_{2} ⊕ ℂ) to P_{1} := ℙ_{Y} (N_{1} ⊕ ℂ) and P_{2} := ℙ_{Y} (N_{2} ⊕ ℂ) respectively. Let \overline{K}_{1} = K(\overline{O}_{Y} , \overline{N}_{1}) and \overline{K}_{2} = K(\overline{O}_{Y} , \overline{N}_{2}) be the Koszul resolutions in P_{1} and P_{2} respectively. Then,

\overline{K} = π_{1}^{∗}K_{1} ⊗ π_{2}^{∗}K_{2}

is a resolution of O_{Y} in P. Then the theory T is transitive in this case if

T (\overline{K}) = π_{2}^{∗}T (\overline{K}_{2}) • π_{1}^{∗}(c_{r_{1}}(\overline{Q}_{1}) • Td^{−1}(\overline{Q}_{1})) + (i_{1})_{∗}(T (\overline{K}_{1}) • p_{1}^{∗} Td^{−1}(\overline{N}_{2})),

where r_{1} is the rank of N_{1}, \overline{Q}_{1} is the tautological quotient bundle in P_{1} with the induced metric, i_{1} : P_{1} ⟶ P is the inclusion and p_{1} : P_{1} ⟶ Y is the projection. The singular Bott-Chern classes that we have defined depend on the choice of a hermitian metric on the normal bundle and behave well with respect inverse images. Nevertheless, when one is interested in covariant functorial properties and, in particular, in a composition of closed immersions, it might be interesting to consider a variant of singular Bott-Chern classes that depend on the choice of metrics on the tangent bundles to Y and X.

Notation 8.36. Let \overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗} → i_{∗}F) be a hermitian embedded vector bundle. Let \overline{T}_{X} and \overline{T}_{Y} be the tangent bundles to X and Y provided with hermitian metrics. As usual we write Td(Y) = Td(\overline{T}_{Y}) and Td(X) = Td(\overline{T}_{X}). We put

\overline{ξ}_{c} = (i: Y ⟶ X, \overline{T}_{X}, \overline{T}_{Y} , \overline{F} , \overline{E}_{∗} → i_{∗}F).

By abuse of notation we will also say that \overline{ξ}_{c} is a hermitian embedded vector bundle. In this situation we denote by \overline{ξ}_{N_{Y /X}} the exact sequence of hermitian vector bundles

\overline{ξ}_{N_{Y /X}}: 0 ⟶ \overline{T}_{Y} ⟶ i^{∗}\overline{T}_{X} ⟶ \overline{N}_{Y/X} ⟶ 0.

If there is no danger of confusion we will denote \overline{N} = \overline{N}_{Y/X} and therefore \overline{ξ}_{N} = \overline{ξ}_{N_{Y /X}}.

Definition 8.37. Let T be a theory of singular Bott-Chern classes. Then the covariant singular Bott-Chern class associated to T is given by

T_{c}(\overline{ξ}_{c}) = T (\overline{ξ}) + i_{∗}(ch(\overline{F}) • \begin{matrix} \\ Td & −1 \end{matrix}(\overline{ξ}_{N_{Y /X}}) Td(Y))  (8.38)

Proposition 8.39. The covariant singular Bott-Chern classes satisfy the following properties

(i) The class T_{c}(\overline{ξ}_{c}) does not depend on the choice of the metric on N_{Y/X}.

(ii) The differential equation

d_{D} T_{c}(\overline{ξ}_{c}) = ∑_{k}(−1)^{k} ch(\overline{E}_{k}) − i_{∗}(ch(\overline{F}) • Td(Y)) • Td^{−1}(X) (8.40)

holds.

(iii) If the theory T is compatible with the projection formula, then

T_{c}(\overline{ξ}_{c} ⊗ \overline{G}) = T_{c}(\overline{ξ}_{c}) • ch(\overline{G}).

(iv) If, moreover, the theory T is transitive, then, using notation 8.8 adapted to the current setting, we have

T_{c}(\overline{ξ}_{Y ↪M,c}) = ∑_{k}(−1)^{k}T_{c}(\overline{ξ}_{X↪M,k,c})

+ (i_{X/M})_{∗}(T_{c}(\overline{ξ}_{Y ↪X,c}) • Td(X)) • Td^{−1}(M). (8.41)

(v) With the hypothesis of corollary 6.14, we have

T_{c}( ⨁_{j even} \overline{ξ}_{j,c}) − T_{c}( ⨁_{j odd} \overline{ξ}_{j,c}) = [c̃h(\overline{ε})] − i_{∗}([c̃h(\overline{χ}) • Td(Y)]) • Td^{−1}(X).

(8.42)

Proof. All the statements follow from straightforward computations.

9 Homogeneous singular Bott-Chern classes

In this section we will show that, by adding a natural fourth axiom to definition 6.9, we obtain a unique theory of singular Bott-Chern classes that we call homogeneous singular Bott-Chern classes, and we will compare it with the classes previously defined by Bismut, Gillet and Soulé and by Zha. In the paper [6], Bismut, Gillet and Soulé introduced a theory of singular Bott-Chern classes that is the main ingredient in their construction of direct images for closed immersions. Strictly speaking, the construction of [6] only produces a theory of singular Bott-Chern classes in the sense of this paper when the metrics involved satisfy a technical condition, called Condition (A) of Bismut. Nevertheless, there is a unique way to extend the definition of [6] from metrics satisfying Bismut’s condition (A) to general metrics in such a way that one obtains a theory of singular Bott-Chern classes in the sense of this paper. In his thesis [32], Zha gave another definition of singular Bott-Chern classes, and he also used them to define direct images for closed immersions in Arakelov theory. We will recall the construction of both theories of singular Bott-Chern classes and we will show that they agree with the theory of homogeneous singular Bott-Chern classes. We warn the reader that the normalizations we use differ from the normalizations in [6] and [32]. The main difference is that we insist on using the algebro-geometric twist in cohomology, whereas in the other two papers the authors use cohomology with real coefficients. Let r_{F} and r_{N} be two positive integers. Let Y be a complex manifold and let \overline{F} and \overline{N} be two hermitian vector bundles of rank r_{F} and r_{N} respectively. Let P = ℙ(N ⊕ ℂ) and let s be the zero section. We will follow the notations of definition 5.3. Then T (K(\overline{F} , \overline{N})) satisfies the differential equation

d_{D} T (K(\overline{F} , \overline{N})) = c_{r_{N}} (\overline{Q}) Td^{−1}(\overline{Q}) ch(π_{P}^{∗} \overline{F}) − s_{∗}(ch(\overline{F}) Td^{−1}(\overline{N})).

Therefore, the class

ẽ_{T} (\overline{F} , \overline{N}) := T (K(\overline{F} , \overline{N})) • Td(\overline{Q}) • ch^{−1}(π_{P}^{∗} \overline{F})

satisfies the simpler equation

d_{D} ẽ_{T} (\overline{F} , \overline{N}) = [c_{r_{N}} (\overline{Q})] − δ_{Y} .  (9.1)

Observe that the right hand side of this equation belongs to D_{D}^{2r_{N}} (P, r_{N}). Thus it seems natural to introduce the following definition.

Definition 9.2. Let T be a theory of singular Bott-Chern classes of rank r_{F} > 0 and codimension r_{N}. Then the class

ẽ_{T} (\overline{F} , \overline{N}) = T (K(\overline{F} , \overline{N})) • Td(\overline{Q}) • ch^{−1}(π_{P}^{∗} \overline{F})

is called the Euler-Green class associated to T. The class T (K(\overline{F} , \overline{N})) is said to be homogeneous if

ẽ_{T} (\overline{F} , \overline{N}) ∈ D̃_{D}^{2r_{N} −1}(P, r_{N}).

A theory of singular Bott-Chern classes of rank 0 is said to be homogeneous if it agrees with the theory of Bott-Chern classes associated to the Chern character. Finally, a theory of singular Bott-Chern classes is said to be homogeneous if its restrictions to all ranks and codimensions are homogeneous.

The main interest of the above definition is the following result.

Theorem 9.3. Given two positive integers r_{F} and r_{N} there exists a unique theory of homogeneous singular Bott-Chern classes of rank r_{F} and codimension r_{N}.

Proof. The proof of this result is based on the theory of Euler-Green classes. Let P = ℙ(N ⊕ ℂ) be as before, and let s denote the zero section of P. Let D_{∞} be the subvariety of P that parametrizes the lines contained in N. Then D_{∞} = ℙ(N).

Lemma 9.4. There exists a unique class ẽ(P, \overline{Q}, s) ∈ D_{D}^{2r_{N} −1}(P, r_{N}) such that

(i) It satisfies

d_{D} ẽ(P, \overline{Q}, s) = [c_{r_{N}} (\overline{Q})] − δ_{Y} .  (9.5)

(ii) The restriction ẽ(P, \overline{Q}, s)|_{D_{∞}} = 0.

Proof. We first show the uniqueness. Assume that ẽ and e′ are two classes that satisfy the hypothesis of the theorem. Then e′ − ẽ is closed. Hence it determines a cohomology class in H_{D^{an}}^{2r_{N} −1}(P, r_{N}). Since, by theorem 1.2, the restriction

H_{D^{an}}^{2r_{N} −1}(P, r_{N}) ⟶ H_{D^{an}}^{2r_{N} −1}(D_{∞}, r_{N})  (9.6)

is an isomorphism, condition (ii) implies that e′ = ẽ. Now we prove the existence. Since Y is the zero locus of the section s, that is transversal to the zero section of Q, we know that the currents [c_{r_{N}}] and δ_{Y} are cohomologous. Therefore there exists an element ã ∈ D̃_{D}^{2r_{N} −1}(P, r_{N}) such that d_{D} ã = [c_{r_{N}} (\overline{Q})]−δ_{Y}. Since \overline{Q} restricted to D_{∞} splits as an orthogonal direct sum

\overline{Q}|_{D_{∞}} = \overline{S} ⊕ \overline{ℂ}  (9.7)

where the metric on the factor ℂ is trivial, and the section s restricts to the constant section 1, we obtain that ([c_{r_{N}} (\overline{Q})] − δ_{Y})|_{D_{∞}} = 0. Therefore ã determines a class in H_{D^{an}}^{2r_{N} −1}(P, r_{N}). Using again that (9.6) is an isomorphism, we find an element b̃ ∈ H_{D^{an}}^{2r_{N} −1}(P, r_{N}), such that ẽ = ã − b̃ satisfies the conditions of the lemma.

We continue with the proof of theorem 9.3. We first prove the uniqueness. Let T be a theory of homogeneous singular Bott-Chern classes. The splitting (9.7) implies easily that the restriction of the Koszul resolution K(\overline{F} , \overline{N}) to D_{∞} is orthogonally split. By the functoriality of singular Bott-Chern classes, T (K(\overline{F} , \overline{N}))|_{D_{∞}} = 0. Thus the class

ẽ_{T} (\overline{F} , \overline{N}) := T (K(\overline{F} , \overline{N})) • Td(\overline{Q}) • ch^{−1}(π_{P}^{∗} \overline{F}) ∈ D̃_{D}^{2r_{N} −1}(P, r_{N})

satisfies the two conditions of lemma 9.4. Therefore ẽ_{T} (\overline{F} , \overline{N}) = ẽ(P, \overline{Q}, s) and

T (K(\overline{F} , \overline{N})) = ẽ(P, \overline{Q}, s) • Td^{−1}(\overline{Q}) • ch(π_{P}^{∗} \overline{F}),  (9.8)

where the right hand side does not depend on the theory T. In consequence we have that

C_{T} (F, N) = (π_{P})_{∗}T (K(\overline{F} , \overline{N}))  (9.9)

does not depend on the theory T. Thus by the uniqueness in theorem 7.1 we obtain the uniqueness here. For the existence we observe

Lemma 9.10. The current

C(F, N) = (π_{P})_{∗}(ẽ(P, \overline{Q}, s) • Td^{−1}(\overline{Q})) • ch(\overline{F})

is a characteristic class for pairs of vector bundles of rank r_{F} and r_{N}.

Proof. We first compute, using equation (9.5) and corollary 3.8,

d_{D} C(F, N) = (π_{P})_{∗} (d_{D} ẽ(P, \overline{Q}, s) • Td^{−1}(\overline{Q})) • ch(\overline{F})
= (π_{P})_{∗} (([c_{r_{N}} (\overline{Q})] − δ_{Y}) • Td^{−1}(\overline{Q})) • ch(\overline{F})
= (π_{P})_{∗} (c_{r_{N}} (\overline{Q}) • Td^{−1}(\overline{Q})) • ch(\overline{F}) − Td^{−1}(\overline{N}) • ch(\overline{F})
= 0.

Thus C(F, N) determines a cohomology class. This class is functorial by construction. By proposition 1.7 this class does not depend on the metric and defines a characteristic class.

By the existence in theorem 7.1 we obtain a theory of singular Bott-Chern classes T_{C} that is easily seen to be homogeneous.

A reformulation of theorem 9.3 is

Theorem 9.11. There exists a unique way to associate to each hermitian embedded vector bundle \overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗}) a class of currents

T^{h}(\overline{ξ}) ∈ ⨁_{p} D̃_{D}^{2p−1}(X, N_{Y,0}^{∗}, p)

that we call homogeneous singular Bott-Chern class, satisfying the following properties

(i) (Differential equation) The equality

d_{D} T^{h}(\overline{ξ}) = ∑_{i}(−1)^{i}[ch(\overline{E}_{i})] − i_{∗}([Td^{−1}(\overline{N}) ch(\overline{F})])  (9.12)

holds.

(ii) (Functoriality) For every morphism f: X′ ⟶ X of complex manifolds that is transverse to Y,

f^{∗}T^{h}(\overline{ξ}) = T^{h}(f^{∗}\overline{ξ}).

(iii) (Normalization) Let \overline{A} = (A_{∗}, g_{∗}) be a non-negatively graded orthogonally split complex of vector bundles. Write \overline{ξ} ⊕ \overline{A} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗} ⊕ \overline{A}_{∗}). Then T^{h}(\overline{ξ}) = T^{h}(\overline{ξ} ⊕ \overline{A}). Moreover, if X = Spec ℂ is one point, Y = ∅ and \overline{E}_{∗} = 0, then T^{h}(\overline{ξ}) = 0.

(iv) (Homogeneity) If r_{F} = rk(F) > 0 and r_{N} = rk(N) > 0, then, with the notations of definition 9.2,

T^{h}(K(\overline{F} , \overline{N})) • Td(\overline{Q}) • ch^{−1}(π_{P}^{∗} \overline{F}) ∈ D̃_{D}^{2r_{N} −1}(P, r_{N}).
□

The class ẽ(P, \overline{Q}, s) of lemma 9.4 is a particular case of the Euler-Green classes introduced by Bismut, Gillet and Soulé in [6]. The basic properties of the Euler-Green classes are summarized in the following results.

Proposition 9.13. Let X be a complex manifold, let \overline{E} be a hermitian holomorphic vector bundle of rank r and let s be a holomorphic section of E that is transverse to the zero section. Denote by Y the zero locus of s. There is a unique way to assign to each (X, \overline{E}, s) as before a class of currents

ẽ(X, \overline{E}, s) ∈ D̃_{D}^{2r−1}(X, N_{Y,0}^{∗}, r)

satisfying the following properties

(i) (Differential equation)

d_{D} ẽ(X, \overline{E}, s) = c_{r}(\overline{E}) − δ_{Y} .  (9.14)

(ii) (Functoriality) If f: X′ ⟶ X is a morphism transverse to Y then

ẽ(X′, f^{∗}\overline{E}, f^{∗}s) = f^{∗}ẽ(X, \overline{E}, s).  (9.15)

(iii) (Multiplicativity) Let \overline{E}_{1} and \overline{E}_{2} be hermitian holomorphic vector bundles, and let s_{1} and s_{2} be holomorphic sections of \overline{E}_{1} and \overline{E}_{2} respectively that are transverse to the zero section and with zero locus Y_{1} and Y_{2}. We write \overline{E} = \overline{E}_{1} ⊕ \overline{E}_{2} and s = s_{1} ⊕ s_{2}. Assume that s is transverse to the zero section; hence Y_{1} and Y_{2} meet transversely. With this hypothesis we have

ẽ(X, \overline{E}, s) = ẽ(X, \overline{E}_{1}, s_{1}) ∧ c_{r_{2}}(\overline{E}_{2}) + δ_{Y_{1}} ∧ ẽ(X, \overline{E}_{2}, s_{2})

= ẽ(X, \overline{E}_{1}, s_{1}) ∧ δ_{Y_{2}} + c_{r_{1}}(\overline{E}_{1}) ∧ ẽ(X, \overline{E}_{2}, s_{2}).

(iv) (Line bundles) If \overline{L} is a hermitian line bundle and s is a section of L,

then

ẽ(X, \overline{L}, s) = − log ∥s∥.  (9.16)

Proof. Bismut, Gillet and Soulé prove the existence by constructing explicitly an Euler-Green current in the total space of E and pulling it back to X by the section s. For the uniqueness, first we see that properties (i) and (ii) imply that, if h_{0} and h_{1} are two hermitian metrics in E, then

ẽ(X, (E, h_{0}), s) − ẽ(X, (E, h_{1}), s) = c̃_{r}(E, h_{0}, h_{1}).  (9.17)

We now consider π: P = ℙ(E ⊕ℂ) ⟶ X, with the tautological exact sequence

0 ⟶ O(−1) ⟶ π^{∗}E ⊕ ℂ ⟶ Q ⟶ 0

On Q we consider the metric induced by the metric of \overline{E} and the trivial metric on the factor ℂ, and let s_{Q} the section of Q induced by the section 1 of ℂ. Let D_{∞} be as in lemma 9.4. Then properties (ii) to (iv) imply that ẽ(P, \overline{Q}, s_{Q})|_{D_{∞}} = 0. Hence by lemma 9.4 ẽ is uniquely determined. Finally, let f: X ⟶ P be the map given by x ⟼ (s(x) : −1). Then f^{∗}Q ≅ E, although they are not necessarily isometric, and f^{∗}s_{Q} = s. Therefore, the functoriality and equation (9.17) determine ẽ(X, \overline{E}, s).

To prove the existence, we use lemma 9.4, functoriality and equation (9.17) to define the Euler-Green classes. It is easy to show that they are well defined and satisfy properties (i) to (iv).

Equation (9.8) relating homogeneous singular Bott-Chern classes and Euler-Green classes in a particular case can be generalized to arbitrary vector bundles.

Proposition 9.18. Let X be a complex manifold, \overline{E} a hermitian vector bundle over X, s a section of E transversal to the zero section and i: Y ⟶ X the zero locus of s. Let K(\overline{E}) be the Koszul resolution of i_{∗}O_{Y} determined by \overline{E} and s. We can identify N_{Y/X} with i^{∗}E. We denote by \overline{N}_{Y/X} the vector bundle with the metric induced by the above identification. Then

T^{h}(i, \overline{O}_{Y} , \overline{N}_{Y/X}, K(\overline{E})) = ẽ(X, \overline{E}, s) • Td^{−1}(\overline{E}).

Proof. Let P = ℙ(E ⊕ ℂ). We follow the notation of proposition 9.13. We denote by h_{0} the original metric of \overline{E} and by h_{1} the metric induced by the isomorphism E ≅ f^{∗}Q. Observe that h_{0} and h_{1} agree when restricted to Y, because the preimage of \overline{Q} by the zero section agrees with \overline{E}. Hence there is an isometry \overline{N}_{Y/X} ≅ i^{∗}f^{∗}\overline{Q}. We denote T^{h}(K(\overline{E})) = T^{h}(i, \overline{O}_{Y} , \overline{N}_{Y/X}, K(\overline{E})).

Then we have

T^{h}(K(\overline{E})) = f^{∗}T^{h}(K(\overline{O_{X}}, \overline{E})) + ∑_{i}(−1)^{i}c̃h(⋀^{i} E^{∨}, h_{0}, h_{1})
= f^{∗}(ẽ(P, \overline{Q}, s_{Q}) • Td^{−1}(\overline{Q})) + c̃_{r}(E, h_{0}, h_{1}) • Td^{−1}(E, h_{1})

+ c_{r}(E, h_{0}) • \begin{matrix} \\ Td & −1 \end{matrix}(E, h_{0}, h_{1})

= ẽ(X, \overline{E}, s) • Td^{−1}(E, h_{1}) − c̃_{r}(E, h_{0}, h_{1}) • Td^{−1}(E, h_{1})

+ c̃_{r}(E, h_{0}, h_{1}) • Td^{−1}(E, h_{1}) + c_{r}(E, h_{0}) • \begin{matrix} \\ Td & −1 \end{matrix}(E, h_{0}, h_{1})

= ẽ(X, \overline{E}, s) • Td^{−1}(E, h_{0}) − ẽ(X, \overline{E}, s) • d_{D} \begin{matrix} \\ Td & −1 \end{matrix}(E, h_{0}, h_{1})

+ c_{r}(E, h_{0}) • \begin{matrix} \\ Td & −1 \end{matrix}(E, h_{0}, h_{1})

= ẽ(X, \overline{E}, s) • Td^{−1}(E, h_{0}) − d_{D} ẽ(X, \overline{E}, s) • \begin{matrix} \\ Td & −1 \end{matrix}(E, h_{0}, h_{1})

+ c_{r}(E, h_{0}) • \begin{matrix} \\ Td & −1 \end{matrix}(E, h_{0}, h_{1})

= ẽ(X, \overline{E}, s) • Td^{−1}(E, h_{0}) + i_{∗}\begin{matrix} \\ Td & −1 \end{matrix}(E, h_{0}, h_{1})|_{Y}

= ẽ(X, \overline{E}, s) • Td^{−1}(\overline{E}),

which concludes the proof.

Theorem 9.19. The theory of homogeneous singular Bott-Chern classes is compatible with the projection formula and transitive.

Proof. We have

C_{T^{h}}(F, N) = (π_{P})_{∗}T^{h}(K(\overline{F} , \overline{N}))
= (π_{P})_{∗}(ẽ(P, \overline{Q}, s) • Td^{−1}(\overline{Q}) • ch(π_{P}^{∗} \overline{F}))
= (π_{P})_{∗}(ẽ(P, \overline{Q}, s) • Td^{−1}(\overline{Q})) • ch(\overline{F})
= C_{T^{h}}(O_{Y} , N) • ch(F).

Thus C_{T^{h}} is compatible with the projection formula. We now prove the transitivity. Let Y, N_{1} and N_{2} be as in corollary 8.35. We follow the notation after this corollary. Then applying proposition 9.18 we obtain

T^{h}(\overline{K}) = ẽ(P, π_{1}^{∗}\overline{Q}_{1} ⊕ π_{2}^{∗}\overline{Q}_{2}, s_{1} + s_{2}) • Td^{−1}(π_{1}^{∗}\overline{Q}_{1} ⊕ π_{2}^{∗}\overline{Q}_{2}),  (9.20)

where s_{i} denote the tautological section of \overline{Q}_{i} or its preimage by π_{i}. Then, by proposition 9.13 (iii), taking into account that Y_{1} = P_{2},

T^{h}(\overline{K}) = π_{1}^{∗}(c_{r_{1}}(\overline{Q}_{1}) Td^{−1}(\overline{Q}_{1})) • π_{2}^{∗}(ẽ(P_{2}, \overline{Q}_{2}, s_{2}) Td^{−1}(\overline{Q}_{2}))

+ (i_{1})_{∗}(ẽ(P_{1}, \overline{Q}_{1}, s_{1}) Td^{−1}(\overline{Q}_{1}) • p_{1}^{∗} Td^{−1}(\overline{N}_{2})). (9.21)

Applying again proposition 9.18 we obtain

T^{h}(\overline{K}) = π_{1}^{∗}(c_{r_{1}}(\overline{Q}_{1}) Td^{−1}(\overline{Q}_{1})) • π_{2}^{∗}(T^{h}(\overline{K}_{2})) + (i_{1})_{∗}(T^{h}(\overline{K}_{1}) • p_{1}^{∗} Td^{−1}(\overline{N}_{2})).

(9.22)

Thus, by corollary 8.35 the theory of homogeneous singular Bott-Chern classes is transitive.

We next recall the construction of singular Bott-Chern classes of Bismut, Gillet and Soulé. Let i: Y ⟶ X be a closed immersion of complex manifolds and let \overline{ξ} = (i, \overline{N}, \overline{F} , \overline{E}_{∗}) be a hermitian embedded vector bundle. We consider the associated complex of sheaves

0 → E_{n} →^{v} … →^{v} E_{0} → 0,

where we denote by v the differential of this complex. This complex is exact for all x ∈ X \ Y. The cohomology sheaves of this complex are holomorphic vector bundles on Y which we denote by

H_{n} = H_{n}(E_{∗}|_{Y}), H = ⨁_{n} H_{n}.

For each x ∈ Y and U ∈ T_{x}X we denote by ∂_{U}v(x) the derivative of the map v calculated in any holomorphic trivialization of E near x. Then ∂_{U}v(x) acts on H_{x}. Moreover, this action only depends on the class y of U in N_{x}. We denote it by ∂_{y}v(x). Moreover (∂_{y}v(x))^{2} = 0; therefore the pull-back of H to the total space of N together with ∂_{y}v is a complex that we denote by (H, ∂_{y}v). On the total space of N, the interior multiplication by y ∈ N turns ⋀ N^{∨} into a Koszul complex. By abuse of notation we denote also by ι_{y} the operator ι_{y} ⊗ 1 acting on ⋀ N^{∨} ⊗ F. There is a canonical isomorphism between the complexes (H, ∂_{y}v) and (⋀ N^{∨} ⊗ F, ι_{y}). An explicit description of this isomorphism can be found in [3] §1. Let v^{∗} be the adjoint of the operator v with respect to the metrics of \overline{E}_{∗}. Then we have an identification of vector bundles over Y

H_{k} = {f ∈ E_{k} | vf = v^{∗}f = 0}.

This identification induces a hermitian metric on H_{k}, and hence on H. Note that the metrics on N and F also induce a hermitian metric on ⋀ N^{∨} ⊗ F.

Definition 9.23. We say that \overline{ξ} = (i, \overline{N}, \overline{F} , \overline{E}_{∗}) satisfies Bismut assumption (A) if the canonical isomorphism between (H, ∂_{y}v) and (⋀ N^{∨} ⊗ F, ι_{y}) is an isometry.

Proposition 9.24. Let \overline{ξ} = (i, \overline{N}, \overline{F} , \overline{E}_{∗}) be as before, with \overline{N} = (N, h_{N}) and \overline{F} = (F, h_{F}). Then there exist metrics h′_{E_{k}} over E_{k} such that the hermitian embedded vector bundle \overline{ξ}′ = (i, \overline{N}, \overline{F} , (E_{∗}, h′_{E_{∗}})) satisfies Bismut assumption (A).

Proof. This is [3] proposition 1.6.

Let ∇^{E} be the canonical hermitian holomorphic connection on E and let V = v + v^{∗}. Then

A_{u} = ∇^{E} + \sqrt{u}V

is a superconnection on E. Let ∇^{H} be the canonical hermitian connection on H. Then

B = ∇^{H} + ∂_{y}v + (∂_{y}v)^{∗}

is a superconnection on H. Let N_{H} be the number operator on the complex (E, v), that is, N_{H} acts on E_{k} by multiplication by k, and let Tr_{s} denote the supertrace. Recall that here we are using the symbol [ ] to denote the current associated to a locally integrable differential form and the symbol δ_{Y} to denote the current integration along a subvariety, both with the normalizations of notation 1.3. For 0 < Re(s) ≤ 1/2 let ζ_{E}(s) be the current on X given by the formula

ζ_{E}(s) = \frac{1}{Γ(s)} ∫_{0}^{∞} u^{s−1}\{ [Tr_{s} (N_{H} exp(−A_{u}^{2}))]

− i_{∗} [∫_{N} Tr_{s} (N_{H} exp(−B^{2}))]} d u. (9.25)

This current is well defined and extends to a current that depends holomorphically on s near 0.

Definition 9.26. Assume that \overline{ξ} = (i, \overline{N}, \overline{F} , \overline{E}_{∗}) satisfies Bismut assumption (A). Then we denote

T^{BGS}(\overline{ξ}) = −\frac{1}{2}ζ′_{E}(0).

By abuse of notation we will denote also by T^{BGS}(\overline{ξ}) its class in ˜⨁_{p}D̃_{D}^{2p−1}(X, p).

Let now \overline{ξ} = (i, \overline{N}, \overline{F} , (E_{∗}, h_{E_{∗}})) be general and let \overline{ξ}′ = (i, \overline{N}, \overline{F} , (E_{∗}, h′_{E_{∗}})) be any hermitian embedded vector bundle satisfying assumption (A) provided by proposition 9.24. Then we denote

T^{BGS}(\overline{ξ}) = T^{BGS}(\overline{ξ}′) + ∑_{i}(−1)^{i}c̃h(E_{i}, h_{E_{i}}, h′_{E_{i}}),

where c̃h(E_{i}, h_{E_{i}}, h′_{E_{i}}) is as in definition 2.13.

Remark 9.27. This definition only agrees (up to a normalization factor) with the definition in [6] for hermitian embedded vector bundles that satisfy assumption (A).

Theorem 9.28. The assignment that, to each hermitian embedded vector bundle \overline{ξ}, associates the current T^{BGS}(\overline{ξ}), is a theory of singular Bott-Chern classes that agrees with T^{h}.

Proof. First we have to show that, when \overline{ξ} does not satisfy assumption (A) then T^{BGS}(\overline{ξ}) is well defined. Assume that \overline{ξ}′′ = (i, \overline{N}, \overline{F} , (E_{∗}, h′_{E_{∗}})) is another choice of hermitian embedded vector bundle satisfying assumption (A). By lemma 2.17 we have that

c̃h(E_{i}, h_{i}, h′_{i}) + c̃h(E_{i}, h′_{i}, h′′_{i}) + c̃h(E_{i}, h′′_{i}, h_{i}) = 0.

By [6] theorem 2.5 we have that

T^{BGS}(\overline{ξ}′) − T^{BGS}(\overline{ξ}′′) = ∑_{i}(−1)^{i}c̃h(E_{i}, h′_{E_{i}}, h′′_{E_{i}}).

Summing up we obtain that T^{BGS}(\overline{ξ}) is well defined. If the hermitian embedded vector bundle \overline{ξ} satisfies Bismut assumption (A) then, by [6] theorem 1.9, T^{BGS}(\overline{ξ}) satisfies equation (6.10). If \overline{ξ} does not satisfy assumption (A) then, combining [6] theorem 1.9 and equation (2.4), we also obtain that T^{BGS}(\overline{ξ}) satisfies equation (6.10). The functoriality property is [6] theorem 1.10. In order to prove the normalization property, let \overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗}) be a hermitian embedded vector bundle that satisfies assumption (A) and let \overline{A} be a non-negatively graded orthogonally split complex of vector bundles on X. Observe that \overline{A} is also a (trivial) hermitian embedded vector bundle. Then \overline{A} and \overline{ξ} ⊕ \overline{A} also satisfy assumption (A). By [6] theorem 2.9

T^{BGS}(\overline{ξ} ⊕ \overline{A}) = T^{BGS}(\overline{ξ}) + T^{BGS}(\overline{A}).

But by [5] remark 2.3, T^{BGS}(\overline{A}) agrees with the Bott-Chern class associated to the Chern character and the exact complex \overline{A}. Since A is orthogonally split we have T^{BGS}(\overline{A}) = 0. Now the case when ξ does not satisfy assumption (A) follows from the definition. By [6] theorem 3.17, with the hypothesis of proposition 9.18, we have that

T^{BGS}(i, \overline{O}_{Y} , \overline{N}_{Y/X}, K(\overline{E})) = ẽ(X, \overline{E}, s) • Td^{−1}(\overline{E})
= T^{h}(i, \overline{O}_{Y} , \overline{N}_{Y/X}, K(\overline{E})).

From this it follows that C_{T^{BGS}} = C_{T^{h}} and by theorem 7.1, T^{BGS} = T^{h}.

We now recall Zha’s construction. Note that, in order to obtain a theory of singular Bott-Chern classes, we have changed the normalization convention from the one used by Zha. Note also that Zha does not define explicitly a singular Bott-Chern class, but such a definition is implicit in his definition of direct images for closed immersions. Let Y be a complex manifold and let \overline{N} = (N, h) be a hermitian vector bundle. We denote P = ℙ(N ⊕ ℂ). Let π: P ⟶ Y denote the projection and let ι: Y ⟶ P denote the inclusion as the zero section. On P we consider the tautological exact sequence

0 ⟶ O(−1) ⟶ π^{∗}N ⊕ O_{P} ⟶ Q ⟶ 0.

Let h_{1} denote the hermitian metric on Q^{∨} induced by the metric of N and the trivial metric on O_{P} and let h_{0} denote the semi-definite hermitian form on Q^{∨} induced by the map Q^{∨} ⟶ O_{P} obtained from the above exact sequence and the trivial metric on O_{P}. Let h_{t} = (1 − t^{2})h_{0} + t^{2}h_{1}. It is a hermitian metric on Q^{∨}. We will denote \overline{Q}_{t}^{∨} = (Q^{∨}, h_{t}). Let ∇_{t} be the associated hermitian holomorphic connection and let N_{t} denote the endomorphism defined by

\frac{d}{d t} ⟨v, w⟩_{t} = ⟨N_{t}v, w⟩ .

For each n ≥ 1, let Det denote the alternate n-linear form on the space of n by n matrices such that

det(A) = Det(A, …, A).

We denote det(B; A) = Det(B, A, …, A). Zha introduced the differential form

ẽ_{Z}(\overline{Q}^{∨}) = \frac{−1}{2} lim_{s→0} ∫_{s}^{1} det(N_{t}, ∇_{t}^{2}) d t  (9.29)

which is a smooth form on P \ ι(Y), locally integrable on P. Hence it defines a current, also denoted by ẽ_{Z}(\overline{Q}^{∨}) on P. The important property of this current is that it satisfies

d_{D} \overline{e}_{Z}(Q^{∨}) = c_{n}(\overline{Q}_{1}) − δ_{Y} .  (9.30)

In [32], Zha denotes by C(\overline{Q}^{∨}) a form that differs from ẽ_{Z} by the normalization factor and the sign. We denote it by ẽ_{Z} because it agrees with the Euler-Green current introduced in [6].

Proposition 9.31. The equality

ẽ_{Z}(Q^{∨}) = ẽ(P, \overline{Q}_{1}, s_{Q})

holds.

Proof. With the notations of lemma 9.4, both classes satisfy equation (9.30) and their restriction to D_{∞} is zero. By lemma 9.4 they agree.

Definition 9.32. Let \overline{ξ} = (i: Y ⟶ X, \overline{N}, \overline{F} , \overline{E}_{∗}) be as in definition 6.9. Let \overline{A}_{∗}, tr_{1}(\overline{E})_{∗} and \overline{η}_{∗} be as in (7.2). Then we define

T^{Z}(\overline{ξ}) = −(p_{W})_{∗} (∑_{k}(−1)^{k}W_{1} • ch(tr_{1}(\overline{E})_{k}))

− ∑_{k}(−1)^{k}(p_{P})_{∗}[c̃h(\overline{η}_{k})] + (p_{P})_{∗}(ch(π_{p}^{∗}\overline{F}) Td^{−1}(\overline{Q}_{1})ẽ_{Z}(\overline{Q}_{1}^{∨})). (9.33)

It follows directly from the definition that T^{Z} is the theory of singular Bott-Chern classes associated to the class

C_{Z}(F, N) = (p_{P})_{∗}(ch(π_{p}^{∗}\overline{F}) Td^{−1}(\overline{Q}_{1})ẽ_{Z}(\overline{Q}_{1}^{∨})).  (9.34)

Theorem 9.35. The theory of singular Bott-Chern classes T^{Z} agrees with the theory of homogeneous singular Bott-Chern classes T^{h}.

Proof. The result follows directly from theorem 7.1, equation (9.34) and proposition 9.18.

Next we want to use 8.33 to give another characterization of T^{h}. To this end we only need to compute the characteristic class C_{T^{h}}(O_{Y} , L) for a line bundle L as a power series in c_{1}(L).

Theorem 9.36. The theory of homogeneous singular Bott-Chern classes of algebraic vector bundles is the unique theory of singular Bott-Chern classes of algebraic vector bundles that is compatible with the projection formula and transitive and that satisfies

C_{T^{h}}(O_{Y} , L) = 1_{1} • ϕ(c_{1}(L)),

where ϕ is the power series

ϕ(x) = \frac{1}{2} ∑_{n=0}^{∞} \frac{(−1)^{n+1}H_{n+1}}{(n + 2)!} x^{n},

and where H_{n} = 1 + \frac{1}{2} + \frac{1}{3} + ⋯ + \frac{1}{n}, n ≥ 1 are the harmonic numbers.

We already know that T^{h} is compatible with the projection formula and transitive. Thus it only remains to compute the power series ϕ. Let \overline{L} = (L, h_{L}) be a hermitian line bundle over a complex manifold Y. Let z be a system of holomorphic coordinates of Y. Let e be a local section of L and let h(z) = h(e_{z}, e_{z}). Let P = ℙ(L ⊕ ℂ), with π: P ⟶ Y the projection and ι: Y ⟶ P the zero section. We choose homogeneous coordinates on P given by (z, (x : y)), here (x : y) represents the line of L_{z} ⊕ ℂ generated by xe(z) + y1, where 1 is a generator of ℂ of norm 1. On the open set y ≠ 0 we will use the absolute coordinate t = x/y. Let

0 ⟶ O(−1) ⟶ π^{∗}(L ⊕ ℂ) ⟶ Q ⟶ 0

be the tautological exact sequence. The section s = {1} is a global section of Q that vanishes along the zero section. Moreover we have

∥s∥_{(z,(x:y))}^{2} = \frac{xx̄h(z)}{yȳ + xx̄h(z)} = \frac{tt̄h}{1 + tt̄h}.

Singular Bott-Chern Classes 157

Then (recall that we are using the algebro-geometric normalization)

c_{1}(\overline{Q}) = ∂ ¯∂log ∥s∥^{2}  (9.37)
= ∂ ¯∂log \frac{tt̄h}{1 + tt̄h}  (9.38)
= ∂ (\frac{1 + tt̄h}{tt̄h} \frac{t¯∂(t̄h)(1 + tt̄h) − t^{2}t̄h¯∂(t̄h)}{(1 + tt̄h)^{2}})  (9.39)
= ∂ (\frac{t¯∂(t̄h)}{tt̄h(1 + tt̄h)})  (9.40)
= ∂ (\frac{¯∂(t̄h)}{t̄h}) \frac{1}{1 + tt̄h} − \frac{t̄∂(ht) ∧ ¯∂(t̄h)}{t̄h(1 + tt̄h)^{2}}  (9.41)
= \frac{π^{∗}c_{1}(\overline{L})}{1 + tt̄h} − \frac{∂(th) ∧ ¯∂(t̄h)}{h(1 + tt̄h)^{2}}.  (9.42)

We now consider the Koszul resolution

\overline{K}: 0 ⟶ Q^{∨} ⟶^{s} O_{p} ⟶ ι_{∗}O_{X} ⟶ 0.

We denote by T^{h}(\overline{K}) the singular Bott-Chern class associated to this Koszul complex. Then, by proposition 9.13 and proposition 9.18,

T^{h}(\overline{K}) = −\frac{1}{2} Td^{−1}(\overline{Q}) log ∥s∥^{2}.

In order to compute π_{∗}T^{h}(\overline{K}) we have to compute first π_{∗}c_{1}(\overline{Q})^{n} log ∥s∥^{2}. But

c_{1}(\overline{Q})^{n} = \frac{π^{∗}c_{1}(\overline{L})^{n}}{(1 + tt̄h)^{n}} − n (\frac{π^{∗}c_{1}(\overline{L})}{(1 + tt̄h)})^{n−1} \frac{∂(th) ∧ ¯∂(t̄h)}{h(1 + tt̄h)^{2}}.

Therefore

π_{∗}c_{1}(\overline{Q})^{n} log ∥s∥^{2} = −nc_{1}(\overline{L})^{n−1} \frac{1}{2πi} ∫_{ℙ^{1}} \frac{∂(th) ∧ ¯∂(t̄h)}{h(1 + tt̄h)^{n+1}} log \frac{tt̄h}{1 + tt̄h}

= −nc_{1}(\overline{L})^{n−1} \frac{1}{2πi} ∫_{0}^{2π} ∫_{0}^{∞} log \frac{r^{2}}{1 + r^{2}} \frac{−2ir d θ d r}{(1 + r^{2})^{n+1}}

= nc_{1}(\overline{L})^{n−1} ∫_{0}^{1} log(1 − w)w^{n−1} d w

= −c_{1}(\overline{L})^{n−1}H_{n},

where H_{n}, n ≥ 1 are the harmonic numbers. Since

T d^{−1}(\overline{Q}) = \frac{1 − exp(−c_{1}(\overline{Q}))}{c_{1}(\overline{Q})} = ∑_{n=0}^{∞} \frac{(−1)^{n}}{(n + 1)!}c_{1}(\overline{Q})^{n},

Documenta Mathematica 15 (2010) 73–176

we obtain

C_{T^{h}}(O_{Y} , L) = π_{∗}T^{h}(\overline{K}) = \frac{1}{2} ∑_{n=0}^{∞} \frac{(−1)^{n+1}H_{n+1}}{(n + 2)!} c_{1}(\overline{L})^{n}1_{1}.

Then, a reformulation of proposition 8.31 is

Corollary 9.43. Let T be a theory of singular Bott-Chern classes for algebraic vector bundles that is compatible with the projection formula and transitive. Then there is a unique additive genus S_{T} such that

C_{T} (F, N) − C_{T^{h}}(F, N) = ch(F) • Td(N)^{−1} • S_{T} (N).  (9.44)

Conversely, any additive genus determines a theory of singular Bott-Chern classes by the formula (9.44).

10 The arithmetic Riemann-Roch theorem for regular closed immersions

In this section we recall the definition of arithmetic Chow groups and arithmetic K-groups. We see that each choice of an additive theory of singular Bott-Chern classes allows us to define direct images for closed immersions in arithmetic Ktheory. Once the direct images for closed immersions are defined, we prove the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions. A version of this theorem was proved earlier by Bismut, Gillet and Soulé [6] when there is a commutative diagram

\xymatrix{Y \ar[r]^{i} \ar[dr]^{f} & X \ar[d]^{g} \\ & Z},

where i is a closed immersion and f and g are smooth over ℂ. The version of this theorem given in this paper is due to Zha [32], but still unpublished. The theorem of Bismut, Gillet and Soulé compares g_{∗} ĉh(i_{∗}\overline{E}) with f_{∗} ĉh(\overline{E}), whereas the theorem of Zha compares directly ĉh(i_{∗}\overline{E}) with i_{∗} ĉh(\overline{E}). The main difference between the theorem of Bismut, Gillet and Soulé and that of Zha is the kind of arithmetic Chow groups they use. In the first case these groups are only covariant for proper morphisms that are smooth over ℂ; thus the Grothendieck-Riemann-Roch can only be stated for a diagram as above, while in the second case a version of these groups that are covariant for arbitrary proper morphisms is used. Since each choice of a theory of singular Bott-Chern classes gives rise to a different definition of direct images for closed immersions, the arithmetic Grothendieck-Riemann-Roch theorem will have a correction term that depends on the theory of singular Bott-Chern classes used. In the particular case of the homogeneous singular Bott-Chern classes, which are the theories used by Bismut, Gillet and Soulé and by Zha, this correction term vanishes and we obtain the simplest formula. In this case the arithmetic Grothendieck-Riemann-Roch theorem is formally identical to the classical one. Let (A, Σ, F_{∞}) be an arithmetic ring [18]. Since we will allow the arithmetic varieties to be non regular and we will use Chow groups indexed by dimension, following [20] we will assume that the ring A is equidimensional and Jacobson. Let F be the field of fractions of A. An arithmetic variety X is a scheme flat and quasi-projective over A such that X_{F} = X × Spec F is smooth. Then X := X_{Σ} is a complex algebraic manifold, which is endowed with an anti-holomorphic automorphism F_{∞}. One also associates to X the real variety X_{ℝ} = (X, F_{∞}). Following [13], to each regular arithmetic variety we can associate different kinds of arithmetic Chow groups. Concerning arithmetic Chow groups, we shall use the terminology and notation in op. cit. §4 and §6. Let D_{log} be the Deligne complex of sheaves defined in [13] section 5.3; we refer to op. cit. for the precise definition and properties. A D_{log}-arithmetic variety is a pair (X , C) consisting of an arithmetic variety X and a complex of sheaves C on X_{ℝ} which is a D_{log}-complex (see op. cit. section 3.1). We are interested in the following D_{log}-complexes of sheaves:

(i) The Deligne complex D_{l,a,X} of differential forms on X with logarithmic and arbitrary singularities. That is, for every Zariski open subset U of X, we write

E_{l,a,X}^{∗}(U) = lim_{\overrightarrow{\overline{U}}} Γ(\overline{U}, ℰ ∗ ; \overline{U}(log B)),

where the limit is taken over all diagrams

\xymatrix{U \ar[r]^{\overline{ι}} \ar[dr]^{ι} & \overline{U} \ar[d]^{β} \\  & X}

such that \overline{ι} is an open immersion, β is a proper morphism, B = \overline{U} \ U, is a normal crossing divisor and ℰ ∗ ; \overline{U}(log B) denotes the sheaf of smooth differential forms on U with logarithmic singularities along B introduced in [8] .

For any Zariski open subset U ⊆ X, we put

D_{l,a,X}^{∗}(U, p) = (D_{l,a,X}^{∗}(U, p), d_{D}) = (D^{∗}(E_{l,a,X}(U), p), d_{D}).

If U is now a Zariski open subset of X_{ℝ}, then we write

D_{l,a,X}^{∗}(U, p) = (D_{l,a,X}^{∗}(U, p), d_{D}) = (D_{l,a,X}^{∗}(U_{ℂ}, p)^{σ}, d_{D}),

where σ is the involution σ(η) = \overline{F_{∞}^{∗}η} as in [13] notation 5.65.

Note that the sections of D_{l,a,X}^{∗} over an open set U ⊂ X are differential forms on U with logarithmic singularities along X \ U and arbitrary singularities along \overline{X} \ X, where \overline{X} is an arbitrary compactification of X. Therefore the complex of global sections satisfy

D_{l,a,X}^{∗}(X, ∗) = D^{∗}(X, ∗),

where the right hand side complex has been introduced in section §1. The complex D_{l,a,X}^{∗} is a particular case of the construction of [12] section 3.6.

(ii) The Deligne complex D_{cur,X} of currents on X. This is the complex introduced in [13] definition 6.30.

When X is regular, applying the theory of [13] we can define the arithmetic Chow groups ĈH^{∗}(X , D_{l,a,X}) and ĈH^{∗}(X , D_{cur,X}). These groups satisfy the following properties

(i) There are natural morphisms

ĈH^{∗}(X , D_{l,a,X}) ⟶ ĈH^{∗}(X , D_{cur,X})

and, when applicable, all properties below will be compatible with these morphisms.

(ii) There is a product structure that turns ĈH^{∗}(X , D_{l,a,X})_{ℚ} into an associative and commutative algebra. Moreover, it turns ĈH^{∗}(X , D_{cur,X})_{ℚ} into a ĈH^{∗}(X , D_{l,a,X})_{ℚ}-module.

(iii) If f: Y ⟶ X is a map of regular arithmetic varieties, there are pull-back

morphisms

f^{∗}: ĈH^{∗}(X , D_{l,a,X}) ⟶ ĈH^{∗}(Y, D_{l,a,Y}).

If moreover, f is smooth over F, there are pull-back morphisms

f^{∗}: ĈH^{∗}(X , D_{cur,X}) ⟶ ĈH^{∗}(Y, D_{cur,Y}).

The inverse image is compatible with the product structure.

(iv) If f: Y ⟶ X is a proper map of regular arithmetic varieties of relative dimension d, there are push-forward morphisms

f_{∗}: ĈH^{∗}(Y, D_{cur,Y}) ⟶ ĈH^{∗−d}(X , D_{cur,X}).

If moreover, f is smooth over F, there are push-forward morphisms

f_{∗}: ĈH^{∗}(Y, D_{l,a,Y}) ⟶ ĈH^{∗−d}(X , D_{l,a,X}).

The push-forward morphism satisfies the projection formula and is compatible with base change.

(v) The groups ĈH^{∗}(X , D_{l,a,X}) are naturally isomorphic to the groups defined by Gillet and Soulé in [18] (see [12] theorem 3.33). When X is generically projective, the groups ĈH^{∗}(X , D_{cur,X}) are isomorphic to analogous groups introduced by Kawaguchi and Moriwaki [27] and are very similar to the weak arithmetic Chow groups introduced by Zha (see [11]).

(vi) There are well-defined maps

ζ: ĈH^{p}(X , C) ⟶ CH^{p}(X),
a: C̃^{2p−1}(X_{ℝ}, p) ⟶ ĈH^{p}(X , C),
ω: ĈH^{p}(X , C) ⟶ ZC^{2p}(X_{ℝ}, p),

where C is either D_{l,a,X} or D_{cur,X}. For the precise definition of these maps see [13] notation 4.12.

When X is not necessarily regular, following [20] and combining with the definition of [13] we can define the arithmetic Chow groups indexed by dimension ĈH_{∗}(X , D_{l,a,X}) and ĈH_{∗}(X , D_{cur,X}) (see [12] section 5.3). They have the following properties (see [20]).

(i) If X is regular and equidimensional of dimension n then there are isomorphisms

ĈH_{∗}(X , D_{l,a,X}) ≅ ĈH^{n−∗}(X , D_{l,a,X}),
ĈH_{∗}(X , D_{cur,X}) ≅ ĈH^{n−∗}(X , D_{cur,X}).

(ii) If f: Y ⟶ X is a proper map between arithmetic varieties then there is a push-forward map

f_{∗}: ĈH_{∗}(Y, D_{cur,Y}) ⟶ ĈH_{∗}(X , D_{cur,X}).

If f is smooth over F then there is a push-forward map

f_{∗}: ĈH_{∗}(Y, D_{l,a,Y}) ⟶ ĈH_{∗}(X , D_{l,a,X}).

(iii) If f: Y ⟶ X is a flat map or, more generally, a local complete intersection (l.c.i) map of relative dimension d, there are pull-back morphisms

f^{∗}: ĈH_{∗}(X , D_{l,a,X}) ⟶ ĈH_{∗+d}(Y, D_{l,a,Y}).

If moreover, f is smooth over F, there are pull-back morphisms

f^{∗}: ĈH_{∗}(X , D_{cur,X}) ⟶ ĈH_{∗+d}(Y, D_{cur,Y}).

(iv) If f: Y ⟶ X is a morphism of arithmetic varieties with X regular, then there is a cap product

ĈH^{p}(X , D_{l,a,X}) ⊗ ĈH_{d}(Y, D_{l,a,Y}) ⟶ ĈH_{d−p}(Y, D_{l,a,Y})_{ℚ},

and a similar cap-product with the groups ĈH_{d}(Y, D_{cur,Y}). This product is denoted by y ⊗ x ↦ y._{f}x,

For more properties of these groups see [20]. We will define now the arithmetic K-groups in this context. As a matter of convention, in the sequel we will use slanted letters to denote a object defined over A and the same letter in roman type for the corresponding object defined over ℂ. For instance we will denote a vector bundle over X by E and the corresponding vector bundle over X by E.

Definition 10.1. A hermitian vector bundle on an arithmetic variety X, \overline{E}, is a locally free sheaf E with a hermitian metric h_{E} on the vector bundle E induced on X, that is invariant under F_{∞}. A sequence of hermitian vector bundles on X

(\overline{ε}) … ⟶ \overline{E}_{n+1} ⟶ \overline{E}_{n} ⟶ \overline{E}_{n−1} ⟶ …

is said to be exact if it is exact as a sequence of vector bundles. A metrized coherent sheaf is a pair \overline{F} = (F, \overline{E}_{∗} → F), where F is a coherent sheaf on X and \overline{E}_{∗} → F is a resolution of the coherent sheaf F = F_{ℂ} by hermitian vector bundles, that is defined over ℝ, hence is invariant under F_{∞}. We assume that the hermitian metrics are also invariant under F_{∞}.

Recall that to every hermitian vector bundle we can associate a collection of Chern forms, denoted by c_{p}. Moreover, the invariance of the hermitian metric under F_{∞} implies that the Chern forms will be invariant under the involution σ. Thus

c_{p}(\overline{E}) ∈ D_{l,a,X}^{2p}(X_{ℝ}, p) = D^{2p}(X, p)^{σ}.

We will denote also by c_{p}(\overline{E}) its image in D_{cur,X}^{2p}(X_{ℝ}, p). In particular we have defined the Chern character ch(\overline{E}) in either of the groups ⨁_{p} D_{l,a,X}^{2p}(X_{ℝ}, p) or ⨁_{p} D_{cur,X}^{2p}(X_{ℝ}, p). Moreover, to each finite exact sequence (\overline{ε}) of hermitian vector bundles on X we can attach a secondary Bott-Chern class c̃h(\overline{ε}). Again, the fact that the sequence is defined over A and the invariance of the metrics with respect to F_{∞} imply that

c̃h(\overline{ε}) ∈ ⨁_{p} D̃_{l,a,X}^{2p−1}(X_{ℝ}, p) = ⨁_{p} D̃^{2p−1}(X, p)^{σ}.

We will denote also by c̃h(\overline{ε}) its image in ⨁_{p} D̃_{cur,X}^{2p−1}(X_{ℝ}, p). The Bott-Chern classes associated to exact sequences of metrized coherent sheaves enjoy the same properties.

Definition 10.2. Let X be an arithmetic variety and let C^{∗}(∗) be one of the two D_{log}-complexes D_{l,a,X} or D_{cur,X}. The arithmetic K-group associated to the D_{log}-arithmetic variety (X , C) is the abelian group K̂(X , C) generated by pairs (\overline{E}, η), where \overline{E} is a hermitian vector bundle on X and η ∈ ⨁_{p≥0} C̃^{2p−1}(X_{ℝ}, p), modulo relations

(\overline{E}_{1}, η_{1}) + (\overline{E}_{2}, η_{2}) = (\overline{E}, c̃h(\overline{ε}) + η_{1} + η_{2})  (10.3)

for each short exact sequence

(\overline{ε}) 0 ⟶ \overline{E}_{1} ⟶ \overline{E} ⟶ \overline{E}_{2} ⟶ 0 .

The arithmetic K′-group associated to the D_{log}-arithmetic variety (X , C) is the abelian group K̂′(X , C) generated by pairs (\overline{F}, η), where \overline{F} is a metrized coherent sheaf on X and η ∈ ⨁_{p≥0} C̃^{2p−1}(X_{ℝ}, p), modulo relations

(\overline{F}_{1}, η_{1}) + (\overline{F}_{2}, η_{2}) = (\overline{F}, c̃h(\overline{ε}) + η_{1} + η_{2})  (10.4)

for each short exact sequence of metrized coherent sheaves

(\overline{ε}) 0 ⟶ \overline{F}_{1} ⟶ \overline{F} ⟶ \overline{F}_{2} ⟶ 0 .

We now give some properties of the arithmetic K-groups. As their proofs are similar, in the essential points, to those of analogous statements in, for example,

[18] in the regular case and [20] in the singular case, we omit them.

(i) We have natural morphisms

K̂(X , D_{l,a,X}) ⟶ K̂(X , D_{cur,X}) and K̂′(X , D_{l,a,X}) ⟶ K̂′(X , D_{cur,X}).

When applicable, all properties below will be compatible with these morphisms.

(ii) K̂(X , D_{l,a,X}) is a ring. The product structure is given by

(\overline{F}_{1}, η_{1})·(\overline{F}_{2}, η_{2}) = (\overline{F}_{1}⊗\overline{F}_{2}, ch(\overline{F}_{1})•η_{2}+η_{1}•ch(\overline{F}_{2})+d_{D} η_{1}•η_{2}) (10.5)

(iii) K̂(X , D_{cur,X}) is a K̂(X , D_{l,a,X})-module.

(iv) There are natural maps

K̂(X , C) ⟶ K̂′(X , C)

that, when X is regular, are isomorphisms.

(v) The groups K̂′(X , D_{l,a,X}) and K̂′(X , D_{cur,X}) are K̂(X , D_{l,a,X})-modules.

(vi) There are natural maps

ω: K̂′(X , C) ⟶ ⨁_{p} ZC^{2p}(p)

that send the class of a pair (\overline{F}, η) with \overline{F} = (F, \overline{E}_{∗} → F_{ℂ}) to the form (or current)

ω(\overline{F}, η) = ∑_{i}(−1)^{i} ch(\overline{E}_{i}) + d_{D} η.

(vii) When X is regular, there exists a Chern character,

ĉh: K̂(X , C)_{ℚ} ⟶ ⨁_{p} ĈH^{p}(X , C)_{ℚ},

that is an isomorphism. Moreover, if C = D_{l,a,X} this isomorphism is compatible with the product structure. If X is not regular, there is a biadditive pairing

K̂(X , D_{l,a,X}) ⊗ ĈH_{∗}(X , D_{l,a,X}) ⟶ ĈH_{∗}(X , D_{l,a,X})_{ℚ},

and a similar pairing with the groups ĈH_{∗}(X , D_{cur,X}), which is denoted in both cases by α ⊗ x ↦ ĉh(α) ∩ x. For the properties of this product see [20] pg. 496.

(viii) If Y and X are arithmetic varieties and f: Y → X is a morphism of arithmetic varieties, f induces a morphism of rings:

f^{∗}: K̂(X , D_{l,a,X}) → K̂(Y, D_{l,a,Y}).

When f is flat, the inverse image is also defined for the groups K̂′(X , D_{l,a,X}). Moreover, if f_{ℂ} is smooth, the inverse image can be defined for the groups K̂(X , D_{cur,X}) and, when in addition f is flat, for the groups K̂′(X , D_{cur,X}).

In what follows we will be interested in direct images for closed immersions. Since the direct images in arithmetic K-theory will depend on the choice of a metric, we have the following

Definition 10.6. A metrized arithmetic variety is a pair (X , h_{X}) consisting of an arithmetic variety X and a hermitian metric on the complex tangent bundle T_{X} that is invariant under F_{∞}.

Let (X , h_{X}) and (Y, h_{Y}) be metrized arithmetic varieties and let i: Y ⟶ X be a closed immersion. Over the complex numbers, we are in the situation of notation 8.36. In particular we have a canonical exact sequence of hermitian vector bundles

\overline{ξ}_{N}: 0 ⟶ \overline{T}_{Y} ⟶ i^{∗}\overline{T}_{X} ⟶ \overline{N}_{Y/X} ⟶ 0  (10.7)

where the tangent bundles T_{Y}, T_{X} are endowed with the hermitian metrics h_{Y}, h_{X} respectively and the normal bundle N_{Y/X} is endowed with an arbitrary hermitian metric h_{N}. We will follow the conventions of notation 8.36. We next define push-forward maps, via a closed immersion, for the elements of the arithmetic K-group of a metrized arithmetic variety. We will define two kinds of push-forward maps. One will depend only on a metric on the complex normal bundle N_{Y/X}. By contrast, the second will depend on the choice of metrics on the complex tangent bundles T_{X} and T_{Y}. The second definition allows us to see K′( , D_{cur,Y}) as a functor from the category whose objects are metrized arithmetic varieties and whose morphisms are closed immersions to the category of abelian groups. As we deal with hermitian vector bundles and metrized coherent sheaves, both definitions will involve the choice of a theory of singular Bott-Chern classes. In order for the push forward to be well defined in K-theory we need a minimal additivity property for the singular Bott-Chern classes.

Definition 10.8. A theory of singular Bott-Chern classes T is called additive if for any closed embedding of complex manifolds i: Y ↪ X and any hermitian embedded vector bundles \overline{ξ}_{1} = (i, \overline{N}, \overline{F}_{1}, \overline{E}_{1,∗}), \overline{ξ}_{2} = (i, \overline{N}, \overline{F}_{2}, \overline{E}_{2,∗}) the equation

T (\overline{ξ}_{1} ⊕ \overline{ξ}_{2}) = T (\overline{ξ}_{1}) + T (\overline{ξ}_{2})

is satisfied. Let C be a characteristic class for pairs of vector bundles. We say that it is additive (in the first variable) if

C(F_{1} ⊕ F_{2}, N) = C(F_{1}, N) + C(F_{2}, N)

for any vector bundles F_{1}, F_{2}, N on a complex manifold X.

The following statement follows directly from equation 7.5:

Proposition 10.9. A theory of singular Bott-Chern classes T is additive if and only if the corresponding characteristic class C_{T} is additive in the first variable.

Note that a theory of singular Bott-Chern classes consists in joining theories of singular Bott-Chern classes in arbitrary rank and codimension (definition 6.9). The property of being additive gives a compatibility condition for these theories, by respect to the hermitian vector bundles \overline{F} (with the notation used in definition 6.9). Note also that if a theory of singular Bott-Chern classes is compatible with the projection formula then it is additive.

Definition 10.10. Let T be an additive theory of singular Bott-Chern classes, and let T_{c} be the associated covariant class as in definition 8.37. Let i: (Y, h_{Y}) ⟶ (X , h_{X}) be a closed immersion of metrized arithmetic varieties and let \overline{N} = \overline{N}_{Y/X} = (N_{Y/X}, h_{N}) be a choice of a hermitian metric on the complex normal bundle. The push-forward maps

i_{∗}^{T_{c}}, i_{∗}^{T} : K̂(Y, D_{cur,Y}) ⟶ K̂′(X , D_{cur,X}) are defined by

i_{∗}^{T_{c}}(\overline{F}, η) = [((i_{∗}F, \overline{E}_{∗} → (i_{∗}F)_{ℂ}), 0)] − [(0, T_{c}(\overline{ξ}_{c}))]
+ [(0, i_{∗}(η Td(Y)i^{∗} Td^{−1}(X)))]  (10.11)
i_{∗}^{T} (\overline{F}, η) = [((i_{∗}F, \overline{E}_{∗} → (i_{∗}F)_{ℂ}), 0)] − [(0, T (\overline{ξ}))]
+ [(0, i_{∗}(η Td^{−1}(\overline{N}_{Y/X})))].  (10.12)

Here

0 → \overline{E}_{n} → … → \overline{E}_{1} → \overline{E}_{0} → (i_{∗}F)_{ℂ} → 0

is a finite resolution of the coherent sheaf (i_{∗}F)_{ℂ} by hermitian vector bundles, \overline{ξ} = (i, \overline{N}_{X/Y} , \overline{F}_{ℂ}, \overline{E}_{∗}) is the induced hermitian embedded vector bundle on X, and \overline{ξ}_{c} = (i, \overline{T}_{X}, \overline{T}_{Y} , \overline{F}_{ℂ}, \overline{E}_{∗}) as in definition 8.37. We can extend this definition to push-forward maps

i_{∗}^{T_{c}}, i_{∗}^{T} : K̂′(Y, D_{cur,Y}) ⟶ K̂′(X , D_{cur,X})

by the rule

i_{∗}^{T_{c}}(\overline{F}, η) = [((i_{∗}F, Tot(\overline{E}_{∗,∗}) → (i_{∗}F)_{ℂ}), 0)] − ∑_{i}(−1)^{i}[(0, T_{c}(\overline{ξ}_{i,c}))]
+ [(0, i_{∗}(η Td(Y)i^{∗} Td^{−1}(X)))],  (10.13)
i_{∗}^{T} (\overline{F}, η) = [((i_{∗}F, Tot(\overline{E}_{∗,∗}) → (i_{∗}F)_{ℂ}), 0)] − ∑_{i}(−1)^{i}[(0, T (\overline{ξ}_{i}))]
+ [(0, i_{∗}(η Td^{−1}(\overline{N}_{Y/X})))],  (10.14)

where 0 → \overline{E}_{n} → ⋯ → \overline{E}_{0} → F_{ℂ} → 0 is a resolution of F_{ℂ} by hermitian vector bundles, \overline{E}_{∗,∗} is a complex of complexes of vector bundles over X, such that, for each i ≥ 0, \overline{E}_{i,∗} → i_{∗}E_{i} is also a resolution by hermitian vector bundles and \overline{ξ}_{i} = (i, \overline{N}_{X/Y} , \overline{E}_{i}, \overline{E}_{i,∗}) is the induced hermitian embedded vector bundle and \overline{ξ}_{i,c} is as in definition 8.37. We suppose that there is a commutative diagram of resolutions

\xymatrix{… \ar[r] & E_{k+1,∗} \ar[r] \ar[d] & E_{k,∗} \ar[r] \ar[d] & E_{k−1,∗} \ar[r] \ar[d] & … \\ … \ar[r] & i_{∗}E_{k+1} \ar[r] & i_{∗}E_{k} \ar[r] & i_{∗}E_{k−1} \ar[r] & …} .

hence a resolution Tot(\overline{E}_{∗,∗}) ⟶ (i_{∗}F)_{ℂ} by hermitian vector bundles.

Note that, whenever the push-forward i_{∗}^{T} appears, we will assume that we have chosen a metric on N_{Y/X}. The two push-forward maps are related by the equation

i_{∗}^{T_{c}}(\overline{F}, η) = i_{∗}^{T} (\overline{F}, η) − [(0, i_{∗} (ω(\overline{F}, η)\begin{matrix} \\ Td & −1 \end{matrix}(\overline{ξ}_{N}) Td(Y)))] ,  (10.15)

where \overline{ξ}_{N} is the exact sequence (10.7).

Proposition 10.16. The push-forward maps i_{∗}^{T}, i_{∗}^{T_{c}} are well defined. That is, they do not depend on the choice of a representative of a class in K̂, nor on the choice of metrics on the coherent sheaf (i_{∗}F)_{ℂ}. The first one does not depend on the choice of metrics on T_{X} nor on T_{Y}, whereas the second one does not depend on the choice of a metric on the normal bundle N_{Y/X}. Moreover, if i is a regular closed immersion or X is a regular arithmetic variety, then i_{∗}^{T_{c}} and i_{∗}^{T} can be lifted to maps

i_{∗}^{T_{c}}, i_{∗}^{T} : K̂(Y, D_{cur,Y}) ⟶ K̂(X , D_{cur,Y}).

Proof. The fact that i_{∗}^{T} only depends on the metric on \overline{N} and not on the metrics on T_{X} and T_{Y} and that for i_{∗}^{T_{c}} is the opposite, follows directly from the definition in the first case and from proposition 8.39 in the second. We will only prove the other statements for i_{∗}^{T_{c}}, as the other case is analogous. We first prove the independence from the metric chosen on the coherent sheaf (i_{∗}F)_{ℂ}. If \overline{E}_{∗} → (i_{∗}F)_{ℂ}, \overline{E}′_{∗} → (i_{∗}F)_{ℂ} are two such metrics, inducing the hermitian embedded vector bundles \overline{ξ} respectively \overline{ξ}′, then, using corollary 6.14

T_{c}(\overline{ξ}′_{c}) − T_{c}(\overline{ξ}_{c}) = T (\overline{ξ}′) − T (\overline{ξ}) = c̃h(\overline{ε}),

where \overline{ε} is the exact complex of hermitian embedded vector bundles

\overline{ε}: 0 ⟶ \overline{ξ} ⟶ \overline{ξ}′ ⟶ 0,

where \overline{ξ}′ sits in degree zero. Therefore, by equation 10.4,

[((i_{∗}F, \overline{E}_{∗} → (i_{∗}F)_{ℂ}), 0)] − [(0, T_{c}(\overline{ξ_{c}}))]

= [((i_{∗}F, \overline{E}′_{∗} → (i_{∗}F)_{ℂ}), 0)] − [(0, T_{c}(\overline{ξ}′_{c}))].

Since the last term of equation 10.11 does not depend on the metric on (i_{∗}F)_{ℂ}, we obtain that i_{∗}^{T_{c}} does not depend on this metric. For proving that the push-forward map i_{∗}^{T_{c}} is well defined it remains to show the independence from the choice of a representative of a class in K̂(Y, D_{cur,Y}). We consider an exact sequence of hermitian vector bundles on Y

\overline{ε}: 0 ⟶ \overline{F}_{1} ⟶ \overline{F} ⟶ \overline{F}_{2} ⟶ 0

and two classes η_{1}, η_{2} ∈ ⨁_{p≥0} D̃_{cur}^{2p−1}(Y, p). We also denote \overline{ε} the induced exact sequence of hermitian vector bundles on Y. We have to prove

i_{∗}^{T_{c}}([(\overline{F}, η_{1} + η_{2} + c̃h(\overline{ε})]) = i_{∗}^{T_{c}}([(\overline{F_{1}}, η_{1})]) + i_{∗}^{T_{c}}([(\overline{F_{2}}, η_{2})]).  (10.17)

Since it is clear that i_{∗}^{T_{c}}(0, η_{1} + η_{2}) = i_{∗}^{T_{c}}(0, η_{1}) + i_{∗}^{T_{c}}(0, η_{2}), we are led to prove

i_{∗}^{T_{c}}([(\overline{F}, c̃h(\overline{ε})]) = i_{∗}^{T_{c}}([(\overline{F_{1}}, 0)]) + i_{∗}^{T_{c}}([(\overline{F_{2}}, 0)]).  (10.18)

We choose metrics on the coherent sheaves (i_{∗}F_{1})_{ℂ}, (i_{∗}F_{2})_{ℂ} and (i_{∗}F)_{ℂ} respectively:

\overline{E}_{1,∗} ⟶ (i_{∗}F_{1})_{ℂ} , \overline{E}_{2,∗} ⟶ (i_{∗}F_{2})_{ℂ} , \overline{E}_{∗} ⟶ (i_{∗}F)_{ℂ}.

We denote \overline{ξ}_{1}, \overline{ξ}_{2}, \overline{ξ} the induced hermitian embedded vector bundles. We obtain an exact sequence of metrized coherent sheaves on X:

\overline{ν}: 0 ⟶ \overline{i_{∗}F_{1}} ⟶ \overline{i_{∗}F} ⟶ \overline{i_{∗}F_{2}} ⟶ 0.

Then, using the fact that the theory T is additive and equation (8.42) we have

T_{c}(\overline{ξ}_{1,c}) + T_{c}(\overline{ξ}_{2,c}) − T_{c}(\overline{ξ}_{c}) = [c̃h(\overline{ν})] − i_{∗}([c̃h(\overline{ε}) • Td(Y)]) • Td^{−1}(X). (10.19)

Moreover, by the relation (10.4),

[(\overline{i_{∗}F_{1}}, 0)] + [(\overline{i_{∗}F_{2}}, 0)] = [(\overline{i_{∗}F}, c̃h(\overline{ν}))].  (10.20)

Hence, we compute,

i_{∗}^{T_{c}}([(\overline{F}, c̃h(\overline{ε})]) − i_{∗}^{T_{c}}([(\overline{F_{1}}, 0)]) − i_{∗}^{T_{c}}([(\overline{F_{2}}, 0)])
= [(i_{∗}\overline{F}, 0)] − [(i_{∗}\overline{F_{1}}, 0)] − [(i_{∗}\overline{F_{2}}, 0)]
− [(0, T_{c}(\overline{ξ}_{c}))] + [(0, T_{c}(\overline{ξ_{1,c}}))] + [(0, T_{c}(\overline{ξ_{2,c}}))]
+ [(0, i_{∗}([c̃h(\overline{ε})] • Td(Y) • i^{∗} Td^{−1}(X)))]
= −[(0, i_{∗}([c̃h(\overline{ε})] • Td(Y) • i^{∗} Td^{−1}(X))))]
+ [(0, i_{∗}([c̃h(\overline{ε})] • Td(Y) • i^{∗} Td^{−1}(X))))]
= 0.

The proof that i_{∗}^{T_{c}} for metrized coherent sheaves is well defined is similar. The proof of its independence from choice of a metric on N_{Y/X} or from the choice of the resolutions and metrics in X is the same as before. Now let

0 ⟶ \overline{F}′ ⟶ \overline{F} ⟶ \overline{F}′′ ⟶ 0

be a short exact sequence of metrized coherent sheaves on Y. This means that we have resolutions \overline{E}′_{∗} → F′_{ℂ}, \overline{E}_{∗} → F_{ℂ} and \overline{E}′′_{∗} → F′′_{ℂ}. Using theorem 2.24 we can suppose that there is a commutative diagram of resolutions

\begin{aligned}0 & → & \overline{E}′_{∗} & → & \overline{E}_{∗} & → & \overline{E}′′_{∗} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & F′_{ℂ} & → & F_{ℂ} & → & F′′_{ℂ} & → & 0, \end{aligned}  (10.21)

with exact rows. Moreover, we can assume that the complexes of complexes \overline{E}′_{∗,∗}, \overline{E}_{∗,∗}, \overline{E}′′_{∗,∗} used in definition 10.10 are chosen compatible with diagram (10.21). Thus we obtain a commutative diagram

0 → Tot \overline{E}′_{∗,∗} → Tot \overline{E}_{∗,∗} → Tot \overline{E}′′_{∗,∗} → 0
↓ ↓ ↓
0 → i_{∗}F′_{ℂ} → i_{∗}F_{ℂ} → i_{∗}F′′_{ℂ} → 0.

(10.22)

We denote by \overline{ν} the exact sequence of metrized coherent sheaves on X defined by diagram (10.22). We denote \overline{χ}_{i} the exact sequence of hermitian vector bundles on Y

\overline{χ}_{i}: 0 ⟶ \overline{E}′_{i} ⟶ \overline{E}_{i} ⟶ \overline{E}′′_{i} ⟶ 0,

and by \overline{ε} the exact sequence of metrized coherent sheaves on X

\overline{ε}_{i}: 0 ⟶ \overline{i_{∗}E}′_{i} ⟶ \overline{i_{∗}E}_{i} ⟶ \overline{i_{∗}E}′′_{i} ⟶ 0.

Moreover, let \overline{ξ}_{i}, \overline{ξ}′_{i} and \overline{ξ}′′_{i} denote the hermitian embedded vector bundles defined by the above resolutions and \overline{E}_{i}, \overline{E}′_{i} and \overline{E}′′_{i} respectively and let \overline{ξ}_{i,c}, \overline{ξ}′_{i,c} and \overline{ξ}′′_{i,c} be as in definition 8.37. Then, using proposition 2.38 and equation (8.42) we obtain

c̃h(\overline{ν}) = ∑_{i}(−1)^{i}c̃h(\overline{ε})
= ∑_{i}(−1)^{i}(T_{c}(\overline{ξ}′_{i,c}) + T_{c}(\overline{ξ}′′_{i,c}) − T_{c}(\overline{ξ}_{i,c}))  (10.23)

+ ∑_{i}(−1)^{i}i_{∗}(c̃h(\overline{χ}_{i}) • Td(Y)) • Td^{−1}(X)

Now the proof follows as before, but using equation (10.23) instead of equation (10.19). If X is a regular arithmetic variety, the lifting property follows from the isomorphism between the K̂-groups and the K̂′-groups. Suppose now that i: Y ⟶ X is a regular closed immersion and let [\overline{F}, η] ∈ K̂(Y, D_{cur,Y}). Then it follows from [2] III that the coherent sheaf i_{∗}F can be resolved

0 ⟶ E_{n} ⟶ … ⟶ E_{0} ⟶ i_{∗}F ⟶ 0

with E_{i} locally free sheaves on X. Moreover we endow the vector bundles E_{i} induced on X with hermitian metrics and so we obtain a metric on the coherent sheaf i_{∗}F and the corresponding hermitian embedded vector bundle \overline{ξ}. Using the independence from the resolutions and on the metrics we see that the equation 10.11 defines an element in K̂(X , D_{cur,X}).

Proposition 10.24. For any element α ∈ K̂′(Y, D_{cur,Y}) we have

ω(i_{∗}^{T_{c}}(α)) Td(X) = i_{∗}(ω(α) Td(Y))  (10.25)
ω(i_{∗}^{T} (α)) = i_{∗}(ω(α) Td^{−1}(N_{Y/X}))  (10.26)

Proof. We will prove the statement only for i_{∗}^{T_{c}}. We consider first a class of the form [\overline{F}, 0]. Using equation (8.38) we obtain, after choosing a metric \overline{E}_{i} ⟶ (i_{∗}F)_{ℂ}, and considering the induced hermitian embedded vector bundle \overline{ξ_{c}}:

ω(i_{∗}^{T_{c}}([\overline{F}, 0])) Td(X) = (∑(−1)^{i} ch(\overline{E_{i}}) − d_{D} T_{c}(\overline{ξ_{c}})) Td(X)
= i_{∗}(ch(\overline{F}) • Td(Y) • i^{∗} Td^{−1}(X)i^{∗}(Td(X)))
= i_{∗}(ch(\overline{F}) • Td(Y))
= i_{∗}(ω([\overline{F}, 0]) Td(Y))

Taking now a class of the form [0, η] we obtain:

ω(i_{∗}^{T_{c}}([0, η])) Td(X) = d_{D} (i_{∗}(η Td(Y)i^{∗} Td^{−1}(X))) Td(X)
= i_{∗} d_{D}(η Td(Y))
= i_{∗}(ω([0, η]) Td(Y))

and hence the equality 10.25 is proved.

The next proposition explains the terminology “compatible with the projection formula” and “transitive” that we used for theories of singular Bott-Chern classes. The second statement is the main reason to introduce the push-forward i_{∗}^{T_{c}}.

Proposition 10.27. If the theory of singular Bott-Chern classes is compatible with the projection formula, we have that, for α ∈ K̂′(Y, D_{cur,Y}) and β ∈ K̂(X , D_{l,a,X}) the following equalities hold

i_{∗}^{T_{c}}(αi^{∗}β) = i_{∗}^{T_{c}}(α)β,
i_{∗}^{T} (αi^{∗}β) = i_{∗}^{T} (α)β.

If moreover the theory of singular Bott-Chern classes is transitive and j: (Z, h_{Z}) ⟶ (Y, h_{Y}) is another closed immersion of metrized arithmetic varieties, then

(i ∘ j)_{∗}^{T_{c}} = i_{∗}^{T_{c}} ∘ j_{∗}^{T_{c}}.

Proof. We prove first the projection formula. For simplicity we will treat the case when α ∈ K̂(Y, D_{cur,Y}). Let α = (\overline{F}, η), let \overline{ξ_{c}} = (i, \overline{T}_{X}, \overline{T}_{Y} , \overline{F}_{ℂ}, \overline{E}_{∗}) be a hermitian embedded vector bundle and let β = (\overline{E}, χ). Using equations (10.11) and (10.5), we obtain

i_{∗}^{T_{c}}(αi^{∗}β) − i_{∗}^{T_{c}}(α)β = − ∑_{i}(−1)^{i} ch(\overline{E}_{i}) • χ + d_{D}(T_{c}(\overline{ξ}_{c})) • χ
+ i_{∗}(ch((\overline{F})_{ℂ}) • Td(Y))) • Td^{−1}(X) • χ
+ T_{c}(\overline{ξ}_{c}) • ch(\overline{E}_{ℂ}) − T_{c}(\overline{ξ}_{c} ⊗ \overline{E}_{ℂ})
= T_{c}(\overline{ξ}_{c} ⊗ \overline{E}_{ℂ}) − T_{c}(\overline{ξ}_{c}) • ch(\overline{E}_{ℂ}).

Therefore, if T is compatible with the projection formula, then the projection formula holds. The fact that, if moreover T is transitive then (i ∘ j)_{∗}^{T_{c}} = i_{∗}^{T_{c}} ∘ j_{∗}^{T_{c}} follows directly from the definition and equation (8.41).

If i: Y ⟶ X is a regular closed immersion between arithmetic varieties, then the normal cone N_{Y/X} is a locally free sheaf. The choice of a hermitian metric on N_{Y/X} determines a hermitian vector bundle \overline{N}_{Y/X}. If now i: (Y, h_{Y}) ⟶ (X , h_{X}) is a closed immersion between regular metrized arithmetic varieties, then the tangent bundles T_{Y} and T_{X} are virtual vector bundles. Since over ℂ they define vector bundles, we can provide them with hermitian metrics and denote the hermitian virtual vector bundles by \overline{T}_{X} and \overline{T}_{Y}. There are well defined clases T̂d(Y) = T̂d(\overline{T}_{Y}) and T̂d(X) = T̂d(\overline{T}_{X}). The arithmetic Grothendieck-Riemann-Roch theorem for closed immersions compares the direct images in the arithmetic K-groups with the direct images in the arithmetic Chow groups.

Theorem 10.28 ([6], [32]). Let T be a theory of singular Bott-Chern classes and let S_{T} be the additive genus of corollary 9.43.

(i) Let i: Y ⟶ X be a regular closed immersion between arithmetic varieties. Assume that we have chosen a hermitian metric on the complex bundle N_{Y/X}. Then, for any α = (\overline{F}, η) ∈ K̂(Y, D_{cur,Y}) the equation

ĉh(i_{∗}^{T} (α)) = i_{∗}(ĉh(α)T̂d^{−1}(\overline{N}_{Y/X})) − a(i_{∗}(ch(F_{ℂ}) Td^{−1}(N_{Y/X})S_{T} (N))

(10.29)

holds.

(ii) Let i: (Y, h_{Y}) ⟶ (X , h_{X}) be a closed immersion between regular metrized arithmetic varieties. Then, for any α = (\overline{F}, η) ∈ K̂(Y, D_{cur,Y}) the equation

ĉh(i_{∗}^{T_{c}}(α))T̂d(X) = i_{∗}(ĉh(α)T̂d(Y)) − a(i_{∗}(ch(F_{ℂ}) Td(Y)S_{T} (N)))

(10.30)

holds.

Proof. The proof follows the classical pattern of the deformation to the normal cone as in [6] and [32]. Let W be the deformation to the normal cone to Y in X. We will follow the notation of section 5. Since i is a regular closed immersion, there is a finite resolution by locally free sheaves

0 → E_{n} → ⋯ → E_{1} → E_{0} → i_{∗}F → 0.

We choose hermitian metrics on the complex bundles E_{i} = (E_{i})_{ℂ}. The immersion j: Y × ℙ^{1} ⟶ W is also a regular immersion. The construction of theorem 5.4 is valid over the arithmetic ring A. Therefore we have a resolution by hermitian vector bundles

0 → G̃_{n} → ⋯ → G̃_{1} → G̃_{0} → i_{∗}F → 0. such that its restriction to X × {0} is isometric to E_{∗}. Its restriction to X̃ is orthogonally split, and its restriction to P = ℙ(N_{Y/X} ⊕ O_{Y}) fits in a short exact sequence

0 ⟶ \overline{A}_{∗} ⟶ Ẽ_{∗}|_{P} ⟶ K(\overline{F}, \overline{N}_{Y/X}) ⟶ 0,

where \overline{A}_{∗} is orthogonally split and K(\overline{F}, \overline{N}_{Y/X}) is the Koszul resolution. We denote by \overline{η}_{k} the piece of degree k of this exact sequence. Let t be the absolute coordinate of ℙ^{1}. It defines a rational function in W and

d̂iv(t) = (X_{0} + P + X̃, (0, −\frac{1}{2} log t\overline{t}))

The key point of the proof of the theorem is that, in the group ĈH^{∗}(X , D_{cur,X}), we have

(p_{W})_{∗}(ĉh(Ẽ_{∗})d̂iv(t)) = 0.

Using the definition of the product in the arithmetic Chow rings we obtain

(p_{W})_{∗}(ĉh(Ẽ_{∗})d̂iv(t)) = ĉh(\overline{E}_{∗}) − (pX̃)_{∗}ĉh(Ẽ_{∗}|X̃) − (pP̃)_{∗}ĉh(Ẽ_{∗}|_{P})

+ a((p_{W})_{∗}(ch((Ẽ_{∗})_{ℂ}) • W_{1})). (10.31)

But we have

ĉh(\overline{E}_{∗}) = ĉh(i_{∗}^{T} (\overline{F})) + a(T (\overline{ξ})),  (10.32)
(pX̃)_{∗}ĉh(Ẽ_{∗}|X̃) = 0,  (10.33)
(pP̃)_{∗}ĉh(Ẽ_{∗}|_{P}) = i_{∗}(π_{P})_{∗}(ĉh(K(\overline{F}, \overline{N}_{Y/X})) − ∑_{k}(−1)^{k} a(c̃h(\overline{η}_{k}))). (10.34)

Moreover, by equation (7.3),

a((p_{W})_{∗}(ch((Ẽ_{∗})_{ℂ}) • W_{1})) = − a(T (\overline{ξ})) − ∑_{k}(−1)^{k} a(c̃h(\overline{η}_{k})))

+ a(i_{∗}C_{T} (F_{ℂ}, N_{ℂ})). (10.35)

Thus we are led to compute i_{∗}(π_{P})_{∗}ĉh(K(\overline{F}, \overline{N}_{Y/X})). This is done in the following two lemmas.

Lemma 10.36. Let Y be an arithmetic variety, \overline{N} a rank r hermitian vector bundle over Y and denote P = ℙ^{1}(N ⊕ O_{Y}), and \overline{Q} the tautological quotient bundle. Let Y_{0} be the cycle defined by the zero section of P. Then

ĉ_{r}(\overline{Q}) = (Y_{0}, (c_{r}(\overline{Q}_{ℂ}), ẽ(P_{ℂ}, \overline{Q}_{ℂ}, s))),  (10.37)

where ẽ(P_{ℂ}, \overline{Q}_{ℂ}, s) is the Euler-Green current of lemma 9.4.

Proof. We know that ĉ_{r}(\overline{Q}) = (Y_{0}, (c_{r}(\overline{Q}_{ℂ}), ẽ)) for certain Green current ẽ. By definition this Green current satisfies

d_{D} ẽ = c_{r}(\overline{Q}_{ℂ}) − δ_{Y_{ℂ}}.

Moreover, since the restriction of \overline{Q}_{ℂ} to D_{∞} has a global section of constant norm we have that ẽ|_{D_{∞}} = 0. Therefore, by lemma 9.4,

ẽ = ẽ(P_{ℂ}, \overline{Q}_{ℂ}, s).

Lemma 10.38. The following equality hold:

(π_{P})_{∗}ĉh(K(\overline{F}, \overline{N})_{∗}) =

ĉh(\overline{F})\begin{matrix} \\ Td & −1 \end{matrix}(\overline{N}) + a(C_{T} (\overline{F}, \overline{N}) − ch(F_{ℂ}) Td^{−1}(N_{Y/X})S_{T} (N)). (10.39)

Proof. We just compute, using lemma 10.36,

(π_{P})_{∗}ĉh(K(\overline{F}, \overline{N})_{∗}) = (π_{P})_{∗} ∑_{k}(−1)^{k}ĉh(⋀^{k} \overline{Q}^{∨})ĉh(π_{P}^{∗} \overline{F})

= (π_{P})_{∗}(ĉ_{r}(\overline{Q})\begin{matrix} \\ Td & −1 \end{matrix}(\overline{Q}))ĉh(\overline{F})

= \begin{matrix} \\ Td & −1 \end{matrix}(\overline{N})ĉh(\overline{F}) + a((π_{P})_{∗}(ẽTd^{−1}(\overline{Q})) ch(\overline{F}))

= \begin{matrix} \\ Td & −1 \end{matrix}(\overline{N})ĉh(\overline{F}) + a((π_{P})_{∗}(T^{h}(K(\overline{F} , \overline{N}))) ch(\overline{F}))

= \begin{matrix} \\ Td & −1 \end{matrix}(\overline{N})ĉh(\overline{F}) + a(C_{T^{h}}(F, N))

= \begin{matrix} \\ Td & −1 \end{matrix}(\overline{N})ĉh(\overline{F}) + C_{T} (F, N) − a(Td^{−1}(N) ch(F)S_{T} (N)).

The equation (10.29) follows by combining equations (10.31), (10.32), (10.33), (10.34), (10.35) and (10.39). The equation (10.30) follows from equation (10.29) by a straightforward computation.

Since T is homogeneous if and only if S_{T} = 0, in view of this result, the theory of homogeneous singular Bott-Chern classes is characterized for being the unique theory of singular Bott-Chern classes that provides an exact arithmetic Grothendieck-Riemann-Roch theorem for closed immersions. By contrast, if one uses a theory of singular Bott-Chern classes that is not homogeneous, there is an analogy between the genus S_{T} and the R-genus that appears in the arithmetic Grothendieck-Riemann-Roch theorem for submersions. Since there is a unique theory of homogeneous singular Bott-Chern classes, the following definition is natural.

Definition 10.40. Let i: (Y, h_{Y}) ⟶ (X , h_{X}) be a closed immersion of metrized arithmetic varieties, the push-forward map

i_{∗}: K̂′(Y, D_{cur,Y}) ⟶ K̂′(X , D_{cur,Y})

is defined as i_{∗} = i_{∗}^{T_{c}^{h}}.

Corollary 10.41. The push-forward map makes K̂′( , D_{cur,Y}) and K̂( , D_{cur,Y}) functors from the category of regular metrized arithmetic varieties and closed immersions to the category of abelian groups.

Corollary 10.42. Let i: (Y, h_{Y}) ⟶ (X , h_{X}) be a closed immersion of regular metrized arithmetic varieties, then

ĉh(i_{∗}^{T} (α))T̂d(X) = i_{∗}(ĉh(α)T̂d(Y)).  (10.43)

Remark 10.44. Combining theorem 10.28 with [16] we can obtain an arithmetic Grothendieck-Riemann-Roch theorem for projective morphisms of regular arithmetic varieties. In a forthcoming paper we will show that the higher torsion forms used to define the direct images for submersions can also be characterized axiomatically.

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José I. Burgos Gil Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) burgos@icmat.es, jiburgosgil@gmail.com Temporary Address: Centre de Recerca Matemática CRM UAB Science Faculty 08193 Bellaterra Barcelona, Spain Răzvan Liţcanu University Al. I. Cuza Faculty of Mathematics Bd. Carol I, 11 700506 Iaşi Romania litcanu@uaic.ro