A Lefschetz Fixed Point Formula

for Singular Arithmetic Schemes

with Smooth Generic Fibres

Shun Tang

Received: May 31, 2010 Revised: October 11, 2010

Communicated by Takeshi Saito

Abstract. In this article, we consider singular equivariant arithmetic schemes whose generic fibres are smooth. For such schemes, we prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. This formula is an analog, in the arithmetic case, of the Lefschetz formula proved by R. W. Thomason in [31]. In particular, our result implies a fixed point formula which was conjectured by V. Maillot and D. Rössler in [25].

2010 Mathematics Subject Classification: 14C40, 14G40, 14L30, 58J20, 58J52 Keywords and Phrases: fixed point formula, singular arithmetic scheme, Arakelov geometry

1 Introduction

It is the aim of this article to prove a singular Lefschetz fixed point formula for some schemes which admit the actions of a diagonalisable group scheme, in the context of Arakelov geometry. We first roughly describe the history of the study of such Lefschetz fixed point formulae and relative Lefschetz-Riemann-Roch problems. Let k be an algebraically closed field and let n be an integer which is prime to the characteristic of k. A projective k−variety X which admits an automorphism g of order n will be called an equivariant variety. An equivariant coherent sheaf on X is a coherent sheaf F on X together with a homomorphism φ : g^{∗}F → F. It is clear that this homomorphism induces a family of endomorphisms H^{i}(φ) on cohomology spaces H^{i}(X, F).

A classical Lefschetz fixed point formula is to give an expression of the alternating sum of the traces of H^{i}(φ), as a sum of the contributions from the components of the fixed point subvariety X_{g}. On the other hand, roughly speaking, a Lefschetz-Riemann-Roch theorem is a commutative diagram in equivariant K−theory which can be regarded as a Grothendieck type generalization of the Lefschetz fixed point formula. Indeed, when we choose the base variety in such a commutative diagram to be a point, we will get the ordinary Lefschetz fixed point formula. If X is nonsingular, P. Donovan has proved such a theorem in [12] by using the results and some of the methods of the paper of A. Borel and J. P. Serre on the Grothendieck-Riemann-Roch theorem (cf. [10]). In [1], P. Baum, W. Fulton and G. Quart generalized Donovan’s theorem to singular varieties, the key step of their proof heavily relies on an elegant method called the deformation to the normal cone. Denote by G_{0}(X, g) (resp. K_{0}(X, g)) the Quillen’s algebraic K-group associated to the category of equivariant coherent sheaves (resp. vector bundles of finite rank) on X, then K_{0}(Pt, g) is isomorphic to the group ring ℤ[k] and G_{0}(X, g) (resp. K_{0}(X, g)) has a natural K_{0}(Pt, g)-module (resp. K_{0}(Pt, g)-algebra) structure. Let f be an equivariantly projective morphism between two equivariant varieties X and Y, then it is possible to define a push-forward morphism f_{∗} from G_{0}(X, g) to G_{0}(Y, g) in a rather standard way. Let R be any flat K_{0}(Pt, g)-algebra in which 1 − ζ is invertible for each non-trivial n-th root of unity ζ in k. The main result of Baum, Fulton and Quart reads: there exists a family of group homomorphisms L. between K-groups making the following diagram

\xymatrix{G_{0}(X, g) \ar[r]^{L.} \ar[d]_{f_{∗}} & G_{0}(X_{g}, g) ⊗_{ℤ[k]} R \ar[d]^{_{f_{g}_{∗}}} \\ G_{0}(Y, g) \ar[r]^{L.} & G_{0}(Y_{g}, g) ⊗_{ℤ[k]} R}

commutative. If Z is a nonsingular equivariant variety such that there exists an equivariant closed immersion from X to Z, then for every equivariant coherent sheaf E on X the homomorphism L. is exactly given by the formula

L.(E) = λ_{−1}^{−1}(N_{Z/Z_{g}}^{∨}) · ∑_{j}(−1)^{j}Tor_{O_{Z}}^{j} (i_{∗}E, O_{Z_{g}})

where N_{Z/Z_{g}} stands for the normal bundle of Z_{g} in Z and λ_{−1}(N_{Z/Z_{g}}^{∨}) := ∑(−1)^{j} ∧^{j} N_{Z/Z_{g}}^{∨}. We would like to indicate that one can use the same method so called the deformation to the normal cone to extend Baum, Fulton and Quart’s result to general scheme case where X and Y are Noetherian, separated schemes endowed with projective actions of the diagonalisable group scheme μ_{n} associated to ℤ/nℤ. Here by a μ_{n}-action on X we understand a morphism m_{X} : μ_{n} × X → X which statisfies some compatibility properties. Denote by p_{X} the projection from μ_{n} × X to X. For a coherent O_{X}-module E on X, a μ_{n}-action on E we mean an isomorphism m_{E} : p_{X}^{∗}E → m_{X}^{∗}E which satisfies certain associativity properties. We refer to [20] and [21, Section 2] for the group scheme action theory we are talking about. In [31], R. W. Thomason used another way to generalize Baum, Fulton and Quart’s result to the scheme case, and he removed the condition of projectivity. The strategy Thomason followed was to use Quillen’s localization sequence for higher equivariant K-groups to prove an algebraic concentration theorem. Let D be an integral Noetherian ring, and let μ_{n} be the diagonalisable group scheme over D associated to ℤ/nℤ. Denote the ring K_{0}(D, μ_{n}) ≅ K_{0}(D)[T]/(1 − T^{n}) by R(μ_{n}). We consider the prime ideal ρ in R(μ_{n}) with which the intersection of ℤ[ℤ/nℤ] is exactly the kernel of the canonical morphism ℤ[T]/(1 − T^{n}) → ℤ[T]/(Φ_{n}) where Φ_{n} stands for the nth cyclotomic polynomial (cf. [31, Lem. 1.6]). By construction the elements 1 − T^{k} for k = 1, …, n − 1 are not contained in ρ. Let X be a μ_{n}-equivariant scheme over D, then G_{0}(X, μ_{n}) (resp. K_{0}(X, μ_{n})) has a natural R(μ_{n})-module (resp. R(μ_{n})-algebra) structure. Denote by i the inclusion from X_{μ_{n}} to X. The algebraic concentration theorem reads: there exists a natural group homomorphism i_{∗} from G_{0}(X_{μ_{n}}, μ_{n})_{ρ} to G_{0}(X, μ_{n})_{ρ} which is an isomorphism. Moreover, if X is regular, the inverse map of i_{∗} is given by λ_{−1}^{−1}(N_{X/X_{μn}}^{∨}) · i^{∗} where N_{X/X_{μn}} is the normal bundle of X_{μ_{n}} in X. This concentration theorem can be used to prove a singular Lefschetz fixed point formula which is an extension of Baum, Fulton and Quart’s result in general scheme case. Thomason’s approach has nothing to do with the construction of the deformation to the normal cone, and the localization he used is slightly weaker than Baum, Fulton and Quart’s in the sense that the complement of the ideal ρ in R(μ_{n}) is not the smallest algebra in which the elements 1 − T^{k} (k = 1, …, n − 1) are invertible. If one exactly chooses R to be the complement of the ideal ρ in R(μ_{n}), then these two localizations are equal to each other. In [21], K. Köhler and D. Rössler generalized the regular case of Baum, Fulton and Quart’s result to Arakelov geometry. To every regular μ_{n}-equivariant arithmetic scheme X, they associate an equivariant arithmetic K_{0}-group K̂_{0}(X, μ_{n}) which contains some smooth form class on X_{μ_{n}}(ℂ) as analytic datum. Such an equivariant arithmetic K_{0}-group has a ring structure so that it is also an R(μ_{n})-algebra. Let \overline{N}_{X/X_{μn}} be the normal bundle with respect to the regular immersion X_{μ_{n}} ↪ X which is endowed with the quotient metric induced by a chosen Kähler metric of X(ℂ), then the main theorem in [21] reads: the element λ_{−1}(\overline{N}_{X/X_{μn}}^{∨}) is invertible in K̂_{0}(X_{μ_{n}}, μ_{n}) ⊗_{R(μ_{n})} R and we have the following commutative diagram

\xymatrix{K̂_{0}(X, μ_{n}) \ar[r]^{Λ_{R}(f)^{−1}·τ} \ar[d]_{f_{∗}} & K̂_{0}(X_{μ_{n}}, μ_{n}) ⊗_{R(μ_{n})} R \ar[d]^{_{f_{μn}_{∗}}} \\ K̂_{0}(D, μ_{n}) \ar[r]^{ι} & K̂_{0}(D, μ_{n}) ⊗_{R(μ_{n})} R}

Documenta Mathematica 15 (2010) 1049–1108

where Λ_{R}(f) := λ_{−1}(\overline{N}_{X/X_{μn}}^{∨}) · (1 + R_{g}(N_{X/X_{μn}})), τ stands for the restriction map and ι is the natural morphism from a ring or a module to its localization which sends an element e to \frac{e}{1}. Here R_{g}(·) is the equivariant R-genus, the definition of the two push-forward morphisms f_{∗} and f_{μ_{n}}_{∗} involves an important analytic datum which is called the equivariant analytic torsion. The strategy Köhler and Rössler followed to prove such an arithmetic Lefschetz-Riemann-Roch theorem was to use the construction of the deformation to the normal cone to prove an analog of this theorem for equivariant closed immersions. After that, they decompose the morphism f to a closed immersion h from X to some projective space ℙ_{D}^{r} followed by a smooth morphism p from ℙ_{D}^{r} to Spec(D). Then the theorem in general situation follows from an argument of investigating the behavior of the error term under the morphisms h and p. Provided X. Ma’s extension of equivariant analytic torsion to higher equivariant analytic torsion form, it was conjectured by Köhler and Rössler in [22] that an analog of [21, Theorem 4.4] in relative setting holds. We have already proved this conjecture in [29]. Our method is similar to Thomason’s, we first show that there exists an arithmetic concentration theorem in Arakelov geometry and then deduce from it the relative Lefschetz fixed point formula. The same as Thomason’s approach, our method has nothing to do with the construction of the deformation to the normal cone, but unfortunately it only works for regular arithmetic schemes. One may naturally asked that whether it is possible to construct a more general arithmetic Ĝ_{0}-theory and prove a relative Lefschetz fixed point formula for singular arithmetic schemes which is entirely an analog of Thomason’s singular Lefschetz formula in Arakelov geometry. The answer is Yes, and this is what we have done in this article. To do this, one needs a Ĝ_{0}-theoretic vanishing theorem which can be viewed as an extension of Köhler and Rössler’s fixed point formula for closed immersions to the singular case. The proof of such a vanishing theorem occupies a lot of space in this article. Let X and Y be two singular equivariant arithmetic schemes with smooth generic fibres, and let f : X → Y be an equivariant morphism which is smooth on the complex numbers. Assume that the μ_{n}-action on Y is trivial and f can be decomposed to be h ∘ i where i is an equivariant closed immersion from X to some regular arithmetic scheme Z and h : Z → Y is equivariant and smooth on the complex numbers. Let \overline{η} be an equivariant hermitian sheaf on X. Referring to Section 6.1 for the explanations of various notations, we announce that our main theorem in this article is the following equality which holds in Ĝ_{0}(Y, μ_{n}, ∅)_{ρ}:

f_{∗}(\overline{η}) =f_{μ_{n}}^{Z}_{∗}(i_{μ_{n}}^{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨})) · ∑_{k}(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}))

+ ∫_{X_{g}/Y} Td(T f_{g}, ω^{X})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω^{X})Td_{g}^{−1}(\overline{F})

− ∫_{X_{g}/Y} Td_{g}(T f)ch_{g}(η)R_{g}(N_{X/X_{g}})

+ ∫_{X_{g}/Y} T̃d(T f_{g}, ω^{X}, ω_{X}^{Z})ch_{g}(\overline{η})Td_{g}(\overline{N}_{Z/Z_{g}})Td_{g}^{−1}(\overline{F}).

The structure of this article is as follows. In Section 2, we recall some differential-geometric facts for the convenience of the reader. In Section 3, we formulate and prove a vanishing theorem for equivariant closed immersions in a purely analytic setting. In Section 4, we define the arithmtic G_{0}-groups with respect to fixed wave front sets which are necessary for our later arguments. In Section 5 and Section 6, we formulate and prove the arithmetic concentration theorem and the relative Lefschetz fixed point formula for singular arithmetic schemes. Acknowledgements. The author wishes to thank his thesis advisor Damian Rössler for providing such an interesting topic, also for his constant encouragement and for many fruitful discussions between them. The author is greatful to Xiaonan Ma, from whom he gets many meaningful comments and suggestions. Finally, thanks to the referee, for his/her excellent work.

2 Differential-geometric preliminaries

2.1 Equivariant Chern-Weil theory

Let G be a compact Lie group and let M be a compact complex manifold which admits a holomorphic G-action. By an equivariant hermitian vector bundle on M, we understand a hermitian vector bundle on M which admits a G-action compatible with the G-structure of M and whose metric is G-invariant. Let g ∈ G be an automorphism of M, we shall denote by M_{g} = {x ∈ M | g · x = x} the fixed point submanifold. M_{g} is also a compact complex manifold. Now let \overline{E} be an equivariant hermitian vector bundle on M, it is well known that the restriction of \overline{E} to M_{g} splits as a direct sum

\overline{E} |_{M_{g}}= ⨁_{ζ∈S^{1}} \overline{E}_{ζ}

where the equivariant structure g^{E} of E acts on \overline{E}_{ζ} as multiplication by ζ. We often write \overline{E}_{g} for \overline{E}_{1} and call it the 0-degree part of \overline{E} |_{M_{g}}. As usual, A^{p,q}(M) stands for the space of (p, q)-forms Γ^{∞}(M, Λ^{p}T^{∗(1,0)}M ∧Λ^{q}T^{∗(0,1)}M), we define

Ã(M) = ⨁_{p=0}^{dimM}(A^{p,p}(M)/(Im∂ + Im\overline{∂})).

We denote by Ω^{\overline{E}^{ζ}} ∈ A^{1,1}(M_{g}) the curvature form associated to \overline{E}_{ζ}. Let (ϕ_{ζ})_{ζ∈S^{1}} be a family of GL(ℂ)-invariant formal power series such that ϕ_{ζ} ∈ ℂ[[gl_{rkE_{ζ}}(ℂ)]] where rkE_{ζ} stands for the rank of E_{ζ} which is a locally constant function on M_{g}. Moreover, let ϕ ∈ ℂ[[⨁_{ζ∈S^{1}} ℂ]] be any formal power series. We have the following definition.

Definition 2.1. The way to associate a smooth form to an equivariant hermitian vector bundle \overline{E} by setting

ϕ_{g}(\overline{E}) := ϕ((ϕ_{ζ}(−\frac{Ω^{\overline{E}^{ζ}}}{2πi}))_{ζ∈S^{1}})

is called an g-equivariant Chern-Weil theory associated to (ϕ_{ζ})_{ζ∈S^{1}} and ϕ. The class of ϕ_{g}(\overline{E}) in Ã(M_{g}) is independent of the metric.

Write dd^{c} for the differential operator \frac{\overline{∂}∂}{2πi}. The theory of equivariant secondary characteristic classes is described in the following theorem.

Theorem 2.2. To every short exact sequence \overline{ε} : 0 → \overline{E}′ → \overline{E} → \overline{E}′′ → 0 of equivariant hermitian vector bundles on M, there is a unique way to attach a class ϕ̃_{g}(\overline{ε}) ∈ Ã(M_{g}) which satisfies the following three conditions: (i). ϕ̃_{g}(\overline{ε}) satisfies the differential equation

dd^{c}ϕ̃_{g}(\overline{ε}) = ϕ_{g}(\overline{E}′ ⊕ \overline{E}′′) − ϕ_{g}(\overline{E});

(ii). for every equivariant holomorphic map f : M′ → M, ϕ̃_{g}(f^{∗}\overline{ε}) = f_{g}^{∗}ϕ̃_{g}(\overline{ε}); (iii). ϕ̃_{g}(\overline{ε}) = 0 if \overline{ε} is equivariantly and orthogonally split.

Proof. This is [21, Theorem 3.4].

We now give some examples of equivariant character forms and their corresponding secondary characteristic classes.

Example 2.3. (i). The equivariant Chern character form ch_{g}(\overline{E}) := ∑_{ζ∈S^{1}} ζch(\overline{E}_{ζ}).

(ii). The equivariant Todd form Td_{g}(\overline{E}) := \frac{c_{rkEg} (\overline{E}_{g})}{ch_{g}(∑_{j=0}^{rkE}(−1)^{j}∧^{j} \overline{E}^{∨})}. As in [18,

Thm. 10.1.1] one can show that

Td_{g}(\overline{E}) = Td(\overline{E}_{g}) ∏_{ζ≠1} det(\frac{1}{1 − ζ^{−1}e^{\frac{Ω^{\overline{E}^{ζ}}}{2πi}}} ).

(iii). Let \overline{ε} : 0 → \overline{E}′ → \overline{E} → \overline{E}′′ → 0 be a short exact sequence of hermitian vector bundles. The secondary Bott-Chern characteristic class is given by c̃h_{g}(\overline{ε}) = ∑_{ζ∈S^{1}} ζc̃h(\overline{ε}_{ζ}). (iv). If the equivariant structure g^{ε} has the eigenvalues ζ_{1}, ⋯, ζ_{m}, then the secondary Todd class is given by

T̃d_{g}(\overline{ε}) = ∑_{i=1}^{m} ∏_{j=1}^{i−1} Td_{g}(\overline{E}_{ζ_{j}}) · T̃d(\overline{ε}_{ζ_{i}}) · ∏_{j=i+1}^{m} Td_{g}(\overline{E}′_{ζ_{j}} + \overline{E}′′_{ζ_{j}}).

Remark 2.4. One can use Theorem 2.2 to give a proof of the statements (iii) and (iv) in the examples above.

Let E be an equivariant hermitian vector bundle with two different hermitian metrics h_{1} and h_{2}, we shall write ϕ̃_{g}(E, h_{1}, h_{2}) for the equivariant secondary characteristic class associated to the exact sequence

0 → (E, h_{1}) → (E, h_{2}) → 0 → 0

where the map from (E, h_{1}) to (E, h_{2}) is the identity map.

2.2 Equivariant analytic torsion forms

In [7], J.-M. Bismut and K. Köhler extended the Ray-Singer analytic torsion to the higher analytic torsion form T for a holomorphic submersion. The purpose of making such an extension is that the differential equation on dd^{c}T gives a refinement of the Grothendieck-Riemann-Roch theorem. Later, in his article [23], X. Ma generalized J.-M. Bismut and K. Köhler’s results to the equivariant case. In this subsection, we shall briefly recall Ma’s construction of the equivariant analytic torsion form. This construction is not very important for understanding the rest of this article, but the equivariant analytic torsion form itself will be used to define a reasonable push-forward morphism between equivariant arithmetic G_{0}-groups. We first fix some notations and assumptions. Let f : M → B be a proper holomorphic submersion of complex manifolds, and let T M, T B be the holomorphic tangent bundle on M, B. Denote by J^{T f} the complex structure on the real relative tangent bundle T_{ℝ}f, and assume that h^{T f} is a hermitian metric on T f which induces a Riemannian metric g^{T f}. Let T^{H}M be a vector subbundle of T M such that T M = T^{H}M ⊕T f, the following definition of Kähler fibration was given in [4, Def. 1.4].

Definition 2.5. The triple (f, h^{T f}, T^{H}M) is said to define a Kähler fibration if there exists a smooth real (1, 1)−form ω which satisfies the following three conditions: (i). ω is closed; (ii). T_{ℝ}^{H}M and T_{ℝ}f are orthogonal with respect to ω; (iii). if X, Y ∈ T_{ℝ}f, then ω(X, Y) = ⟨X, J^{T f}Y ⟩_{g^{T f}}.

It was shown in [4, Thm. 1.5 and 1.7] that for a given Kähler fibration, the form ω is unique up to addition of a form f^{∗}η where η is a real, closed (1, 1)- form on B. Moreover, for any real, closed (1, 1)-form ω on M such that the bilinear map X, Y ∈ T_{ℝ}f ↦ ω(J^{T f}X, Y) ∈ ℝ defines a Riemannian metric and hence a hermitian product h^{T f} on T f, we can define a Kähler fibration whose associated (1, 1)-form is ω. In particular, for a given f, a Kähler metric on M defines a Kähler fibration if we choose T^{H}M to be the orthogonal complement of T f in T M and ω to be the Kähler form associated to this metric. We now recall the Bismut superconnection of a Kähler fibration. Let (ξ, h^{ξ}) be a hermitian complex vector bundle on M. Let ∇^{T f}, ∇^{ξ} be the holomorphic hermitian connections on (T f, h^{T f}) and (ξ, h^{ξ}). Let ∇^{Λ(T^{∗(0,1)}f)} be the connection induced by ∇^{T f} on Λ(T^{∗(0,1)}f). Then we may define a connection on Λ(T^{∗(0,1)}f) ⊗ ξ by setting

∇^{Λ(T^{∗(0,1)}f)⊗ξ} = ∇^{Λ(T^{∗(0,1)}f)} ⊗ 1 + 1 ⊗ ∇^{ξ}.

Let E be the infinite-dimensional bundle on B whose fibre at each point b ∈ B consists of the C^{∞} sections of Λ(T^{∗(0,1)}f) ⊗ ξ |_{f^{−1}b}. This bundle E is a smooth ℤ-graded bundle. We define a connection ∇^{E} on E as follows. If U ∈ T_{ℝ}B, let U^{H} be the lift of U in T_{ℝ}^{H}M so that f_{∗}U^{H} = U. Then for every smooth section s of E over B, we set

∇_{U}^{E}s = ∇_{U^{H}}^{Λ(T^{∗(0,1)}f)⊗ξ}s.

For b ∈ B, let \overline{∂}^{Z_{b}} be the Dolbeault operator acting on E_{b}, and let \overline{∂}^{Z_{b}∗} be its formal adjoint with respect to the canonical hermitian product on E_{b} (cf. [23, 1.2]). Let C(T_{ℝ}f) be the Clifford algebra of (T_{ℝ}f, h^{T f}), then the bundle Λ(T^{∗(0,1)}f) ⊗ ξ has a natural C(T_{ℝ}f)-Clifford module structure. Actually, if U ∈ T f, let U′ ∈ T^{∗(0,1)}f correspond to U defined by U′(·) = h^{T f}(U, ·), then for U, V ∈ T f we set

c(U) = \sqrt{2}U′∧, c(\overline{V}) = −\sqrt{2}i\overline{V}

where i_{(·)} is the contraction operator (cf. [9, Definition 1.6]). Moreover, if U, V ∈ T_{ℝ}B, we set T (U^{H}, V^{H}) = −P^{T f}[U^{H}, V^{H}] where P^{T f} stands for the canonical projection from T M to T f.

Definition 2.6. Let e_{1}, …, e_{2m} be a basis of T_{ℝ}B, and let e^{1}, …, e^{2m} be the dual basis of T_{ℝ}^{∗}B. Then the element

c(T) = \frac{1}{2} ∑_{1≤α,β≤2m} e^{α} ∧ e^{β} ˆ⊗c(T (e_{α}^{H}, e_{β}^{H}))

is a section of (f^{∗}Λ(T_{ℝ}^{∗}B)ˆ⊗End(Λ(T^{∗(0,1)}f) ⊗ ξ))^{odd}.

Definition 2.7. For u > 0, the Bismut superconnection on E is the differential operator

B_{u} = ∇^{E} + \sqrt{u}(\overline{∂}^{Z} + \overline{∂}^{Z∗}) − \frac{1}{2\sqrt{2u}}c(T)

on f^{∗}(Λ(T_{ℝ}^{∗}B))ˆ⊗(Λ(T^{∗(0,1)}f) ⊗ ξ).

Definition 2.8. Let N_{V} be the number operator on Λ(T^{∗(0,1)}f)⊗ξ and on E, namely N_{V} acts as multiplication by p on Λ^{p}(T^{∗(0,1)}f) ⊗ ξ. For U, V ∈ T_{ℝ}B, set ω^{H\overline{H}}(U, V) = ω^{M}(U^{H}, V^{H}) where ω^{M} is the closed form in the definition of Kähler fibration. Furthermore, for u > 0, set N_{u} = N_{V} + \frac{iω^{H\overline{H}}}{u}.

We now turn to the equivariant case. Let G be a compact Lie group, we shall assume that all complex manifolds, hermitian vector bundles and holomorphic morphisms considered above are G−equivariant and all metrics are G-invariant. We will additionally assume that the direct images R^{k}f_{∗}ξ are all locally free so that the G-equivariant coherent sheaf R^{·}f_{∗}ξ is locally free and hence a Gequivariant vector bundle over B. [23, 1.2] gives a G-invariant hermitian metric (the L^{2}-metric) h^{R^{·}f_{∗}ξ} on the vector bundle R^{·}f_{∗}ξ. For g ∈ G, let M_{g} = {x ∈ M | g·x = x} and B_{g} = {b ∈ B | g·b = b} be the fixed point submanifolds, then f induces a holomorphic submersion f_{g} : M_{g} → B_{g}. Let Φ be the homomorphism α ↦ (2iπ)^{−degα/2} of Λ^{even}(T_{ℝ}^{∗}B) into itself. We put

ch_{g}(R^{·}f_{∗}ξ, h^{R^{·}f_{∗}ξ}) = ∑_{k=0}^{dimM−dimB}(−1)^{k}ch_{g}(R^{k}f_{∗}ξ, h^{R^{k}f_{∗}ξ})

and

ch′_{g}(R^{·}f_{∗}ξ, h^{R^{·}f_{∗}ξ}) = ∑_{k=0}^{dimM−dimB}(−1)^{k}kch_{g}(R^{k}f_{∗}ξ, h^{R^{k}f_{∗}ξ}).

Definition 2.9. For s ∈ ℂ with Re(s) > 1, let

ζ_{1}(s) = −\frac{1}{Γ(s)} ∫_{0}^{1} u^{s−1}(ΦTr_{s}[gN_{u}exp(−B_{u}^{2})] − ch′_{g}(R^{·}f_{∗}ξ, h^{R^{·}f_{∗}ξ}))du

and similarly for s ∈ ℂ with Re(s) < \frac{1}{2}, let

ζ_{2}(s) = −\frac{1}{Γ(s)} ∫_{1}^{∞} u^{s−1}(ΦTr_{s}[gN_{u}exp(−B_{u}^{2})] − ch′_{g}(R^{·}f_{∗}ξ, h^{R^{·}f_{∗}ξ}))du.

X. Ma has proved that ζ_{1}(s) extends to a holomorphic function of s ∈ ℂ near s = 0 and ζ_{2}(s) is a holomorphic function of s.

Definition 2.10. The smooth form T_{g}(ω^{M}, h^{ξ}) := \frac{∂}{∂s}(ζ_{1} + ζ_{2})(0) on B_{g} is called the equivariant analytic torsion form.

Theorem 2.11. The form T_{g}(ω^{M}, h^{ξ}) lies in ⨁_{p≥0} A^{p,p}(B_{g}) and satisfies the following differential equation

dd^{c}T_{g}(ω^{M}, h^{ξ}) = ch_{g}(R^{·}f_{∗}ξ, h^{R^{·}f_{∗}ξ}) − ∫_{M_{g}/B_{g}} Td_{g}(T f, h^{T f})ch_{g}(ξ, h^{ξ}).

Here A^{p,p}(B_{g}) stands for the space of smooth forms on B_{g} of type (p, p).

Proof. This is [23, Theorem 2.12].

We define a secondary characteristic class

c̃h_{g}(R^{·}f_{∗}ξ, h^{R^{·}f_{∗}ξ}, h^{′R^{·}f_{∗}ξ}) := ∑_{k=0}^{dimM−dimB}(−1)^{k}c̃h_{g}(R^{k}f_{∗}ξ, h^{R^{k}f_{∗}ξ}, h^{′R^{k}f_{∗}ξ})

such that it satisfies the following differential equation

dd^{c}c̃h_{g}(R^{·}f_{∗}ξ, h^{R^{·}f_{∗}ξ}, h^{′R^{·}f_{∗}ξ}) = ch_{g}(R^{·}f_{∗}ξ, h^{R^{·}f_{∗}ξ}) − ch_{g}(R^{·}f_{∗}ξ, h^{′R^{·}f_{∗}ξ}),

then the anomaly formula can be described as follows.

Theorem 2.12. (Anomaly formula) Let ω′ be the form associated to another Kähler fibration for f : M → B. Let h^{′T f} be the metric on T f in this new fibration and let h^{′ξ} be another metric on ξ. The following identity holds in Ã(B_{g}) := ⨁_{p≥0}(A^{p,p}(B_{g})/(Im∂ + Im\overline{∂})):

T_{g}(ω^{M}, h^{ξ}) − T_{g}(ω^{′M}, h^{′ξ}) =c̃h_{g}(R^{·}f_{∗}ξ, h^{R^{·}f_{∗}ξ}, h^{′R^{·}f_{∗}ξ})
− ∫_{M_{g}/B_{g}}[T̃d_{g}(T f, h^{T f}, h^{′T f})ch_{g}(ξ, h^{ξ})
+ Td_{g}(T f, h^{′T f})c̃h_{g}(ξ, h^{ξ}, h^{′ξ})].

In particular, the class of T_{g}(ω^{M}, h^{ξ}) in Ã(B_{g}) only depends on (h^{T f}, h^{ξ}).

Proof. This is [23, Theorem 2.13].

2.3 Equivariant Bott-Chern singular currents

The Bott-Chern singular current was defined by J.-M. Bismut, H. Gillet and C. Soulé in [5] in order to generalize the usual Bott-Chern secondary characteristic class to the case where one considers the resolutions of hermitian vector bundles associated to the closed immersions of complex manifolds. In [2], J.-M. Bismut generalized this topic to the equivariant case. We shall recall Bismut’s construction of the equivariant Bott-Chern singular current in this subsection. Similar to the equivariant analytic torsion form, the construction itself is not very important for understanding our later arguments, we just recall it for the convenience of the reader. Bismut’s construction was realized via some current valued zeta function which involves the supertraces of Quillen’s superconnections. This is similar to the non-equivariant case. As before, let g be the automorphism corresponding to an element in a compact Lie group G. Let i : Y → X be an equivariant closed immersion of G-equivariant Kähler manifolds, and let \overline{η} be an equivariant hermitian vector bundle on Y. Assume that \overline{ξ}. is a complex (of homological type) of equivariant hermitian vector bundles on X which provides a resolution of i_{∗}\overline{η}. We denote the differential of the complex ξ. by v. Note that ξ. is acyclic outside Y and the homology sheaves of its restriction to Y are locally free and hence they are all vector bundles. We write H_{n} = H_{n}(ξ. |_{Y}) and define a ℤ-graded bundle H = ⨁_{n} H_{n}. For each y ∈ Y and u ∈ T X_{y}, we denote by ∂_{u}v(y) the derivative of v at y in the direction u in any given holomorphic trivialization of ξ. near y. Then the map ∂_{u}v(y) acts on H_{y} as a chain map, and this action only depends on the image z of u in N_{y} where N stands for the normal bundle of i(Y) in X. So we get a chain complex of holomorphic vector bundles (H, ∂_{z}v).

Let π be the projection from the normal bundle N to Y, then we have a canonical identification of ℤ-graded chain complexes

(π^{∗}H, ∂_{z}v) ≅ (π^{∗}(∧N^{∨} ⊗ η), i_{z}).

For this, one can see [3, Section I. b]. Moreover, such an identification is an identification of G-bundles which induces a family of canonical isomorphisms γ_{n} : H_{n} ≅ ∧^{n}N^{∨} ⊗ η. Another way to describe these canonical isomorphisms γ_{n} is applying [13, Exp. VII, Lemma 2.4 and Proposition 2.5]. These two constructions coincide because they are both locally, on a suitable open covering {U_{j}}_{j∈J}, determined by any complex morphism over the identity map of η |_{U_{j}} from (ξ. |_{U_{j}}, v) to the minimal resolution of η |_{U_{j}} (e.g. the Koszul resolution). The advantage of using the construction given in [13] is that it remains valid for arithmetic varieties over any base instead of the complex numbers. Later in [2], for the use of normalization, J.-M. Bismut considered the automorphism of N^{∨} defined by multiplying a constant −\sqrt{−1}, it induces an isomorphism of chain complexes

(π^{∗}(∧N^{∨} ⊗ η), i_{z}) ≅ (π^{∗}(∧N^{∨} ⊗ η), \sqrt{−1}i_{z})

and hence

(π^{∗}H, ∂_{z}v) ≅ (π^{∗}(∧N^{∨} ⊗ η), \sqrt{−1}i_{z}).

This identification induces a family of isomorphisms γ̃_{n} : H_{n} ≅ ∧^{n}N^{∨} ⊗ η. By finite dimensional Hodge theory, for each y ∈ Y, there is a canonical isomorphism

H_{y} ≅ {f ∈ ξ._{y} | vf = 0, v^{∗}f = 0}

where v^{∗} is the dual of v with respect to the metrics on ξ.. This means that H can be regarded as a smooth ℤ-graded G-equivariant subbundle of ξ so that it carries an induced G-invariant metric. On the other hand, we endow ∧N^{∨} ⊗ η with the metric induced from \overline{N} and \overline{η}. J.-M. Bismut introduced the following definition.

Definition 2.13. We say that the metrics on the complex of equivariant hermitian vector bundles \overline{ξ}. satisfy Bismut assumption (A) if the identification (π^{∗}H, ∂_{z}v) ≅ (π^{∗}(∧N^{∨} ⊗ η), \sqrt{−1}i_{z}) also identifies the metrics, it is equivalent to the condition that the identification (π^{∗}H, ∂_{z}v) ≅ (π^{∗}(∧N^{∨} ⊗ η), i_{z}) identifies the metrics.

Proposition 2.14. There always exist G-invariant metrics on ξ. which satisfy Bismut assumption (A) with respect to the equivariant hermitian vector bundles \overline{N} and \overline{η}.

Proof. This is [2, Proposition 3.5].

From now on we always suppose that the metrics on a resolution satisfy Bismut assumption (A). Let ∇^{ξ} be the canonical hermitian holomorphic connection on ξ., then for each u > 0, we may define a G-invariant superconnection

C_{u} := ∇^{ξ} + \sqrt{u}(v + v^{∗})

on the ℤ_{2}-graded vector bundle ξ. Moreover, let Φ be the map α ∈ ∧(T_{ℝ}^{∗}X_{g}) → (2πi)^{−degα/2}α ∈ ∧(T_{ℝ}^{∗}X_{g}) and denote

(Td_{g}^{−1})′(\overline{N}) := \frac{∂}{∂b} |_{b=0} (Td_{g}(b · Id − \frac{Ω\overline{N}}{2πi})^{−1})

where Ω\overline{N} is the curvature form associated to \overline{N}. We recall as follows the construction of the equivariant singular current given in [2, Section VI].

Lemma 2.15. Let N_{H} be the number operator on the complex ξ. i.e. it acts on ξ_{j} as multiplication by j, then for s ∈ ℂ and 0 < Re(s) < \frac{1}{2}, the current valued zeta function

Z_{g}(\overline{ξ}.)(s) := \frac{1}{Γ(s)} ∫_{0}^{∞} u^{s−1}[ΦTr_{s}(N_{H}gexp(−C_{u}^{2})) + (Td_{g}^{−1})′(\overline{N})ch_{g}(\overline{η})δ_{Y_{g}}]du

is well-defined on X_{g} and it has a meromorphic continuation to the complex plane which is holomorphic at s = 0.

Definition 2.16. The equivariant singular Bott-Chern current on X_{g} associated to the resolution \overline{ξ}. is defined as

T_{g}(\overline{ξ}.) := \frac{∂}{∂s} |_{s=0} Z_{g}(\overline{ξ}.)(s).

Theorem 2.17. The current T_{g}(\overline{ξ}.) is a sum of (p, p)-currents and it satisfies the differential equation

dd^{c}T_{g}(\overline{ξ}.) = i_{g}_{∗}ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N}) − ∑_{k}(−1)^{k}ch_{g}(\overline{ξ}_{k}).

Moreover, the wave front set of T_{g}(\overline{ξ}.) is contained in N_{g,ℝ}^{∨} where N_{g,ℝ}^{∨} stands for the underlying real bundle of the dual of N_{g}.

Proof. This follows from [2, Theorem 6.7, Remark 6.8].

Finally, we recall a theorem concerning the relationship of equivariant Bott-Chern singular currents involved in a double complex. This theorem will be used to show that our definition of a general embedding morphism in equivariant arithmetic G_{0}-theory is reasonable.

Theorem 2.18. Let

\overline{χ} : 0 → \overline{η}_{n} → ⋯ → \overline{η}_{1} → \overline{η}_{0} → 0

be an exact sequence of equivariant hermitian vector bundles on Y. Assume that we have the following double complex consisting of resolutions of i_{∗}\overline{χ} such that all rows are exact sequences.

\xymatrix{0 \ar[r] & \overline{ξ}_{n,·} \ar[r] \ar[d] & ⋯ \ar[r] & \overline{ξ}_{1,·} \ar[r] \ar[d] & \overline{ξ}_{0,·} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & i_{∗}\overline{η}_{n} \ar[r] & ⋯ \ar[r] & i_{∗}\overline{η}_{1} \ar[r] & i_{∗}\overline{η}_{0} \ar[r] & 0.}

For each k, we write \overline{ε}_{k} for the exact sequence

0 → \overline{ξ}_{n,k} → ⋯ → \overline{ξ}_{1,k} → \overline{ξ}_{0,k} → 0.

Then we have the following equality in Ũ(X_{g}) := ⨁_{p≥0}(D^{p,p}(X_{g})/(Im∂+Im\overline{∂}))

∑_{j=0}^{n}(−1)^{j}T_{g}(\overline{ξ}_{j,·}) = i_{g}_{∗} \frac{c̃h_{g}(\overline{χ})}{Td_{g}(\overline{N})} − ∑_{k}(−1)^{k}c̃h_{g}(\overline{ε}_{k}).

Here D^{p,p}(X_{g}) stands for the space of currents on X_{g} of type (p, p).

Proof. This is [21, Theorem 3.14].

2.4 Bismut-Ma’s immersion formula

In this subsection, we shall recall Bismut-Ma’s immersion formula which reflects the behaviour of the equivariant analytic torsion forms of a Kähler fibration under composition of an immersion and a submersion. By translating to the equivariant arithmetic G_{0}-theoretic language, such a formula can be used to measure, in arithmetic G_{0}-theory, the difference between a push-forward morphism and the composition formed as an embedding morphism followed by a push-forward morphism. Although Bismut-Ma’s immersion formula plays a very important role in our arguments, we shall not recall its proof since it is rather long and technical. Let i : Y → X be an equivariant closed immersion of G-equivariant Kähler manifolds. Let S be a complex manifold with the trivial G-action, and let f : Y → S, l : X → S be two equivariant holomorphic submersions such that f = l ∘ i. Assume that \overline{η} is an equivariant hermitian vector bundle on Y and \overline{ξ}. provides a resolution of i_{∗}\overline{η} on X whose metrics satisfy Bismut assumption (A). Let ω^{Y}, ω^{X} be the real, closed and G-invariant (1, 1)-forms on Y, X which induce the Kähler fibrations with respect to f and l respectively. We additionally assume that ω^{Y} is the pull-back of ω^{X} so that the Kähler metric on Y is induced by the Kähler metric on X. As before, denote by N the normal bundle of i(Y) in X. Consider the following exact sequence

\overline{N} : 0 → \overline{T f} → \overline{T l} |_{Y} → \overline{N} → 0

where N is endowed with the quotient metric, we shall write T̃d_{g}(\overline{T f}, \overline{T l} |_{Y}) for T̃d_{g}(\overline{N}) the equivariant Bott-Chern secondary characteristic class associated to \overline{N}. It satisfies the following differential equation

dd^{c}T̃d_{g}(\overline{T f}, \overline{T l} |_{Y}) = Td_{g}(T f, h^{T f})Td_{g}(\overline{N}) − Td_{g}(T l |_{Y} , h^{T l}).

For simplicity, we shall suppose that in the resolution ξ., ξ_{j} are all l−acyclic and moreover η is f−acyclic. By an easy argument of long exact sequence, we have the following exact sequence

Ξ : 0 → l_{∗}(ξ_{m}) → l_{∗}(ξ_{m−1}) → … → l_{∗}(ξ_{0}) → f_{∗}η → 0.

By the semi-continuity theorem, all the elements in the exact sequence above are vector bundles. In this case, we recall the definition of the L^{2}-metrics on direct images precisely as follows. We just take f_{∗}h^{η} as an example. Note that the semi-continuity theorem implies that the natural map

(R^{0}f_{∗}η)_{s} → H^{0}(Y_{s}, η |_{Y_{s}})

is an isomorphism for every point s ∈ S where Y_{s} stands for the fibre over s. We may endow H^{0}(Y_{s}, η |_{Y_{s}}) with a L^{2}-metric given by the formula

< u, v >_{L^{2}}:= \frac{1}{(2π)^{d_{s}}} ∫_{Y_{s}} h^{η}(u, v)\frac{ω^{Y} ^{d_{s}}}{d_{s}!}

where d_{s} is the complex dimension of the fibre Y_{s}. It can be shown that these metrics depend on s in a C^{∞} manner (cf. [9, p.278]) and hence define a hermitian metric on f_{∗}η. We shall denote it by f_{∗}h^{η}. In order to understand the statement of Bismut-Ma’s immersion formula, we still have to recall an important concept defined by J.-M. Bismut, the equivariant R-genus. Let W be a G-equivariant complex manifold, and let \overline{E} be an equivariant hermitian vector bundle on W. For ζ ∈ S^{1} and s > 1 consider the zeta function

L(ζ, s) = ∑_{k=1}^{∞} \frac{ζ^{k}}{k^{s}}

and its meromorphic continuation to the whole complex plane. Define the formal power series in x

R̃(ζ, x) := ∑_{n=0}^{∞} (\frac{∂L}{∂s}(ζ, −n) + L(ζ, −n) ∑_{j=1}^{n} \frac{1}{2j} )\frac{x^{n}}{n!} .

Definition 2.19. The Bismut equivariant R-genus of an equivariant hermitian vector bundle \overline{E} with \overline{E} |_{X_{g}}= ∑_{ζ} \overline{E}_{ζ} is defined as

R_{g}(\overline{E}) := ∑_{ζ∈S^{1}} (TrR̃(ζ, −\frac{Ω^{\overline{E}^{ζ}}}{2πi}) − TrR̃(1/ζ, \frac{Ω^{\overline{E}^{ζ}}}{2πi}))

where Ω^{\overline{E}^{ζ}} is the curvature form associated to \overline{E}_{ζ}. Actually, the class of R_{g}(\overline{E}) in Ã(X_{g}) is independent of the metric and we just write R_{g}(E) for it. Furthermore, the class R_{g}(·) is additive.

Theorem 2.20. (Immersion formula) Let notations and assumptions be as above. Then the equality

∑_{i=0}^{m}(−1)^{i}T_{g}(ω^{X}, h^{ξ_{i}}) − T_{g}(ω^{Y} , h^{η}) + c̃h_{g}(Ξ, h^{L^{2}})

= ∫_{X_{g}/S} Td_{g}(T l, h^{T l})T_{g}(\overline{ξ}.) + ∫_{Y_{g}/S} \frac{T̃d_{g}(\overline{T f}, \overline{T l} |_{Y})}{Td_{g}(\overline{N})}ch_{g}(\overline{η})

+ ∫_{X_{g}/S} Td_{g}(T l)R_{g}(T l) ∑_{i=0}^{m}(−1)^{i}ch_{g}(ξ_{i}) − ∫_{Y_{g}/S} Td_{g}(T f)R_{g}(T f)ch_{g}(η)

holds in Ã(S).

Proof. This is the combination of [8, Theorem 0.1 and 0.2], the main theorems in that paper.

3 A vanishing theorem for equivariant closed immersions

3.1 The statement

By a projective manifold we shall understand a compact complex manifold which is projective algebraic, that means a projective manifold is the complex analytic space X(ℂ) associated to a smooth projective variety X over ℂ (cf. [17, Appendix B]). Let μ_{n} be the diagonalisable group variety over ℂ associated to ℤ/nℤ. We say X is μ_{n}-equivariant if it admits a μ_{n}-projective action, this means the associated projective manifold X(ℂ) admits an action by the group of complex n-th roots of unity. Denote by X_{μ_{n}} the fixed point subscheme of X, by GAGA principle, X_{μ_{n}}(ℂ) is equal to X(ℂ)_{g} where g is the automorphism on X(ℂ) corresponding to a fixed primitive n-th root of unity. If no confusion arises, we shall not distinguish between X and X(ℂ) as well as X_{μ_{n}} and X_{g}. Since the classical arguments of locally free resolutions may not be compatible with the equivariant setting, we summarize some crucial facts we need as follows. (i). Every equivariant coherent sheaf on an equivariantly projective scheme is an equivariant quotient of an equivariant locally free coherent sheaf. (ii). Every equivariant coherent sheaf on an equivariantly projective scheme admits an equivariant locally free resolution. It is finite if the equivariant scheme is regular. (iii). An exact sequence of equivariant coherent sheaves on an equivariantly projective scheme admits an exact sequence of equivariant locally free resolutions. (iv). Any two equivariant locally free resolutions of an equivariant coherent sheaf on an equivariantly projective scheme can be dominated by a third one. Now let i : Y → X be a μ_{n}-equivariant closed immersion of projective manifolds with normal bundle N. Let S be a projective manifold with the trivial μ_{n}- action and let h : X → S be an equivariant holomorphic submersion whose restriction f : Y → S is an equivariant holomorphic submersion. According to our assumptions, we may define a Kähler fibration with respect to h by choosing a μ_{n}(ℂ)-invariant Kähler form ω^{X} on X. By restricting ω^{X} to Y we obtain a Kähler fibration with respect to f. The same thing goes to h_{g} : X_{g} → S and f_{g} : Y_{g} → S. Let \overline{η} be an equivariant hermitian holomorphic vector bundle on Y, assume that (\overline{ξ.}, v) is a complex of equivariant hermitian vector bundles on X which provides a resolution of i_{∗}\overline{η}, whose metrics satisfy Bismut assumption (A). Write N_{g} for the 0-degree part of N |_{Y_{g}} which is isomorphic to the normal bundle of i_{g}(Y_{g}) in X_{g} and denote by F the orthogonal complement of N_{g}. According to [13, Exp. VII, Lemma 2.4 and Proposition 2.5] we know that there exists a canonical isomorphism from the homology sheaf H(ξ. |_{X_{g}}) to i_{g}_{∗}(∧^{·}F^{∨} ⊗η |_{Y_{g}}) which is equivariant. Then the restriction of (ξ., v) to X_{g} can always split into a series of short exact sequences in the following way:

(∗) : 0 → Im → Ker → i_{g}_{∗}(∧^{·}F^{∨} ⊗ η |_{Y_{g}}) → 0

and

(∗∗) : 0 → Ker → ξ. |_{X_{g}}→ Im → 0.

Suppose that ∧^{·}F^{∨} ⊗ η |_{Y_{g}} and ξ. |_{X_{g}} are all acyclic (higher direct images vanish). Then according to an easy argument of long exact sequence, these short exact sequences (∗) and (∗∗) induce a series of short exact sequences of direct images:

H(∗) : 0 → R^{0}h_{g}_{∗}(Im) → R^{0}h_{g}_{∗}(Ker) → R^{0}f_{g}_{∗}(∧^{·}F^{∨} ⊗ η |_{Y_{g}}) → 0

and

H(∗∗) : 0 → R^{0}h_{g}_{∗}(Ker) → R^{0}h_{g}_{∗}(ξ. |_{X_{g}}) → R^{0}h_{g}_{∗}(Im) → 0.

By semi-continuity theorem, all elements in the exact sequences above are vector bundles. We endow R^{0}h_{g}_{∗}(ξ. |_{X_{g}}) and R^{0}f_{g}_{∗}(∧^{·}F^{∨} ⊗ η |_{Y_{g}}) with the L^{2}-metrics which are induced by the metrics on ξ., η and F. Here the normal bundle N admits the quotient metric induced from the exact sequence

0 → T f → T h |_{Y} → N → 0

and the bundle F admits the metric induced by the metric on N. Moreover, we endow R^{0}h_{g}_{∗}(Im) and R^{0}h_{g}_{∗}(Ker) with the metrics induced by the L^{2}- metrics of R^{0}h_{g}_{∗}(ξ. |_{X_{g}}) so that H(∗) and H(∗∗) become short exact sequences of equivariant hermitian vector bundles. Denote by c̃h_{g}(\overline{ξ.}, \overline{η}) the alternating sum of the equivariant secondary Bott-Chern characteristic classes of H(∗) and H(∗∗) such that it satisfies the following differential equation

dd^{c}c̃h_{g}(\overline{ξ.}, \overline{η}) = ∑_{j}(−1)^{j}ch_{g}(R^{0}f_{g}_{∗}(∧^{j}\overline{F}^{∨} ⊗ \overline{η} |_{Y_{g}}))

− ∑_{j}(−1)^{j}ch_{g}(R^{0}h_{g}_{∗}(\overline{ξ}_{j} |_{X_{g}})).

Now the difference

δ(i, \overline{η}, \overline{ξ}.) :=c̃h_{g}(\overline{ξ.}, \overline{η}) − ∑_{k}(−1)^{k}T_{g}(ω^{Y_{g}}, h^{∧^{k}F^{∨}⊗η|_{Yg}})

+ ∑_{k}(−1)^{k}T_{g}(ω^{X_{g}}, h^{ξ_{k}|_{Xg}}) − ∫_{X_{g}/S} T_{g}(\overline{ξ.})Td(\overline{T h_{g}})

− ∫_{Y_{g}/S} Td(T f_{g})Td_{g}^{−1}(F)ch_{g}(η)R(N_{g})

− ∫_{Y_{g}/S} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})T̃d(\overline{T f_{g}}, \overline{T h_{g}} |_{Y_{g}})

makes sense and it is an element in ⨁_{p≥0} A^{p,p}(S)/(Im∂ + Im\overline{∂}). Here the symbols T_{g}(·) in the summations stand for analytic torsion forms introduced in Section 2.1, the symbol T_{g}(\overline{ξ.}) in the integral is the equivariant Bott-Chern singular current introduced in Section 2.2. The vanishing theorem for equivariant closed immersions can be formulated as the following.

Theorem 3.1. Let i : Y → X be an equivariant closed immersion of projective manifolds, and let S be a projective manifold with the trivial μ_{n}-action. Assume that we are given two equivariant holomorphic submersions f : Y → S and h : X → S such that f = h ∘ i. Then X admits an equivariant hermitian very ample invertible sheaf \overline{L} relative to the morphism h, and for any equivariant hermitian resolution 0 → \overline{ξ}_{m} → ⋯ \overline{ξ}_{1} → \overline{ξ}_{0} → i_{∗}\overline{η} → 0 we have

δ(i, \overline{η} ⊗ i^{∗}\overline{L}^{⊗n}, \overline{ξ.} ⊗ \overline{L}^{⊗n}) = 0 for n ≫ 0.

Here the metrics on the resolution are supposed to satisfy Bismut assumption (A).

3.2 Deformation to the normal cone

To prove the vanishing theorem for closed immersions, we use a geometric construction called the deformation to the normal cone which allows us to deform a resolution of hermitian vector bundle associated to a closed immersion of projective manifolds to a simpler one. The δ-difference of this new simpler resolution is much easier to compute. Let i : Y ↪ X be a closed immersion of projective manifolds with normal bundle N_{X/Y}. For a vector bundle E on X or Y, the notation ℙ(E) will stand for the projective space bundle Proj(Sym(E^{∨})).

Definition 3.2. The deformation to the normal cone W (i) of the immersion i is the blowing up of X × ℙ^{1} along Y × {∞}. We shall just write W for W (i) if there is no confusion about the immersion.

There are too many geometric objects and morphisms appearing in the construction of the deformation to the normal cone, we have to fix various notations in a clear way. We denote by p_{X} (resp. p_{Y}) the projection X × ℙ^{1} → X (resp. Y × ℙ^{1} → Y) and by π the blow-down map W → X × ℙ^{1}. We also denote by q_{X} (resp. q_{Y}) the projection X × ℙ^{1} → ℙ^{1} (resp. Y × ℙ^{1} → ℙ^{1}) and by q_{W} the composition q_{X} ∘ π. It is well known that the map q_{W} is flat and for t ∈ ℙ^{1}, we have

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

where X̃ is isomorphic to the blowing up of X along Y and P is isomorphic to the projective completion of N_{X/Y} i.e. the projective space bundle ℙ(N_{X/Y} ⊕ O_{Y}). Denote the canonical projection from ℙ(N_{X/Y} ⊕ O_{Y}) to Y by π_{P}, then the morphism O_{Y} → N_{X/Y} ⊕ O_{Y} induces a canonical section i_{∞} : Y ↪ ℙ(N_{X/Y} ⊕ O_{Y}) which is called the zero section embedding. Moreover, let j : Y × ℙ^{1} → W be the canonical closed immersion induced by i × Id, then the component X̃ doesn’t meet j(Y × ℙ^{1}) and the intersection of j(Y × ℙ^{1}) with P is exactly the image of Y under the section i_{∞}. The advantage of the construction of the deformation to the normal cone is that we may control the rational equivalence class of the fibres q_{W}^{−1}(t). More precisely, in the language of line bundles, we have the isomorphisms O(X) ≅ O(P + X̃) ≅ O(P) ⊗ O(X̃) which is an immediate consequence of the isomorphism O(0) ≅ O(∞) on ℙ^{1}. On P = ℙ(N_{X/Y} ⊕ O_{Y}), there exists a tautological exact sequence

0 → O(−1) → π_{P}^{∗} (N_{X/Y} ⊕ O_{Y}) → Q → 0

where Q is the tautological quotient bundle. This exact sequence and the inclusion O_{P} → π_{P}^{∗} (N_{X/Y} ⊕ O_{Y}) induce a section σ : O_{P} → Q which vanishes along the zero section i_{∞}(Y). By duality we get a morphism Q^{∨} → O_{P}, and this morphism induces the following exact sequence

0 → ∧^{n}Q^{∨} → ⋯ → ∧^{2}Q^{∨} → Q^{∨} → O_{P} → i_{∞}_{∗}O_{Y} → 0

where n is the rank of Q. Note that i_{∞} is a section of π_{P} i.e. π_{P} ∘ i_{∞} = Id, the projection formula implies the following definition.

Definition 3.3. For any vector bundle η on Y, the following complex of vector bundles

0 → ∧^{n}Q^{∨} ⊗ π_{P}^{∗} η → ⋯ → ∧^{2}Q^{∨} ⊗ π_{P}^{∗} η → Q^{∨} ⊗ π_{P}^{∗} η → π_{P}^{∗} η → 0

provides a resolution of i_{∞}_{∗}η on P. This complex is called the Koszul resolution of i_{∞}_{∗}η and will be denoted by κ(η, N_{X/Y}). If the normal bundle N_{X/Y} admits some hermitian metric, then the tautological exact sequence induces a hermitian metric on Q. If, moreover, the bundle η also admits a hermitian metric, then the Koszul resolution is a complex of hermitian vector bundles and will be denoted by \overline{κ}(\overline{η}, \overline{N}_{X/Y}).

Now, assume that X is a μ_{n}-equivariant projective manifold and E is an equivariant locally free sheaf on X. Then according to [20, (1.4) and (1.5)], ℙ(E) admits a canonical μ_{n}-equivariant structure such that the projection map ℙ(E) → X is equivariant and the canonical bundle O(1) admits an equivariant structure. Moreover, let Y → X be an equivariant closed immersion of projective manifolds, according to [20, (1.6)] the action of μ_{n} on X can be extended to the blowing up Bl_{Y} X such that the blow-down map is equivariant and the canonical bundle O(1) admits an equivariant structure. So by endowing ℙ^{1} with the trivial μ_{n}-action, the construction of the deformation to the normal cone described above is compatible with the equivariant setting. For the use of our later arguments, the Kähler metric chosen on W should be well controlled on the fibres of the deformation. For this purpose, it is necessary to introduce the following definition.

Definition 3.4. (Rössler) A metric h on W is said to be normal to the deformation if (a). it is invariant and Kähler; (b). the restriction h |_{j_{g}_{∗}(Y_{g}×ℙ^{1})} is a product h′ × h′′, where h′ is a Kähler metric on Y_{g} and h′′ is a Kähler metric on ℙ^{1}; (c). the intersections of X × {0} with j_{∗}(Y × ℙ^{1}) and of P with j_{∗}(Y × ℙ^{1}) are orthogonal at the fixed points.

Lemma 3.5. For any μ_{n}-invariant Kähler metric h^{X} on X which induces an invariant Kähler metric h^{Y} on Y, there exists a metric h^{W} on W which is normal to the deformation and the restriction of h^{W} to X ≅ X × {0} (resp. Y ≅ Y × {∞}) is exactly h^{X} (resp. h^{Y}). Moreover, we may require that the hermitian normal bundles \overline{N}_{Y ×ℙ^{1}/Y ×\{0\}} and \overline{N}_{Y ×ℙ^{1}/Y ×\{∞\}} are both isometric to the trivial bundles with trivial metrics.

Proof. The existence of the metric which is normal to the deformation is the content of [21, Lemma 6.13] and [28, Lemma 6.14], such a metric is constructed via the Grassmannian graph construction. Roughly speaking, according to another description of the deformation to the normal cone via the Grassmannian graph construction, we have an embedding W → X × ℙ^{r} × ℙ^{1} and the metric h^{W} is the μ_{n}-average of the restriction of a product metric on X × ℙ^{r} × ℙ^{1} (cf. [28, Lemma 6.14]). When we endow X in the product with the metric h^{X}, the requirements on restrictions are automatically satisfied since h^{X} is μ_{n}-invariant. To fulfill the requirements on hermitian normal bundles, we may just choose the Fubini-Study metric on ℙ^{1}.

We summarize some very important results about the application of the deformation to the normal cone as follows. Their proofs can be found in [21, Section 2 and 6.2].

Theorem 3.6. Let i : Y → X be an equivariant closed immersion of equivariant projective manifolds, and let W = W (i) be the deformation to the normal cone of i. Assume that \overline{η} is an equivariant hermitian vector bundle on Y. Then (i). there exists an equivariant hermitian resolution of j_{∗}p_{Y}^{∗} (\overline{η}) on W, whose metrics satisfy Bismut assumption (A) and whose restriction to X̃ is equivariantly and orthogonally split; (ii). the natural morphism from the deformation to the normal cone W (i_{g}) to the fixed point submanifold W (i)_{g} is a closed immersion, this closed immersion induces the closed immersions ℙ(N_{X_{g}/Y_{g}} ⊕ O_{Y_{g}}) → ℙ(N_{X/Y} ⊕ O_{Y})_{g} and X̃_{g} → X̃_{g}; (iii). the fixed point submanifold of ℙ(N_{X/Y} ⊕ O_{Y}) is ℙ(N_{X_{g}/Y_{g}} ⊕ O_{Y_{g}}) ∐_{ζ≠1} ℙ((N_{X/Y})_{ζ}); (iv). the closed immersion i_{∞,g} factors through ℙ(N_{X_{g}/Y_{g}} ⊕O_{Y_{g}}) and the other components ℙ((N_{X/Y})_{ζ}) don’t meet Y. Hence the complex κ(O_{Y} , N_{X/Y})_{g}, obtained by taking the 0-degree part of the Koszul resolution, provides a resolution of O_{Y_{g}} on ℙ(N_{X/Y} ⊕ O_{Y})_{g}.

3.3 Proof of the vanishing theorem

We shall first prove the first part of the vanishing theorem for closed immersions i.e. the existence of an equivariant hermitian very ample invertible sheaf on X which is relative to the morphism h : X → S. Generally speaking, such an invertible sheaf can be constructed rather easily since X admits a μ_{n}-projective action and the μ_{n}-action on S is supposed to be trivial, but for the whole proof of the vanishing theorem we would like to construct a special one which is the pull-back of some equivariant hermitian very ample invertible sheaf on W (i) under the identification X ≅ X × {0}. Our starting point is the following.

Definition 3.7. Let M be a μ_{n}-projective manifold, and let ℙ_{M}^{n} be some relative projective space over M. A μ_{n}-action on ℙ_{M}^{n} arising from some μ_{n}- action on the free sheaf O_{M}^{⊕n+1} via the functorial properties of the Proj symbol will be called a global μ_{n}-action.

The advantage of considering global μ_{n}-action is that on a projective space which admits a global μ_{n}-action the twisted line bundle O(1) is naturally μ_{n}- equivariant.

Lemma 3.8. The morphism h : X → S factors though some relative projective space ℙ_{S}^{r} which admits a global μ_{n}-action.

Proof. By assumption, X admits a μ_{n}-projective action. Then [21, Lemma 2.4 and 2.5] imply that there exists an equivariant closed immersion from X to some projective space ℙ^{r} endowed with a global action. By using the universal property of fibre product, we obtain a morphism from X to ℙ_{S}^{r} = S × ℙ^{r} which is equivariant. Moreover, this morphism is clearly a closed immersion. Since the action on S is trivial, the induced action on the fibre product S × ℙ^{r} is still global. So we are done.

Lemma 3.9. Let l : W (i) → S be the composition h∘p_{X} ∘π. Then W (i) admits an equivariant very ample invertible sheaf L which is relative to l.

Proof. By Lemma 3.8, h : X → S factors through some relative projective space ℙ_{S}^{r} which admits a global μ_{n}-action. So X admits an equivariant very ample invertible sheaf relative to h. Since the μ_{n}-action on S is supposed to be trivial, ℙ_{X}^{1} = X×ℙ^{1} ≅ X×_{S}ℙ_{S}^{1} also admits an equivariant very ample invertible sheaf relative to the morphism h ∘ p_{X} which is denoted by G. Moreover, by construction, W (i) admits a very ample invertible sheaf O_{W} (1) ⊗ π^{∗}G^{⊗b} for some b ≥ 0 which is relative to the blow-down map π (cf. [17, II. Proposition 7.10]). Assume that ℙ_{X}^{1} ×_{S} ℙ_{S}^{m} is the relative projective space associated to O_{W} (1) ⊗ π^{∗}G^{⊗b}, and that ℙ_{S}^{n} is the relative projective space associated to G. Then the very ample invertible sheave on ℙ_{X}^{1} ×_{S} ℙ_{S}^{m} with respect to the embedding

ℙ_{X}^{1} ×_{S} ℙ_{S}^{m} ↪ ℙ_{S}^{n} ×_{S} ℙ_{S}^{m}

can be written as G ⊠ O_{ℙ_{S}^{m}}(1) whose restriction to W (i) is equal to O_{W} (1) ⊗ π^{∗}G^{⊗b+1}. Therefore, O_{W} (1)⊗π^{∗}G^{⊗b+1} is a very ample invertible sheaf on W (i) relative to l : W (i) → S, this invertible sheaf is clearly equivariant.

From now on, we shall fix the equivariant very ample invertible sheaf L constructed in Lemma 3.9. We also fix a μ_{n}-invariant hermitian metric on L, note that this metric always exists according to an argument of partition of unity. When we deal with the tensor product of a coherent sheaf F with some power L^{⊗n}, we just write it as F(n) for simplicity. Before we give the proof of the rest of the vanishing theorem, we shall recall the concept of equivariant standard complex and some technical results.

Definition 3.10. Let S be a projective manifold and let \overline{ξ}. be a bounded complex of hermitian vector bundles on S. We say \overline{ξ}. is a standard complex if the homology sheaves of \overline{ξ}. are all locally free and they are endowed with some hermitian metrics. We shall write a standard complex as (\overline{ξ}., h^{H}) to emphasize the choice of the metrics on the homology sheaves.

Definition 3.11. Let S be an equivariant projective manifold. An equivariant standard complex on S is a bounded complex of equivariant hermitian vector bundles on S whose restriction to S_{g} is standard and the metrics on the homology sheaves are g-invariant. Again we shall write an equivariant standard complex as (\overline{ξ}., h^{H}) to emphasize the choice of the metrics on the homology sheaves.

Due to [29, Theorem 5.9], to every equivariant standard complex (\overline{ξ}., h^{H}) on an equivariant projective manifold S, there is a unique axiomatical way to associate an element c̃h_{g}(\overline{ξ.}, h^{H}) in ⨁_{p≥0} A^{p,p}(S_{g})/(Im∂ + Im\overline{∂}) which satisfies the differential equation

dd^{c}c̃h_{g}(\overline{ξ.}, h^{H}) = ∑_{j}(−1)^{j}ch_{g}(H_{j}(\overline{ξ.} |_{S_{g}})) − ∑_{j}(−1)^{j}ch_{g}(\overline{ξ}_{j}).

Let 0 → \overline{ξ.}′ → \overline{ξ.} → \overline{ξ.}′′ → 0 be a short exact sequence of equivariant standard complexes on S. Then by restricting to the fixed point submanifold S_{g}, we get a short exact sequence of standard complexes 0 → \overline{ξ.}′ |_{S_{g}}→ \overline{ξ.} |_{S_{g}}→ \overline{ξ.}′′ |_{S_{g}}→ 0. Hence we obtain a long exact sequence of homology sheaves of these three standard complexes. We shall make a stronger assumption. Suppose that for any j ≥ 0, we have short exact sequence 0 → H_{j}(\overline{ξ.}′ |_{S_{g}}) → H_{j}(\overline{ξ.} |_{S_{g}}) → H_{j}(\overline{ξ.}′′ |_{S_{g}}) → 0 which is denoted by \overline{χ}_{j}. Moreover, for any j ≥ 0, denote by \overline{ε}_{j} the short exact sequence 0 → \overline{ξ}′_{j} → \overline{ξ}_{j} → \overline{ξ}′′_{j} → 0.

Lemma 3.12. Let notations and assumptions be as above. The identity

c̃h_{g}(\overline{ξ.}′, h^{H}) − c̃h_{g}(\overline{ξ.}, h^{H}) + c̃h_{g}(\overline{ξ.}′′, h^{H}) = ∑(−1)^{j}c̃h_{g}(\overline{χ}_{j}) − ∑(−1)^{j}c̃h_{g}(\overline{ε}_{j})

holds in ⨁_{p≥0} A^{p,p}(S_{g})/(Im∂ + Im\overline{∂}).

Proof. On S_{g}, every equivariant standard complex (\overline{ξ.}, h^{H}) splits into a series of short exact sequences of equivariant hermitian vector bundles in the following way

0 → \overline{Im} → \overline{Ker} → \overline{H.} → 0

and

0 → \overline{Ker} → \overline{ξ.} |_{S_{g}}→ \overline{Im} → 0.

According to the argument given after [29, Remark 5.10], c̃h_{g}(\overline{ξ.}, h^{H}) is equal to the alternating sum of the equivariant Bott-Chern secondary characteristic classes of the short exact sequences above. Now since we have supposed that 0 → H_{j}(\overline{ξ.}′ |_{S_{g}}) → H_{j}(\overline{ξ.} |_{S_{g}}) → H_{j}(\overline{ξ.}′′ |_{S_{g}}) → 0 are all exact, by using Snake lemma, we know that 0 → Im(\overline{ξ.}′ |_{S_{g}}) → Im(\overline{ξ.} |_{S_{g}}) → Im(\overline{ξ.}′′ |_{S_{g}}) → 0 and 0 → Ker(\overline{ξ.}′ |_{S_{g}}) → Ker(\overline{ξ.} |_{S_{g}}) → Ker(\overline{ξ.}′′ |_{S_{g}}) → 0 are also all exact sequences. Then the identity in the statement of this lemma immediately follows from the construction of c̃h_{g}(\overline{ξ.}, h^{H}) and the additivity property of the equivariant Bott-Chern secondary characteristic classes.

Corollary 3.13. Let 0 → \overline{ξ.}^{(m)} → ⋯ → \overline{ξ.}^{(1)} → \overline{ξ.}^{(0)} → 0 be an exact sequence of equivariant standard complexes on S such that for every j ≥ 0, 0 → H_{j}(\overline{ξ.}^{(m)} |_{S_{g}}) → ⋯ → H_{j}(\overline{ξ.}^{(1)} |_{S_{g}}) → H_{j}(\overline{ξ.}^{(0)} |_{S_{g}}) → 0 is exact. Then the identity

∑_{k=0}^{m}(−1)^{k}c̃h_{g}(\overline{ξ.}^{(k)}, h^{H}) = ∑(−1)^{j}c̃h_{g}(\overline{χ}_{j}) − ∑(−1)^{j}c̃h_{g}(\overline{ε}_{j})

holds in ⨁_{p≥0} A^{p,p}(S_{g})/(Im∂ + Im\overline{∂}).

Proof. We claim that for every 1 ≤ k ≤ m, the kernel of the complex morphism \overline{ξ.}^{(k)} → \overline{ξ.}^{(k−1)} is still an equivariant standard complex on S. It is clear that we only need to prove this for k = 1. Firstly, the kernel of \overline{ξ.}^{(1)} → \overline{ξ.}^{(0)} is a complex of equivariant hermitian vector bundles, let’s denote it by \overline{K}. By restricting to S_{g} and using an argument of long exact sequence, we know that the homology sheaves of \overline{K} |_{S_{g}} are all equivariant hermitian vector bundles since for any

j ≥ 0 the bundle morphism H_{j}(\overline{ξ.}^{(1)} |_{S_{g}}) → H_{j}(\overline{ξ.}^{(0)} |_{S_{g}}) is already surjective. Therefore, the assumption of exactness on homologies implies that we can split 0 → \overline{ξ.}^{(m)} → ⋯ → \overline{ξ.}^{(1)} → \overline{ξ.}^{(0)} → 0 into a series of short exact sequences of equivariant standard complexes, so the identity in the statement of this corollary follows from Lemma 3.12.

Remark 3.14. A generalized version of Corollary 3.13, in which the exact sequence of (equivariant) standard complexes is replaced by an (equivariant) double standard complex was obtained in Xiaonan Ma’s Ph.D thesis (cf. [24]) where more discussions concerning spectral sequences were involved. Anyway, for arithmetical reason, we only need these special versions as in Lemma 3.12 and Corollary 3.13.

Now we turn back to our proof of the vanishing theorem. As before, let W = W (i) be the deformation to the normal cone associated to an equivariant closed immersion of projective manifolds i : Y → X. For simplicity, denote by P_{g}^{0} the projective space bundle ℙ(N_{X_{g}/Y_{g}} ⊕ O_{Y_{g}}). Moreover, given an invariant Kähler metric on X, we fix an invariant Kähler metric on W which is constructed in Lemma 3.5. In this situation, all normal bundles appearing in the construction of the deformation to the normal cone will be endowed with the quotient metrics. We recall the following lemma.

Lemma 3.15. Over W (i_{g}), there are hermitian metrics on O(X_{g}), O(P_{g}^{0}) and O(X̃_{g}) such that the isometry \overline{O}(X_{g}) ≅ \overline{O}(P_{g}^{0}) ⊗ \overline{O}(X̃_{g}) holds and such that the restriction of \overline{O}(X_{g}) to X_{g} yields the metric of N_{W (i_{g})/X_{g}}, the restriction of \overline{O}(X̃_{g}) to X̃_{g} yields the metric of N_{W (i_{g})/X̃_{g}} and the restriction of \overline{O}(P_{g}^{0}) to P_{g}^{0} induces the metric of N_{W (i_{g})/P_{g}^{0}}.

Proof. This is [21, Lemma 6.15].

Definition 3.16. Let \overline{η} be an equivariant hermitian vector bundle on Y, we say that a resolution

\overline{Ξ} : 0 → \overline{ξ̃}_{m} → ⋯ → \overline{ξ̃}_{0} → j_{∗}p_{Y}^{∗} (\overline{η}) → 0

satisfies the condition (T) if (i). the metrics on ξ̃. satisfy Bismut assumption (A); (ii). the restriction of \overline{Ξ} to X̃ is an equivariantly and orthogonally split exact sequence; (iii). the restrictions of \overline{Ξ}_{∇} to W (i_{g}), X_{g}, P_{g}^{0}, X̃_{g} and P_{g}^{0} ∩ X̃_{g} are complexes with l-acyclic elements and l-acyclic homologies, here \overline{Ξ}_{∇} is the complex of hermitian vector bundles obtained by omitting the last term j_{∗}p_{Y}^{∗} (\overline{η}) in \overline{Ξ}; (iv). the tensor products \overline{Ξ}_{∇} |_{W (i_{g})} ⊗\overline{O}(−X_{g}), \overline{Ξ}_{∇} |_{W (i_{g})} ⊗\overline{O}(−P_{g}^{0}) and \overline{Ξ}_{∇} |_{W (i_{g})} ⊗\overline{O}(−X̃_{g}) are complexes with l-acyclic elements and l-acyclic homologies.

From Theorem 3.6 (i), we already know that there always exists a resolution of j_{∗}p_{Y}^{∗} (\overline{η}) which satisfies the conditions (i) and (ii) in Definition 3.16. Let \overline{Ξ} be such a resolution, we have the following.

Proposition 3.17. For n ≫ 0, \overline{Ξ}(n) satisfies the condition (T).

Proof. The reason is that W (i_{g}), X_{g}, P_{g}^{0}, X̃_{g} and P_{g}^{0} ∩ X̃_{g} are all closed submanifolds of W.

It is well known that both two squares in the following deformation diagram

\xymatrix{Y × \{0\} \ar[r]^{s_{0}} \ar[d]_{i} & Y × ℙ^{1} \ar[d]^{j} & Y × \{∞\} \ar[l]_{s_{∞}} \ar[d]^{i_{∞}} \\ X × \{0\} \ar[r] & W & ℙ(N_{X/Y} ⊕ N_{ℙ^{1}/∞}) \ar[l]}

are Tor-independent. Moreover, according to our choices of the Kähler metrics, we may identify Y × {0} with Y, X × {0} with X, Y × {∞} with Y and ℙ(N_{X/Y} ⊕ N_{ℙ^{1}/∞}) with P = ℙ(N_{X/Y} ⊕ O_{Y}). So if \overline{Ξ} is a resolution of j_{∗}p_{Y}^{∗} (\overline{η}) on W, then the restriction of \overline{Ξ} to X (resp. P) provides a resolution of i_{∗}\overline{η} (resp. i_{∞}_{∗}\overline{η}). The following theorem is the kernel of the whole proof of the vanishing theorem.

Theorem 3.18. (Deformation theorem) Let \overline{Ξ} be a resolution of j_{∗}p_{Y}^{∗} (\overline{η}) on W which satisfies the condition (T), then we have δ(\overline{Ξ} |_{X}) = δ(\overline{Ξ} |_{P}).

Proof. Consider the following tensor product of \overline{Ξ}_{∇} |_{W (i_{g})} with the Koszul resolution associated to the immersion X_{g} ↪ W (i_{g})

0 → \overline{Ξ}_{∇} |_{W (i_{g})} ⊗\overline{O}(−X_{g}) → \overline{Ξ}_{∇} |_{W (i_{g})} ⊗\overline{O}_{W (i_{g})} → \overline{Ξ}_{∇} |_{W (i_{g})} ⊗i_{X_{g}}_{∗}\overline{O}_{X_{g}} → 0.

We have to caution the reader that here the tensor product is not the usual tensor product of two complexes, precisely our resulting sequence is a double complex and we don’t take its total complex. Since we have assumed that \overline{Ξ} satisfies the condition (T), this tensor product induces a short exact sequence of equivariant standard complexes on S by taking direct images. For j ≥ 0, its j-th row is the following short exact sequence

\overline{ε}_{j} : 0 → R^{0}l_{g}^{0}_{∗}(\overline{O}(−X_{g}) ⊗ \overline{ξ̃}_{j} |_{W (i_{g})}) → R^{0}l_{g}^{0}_{∗}(\overline{ξ̃}_{j} |_{W (i_{g})}) → R^{0}h_{g}_{∗}(\overline{ξ̃}_{j} |_{X_{g}}) → 0

where l_{g}^{0} is the composition of the inclusion W (i_{g}) ↪ W with the morphism l.

Note that the j-th homology of \overline{Ξ}_{∇} |_{W (i_{g})} is equal to j_{g}_{∗}(∧^{j}\overline{F̃}^{∨}⊗p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}}) |_{W (i_{g})} where F̃ is the non-zero degree part of the normal bundle associated to the immersion j. Actually j_{g} factors through j_{g}^{0} : Y_{g} × ℙ^{1} ↪ W (i_{g}), then the

j-th homology of \overline{Ξ}_{∇} |_{W (i_{g})} can be rewritten as j_{g}^{0}_{∗}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}}). Write Y_{g,0} := Y_{g} × {0} for simplicity. Using the fact that j_{g}^{0}^{∗}O(−X_{g}) is isomorphic to O(−Y_{g,0}), we deduce from the short exact sequence

0 → j_{g}^{0}_{∗}(\overline{O}(−Y_{g,0}) ⊗ ∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}}) → j_{g}^{0}_{∗}(\overline{O}_{Y_{g}×ℙ^{1}} ⊗ ∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})

→ j_{g}^{0}_{∗}(i_{Y_{g}}_{∗}\overline{O}_{Y_{g}} ⊗ ∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}}) → 0

that the j-th homologies of the induced short exact sequence of equivariant standard complexes form a short exact sequence

\overline{χ}_{j} : 0 → R^{0}u_{g}_{∗}(\overline{O}(−Y_{g,0}) ⊗ ∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}}) → R^{0}u_{g}_{∗}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})

→ R^{0}f_{g}_{∗}(∧^{j}\overline{F}^{∨} ⊗ \overline{η} |_{Y_{g}}) → 0

where u_{g} is the composition of the inclusion Y_{g} × ℙ^{1} ↪ W (i_{g}) with the morphism l_{g}^{0}. The main idea of this proof is that the equivariant Bott-Chern secondary characteristic class of the quotient term of the induced short exact sequence of equivariant standard complexes is nothing but c̃h_{g}(\overline{Ξ}_{∇} |_{X}, h^{H}) which appears in the expression of δ(\overline{Ξ} |_{X}) and the equivariant secondary characteristic classes of \overline{χ}_{j}, \overline{ε}_{j} can be computed by Bismut-Ma’s immersion formula. Precisely, denote by g_{X_{g}} the Euler-Green current associated to X_{g} which was constructed by Bismut, Gillet and Soulé in [6, Section 3. (f)], it satisfies the differential equation dd^{c}g_{X_{g}} = δ_{X_{g}} −c_{1}(\overline{O}(X_{g})). We write Td(\overline{X_{g}}) for Td(\overline{O}(X_{g})), [6, Theorem 3.17] implies that Td^{−1}(\overline{X_{g}})g_{X_{g}} is equal to the singular Bott-Chern current of the Koszul resolution associated to X_{g} ↪ W (i_{g}) modulo Im∂ +Im\overline{∂}. Moreover, write \overline{ξ.} for the restriction \overline{Ξ}_{∇} |_{X}. Then for any j ≥ 0, we compute

c̃h_{g}(\overline{ε}_{j}) =T_{g}(ω^{X_{g}}, h^{ξ_{j}|_{Xg}}) − T_{g}(ω^{W (i_{g})}, h^{ξ̃_{j}|_{W (ig)}})
+ T_{g}(ω^{W (i_{g})}, h^{O(−X_{g})⊗ξ̃_{j}|_{W (ig)}})
+ ∫_{W (i_{g})/S} ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}})Td^{−1}(\overline{X_{g}})g_{X_{g}}
+ ∫_{X_{g}/S} ch_{g}(\overline{ξ}_{j})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})T̃d(\overline{T h_{g}}, \overline{T l_{g}^{0}} |_{X_{g}})
+ ∫_{X_{g}/S} ch_{g}(ξ_{j})R(N_{W (i_{g})/X_{g}})Td(T h_{g}).

Here, one should note that to simplify the last two terms in the right-hand side of Bismut-Ma’s immersion formula, we have used an Atiyah-Segal-Singer type formula for immersion

i_{g}_{∗}(Td_{g}^{−1}(N)ch_{g}(x)) = ch_{g}(i_{∗}(x)).

This formula is the content of [21, Theorem 6.16]. Similarly, for any j ≥ 0, we compute

c̃h_{g}(\overline{χ}_{j}) =T_{g}(ω^{Y_{g}}, h^{∧^{j}F^{∨}⊗η|_{Yg}}) − T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})
+ T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{O(−Y_{g,0})⊗∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})
+ ∫_{Y_{g}×ℙ^{1}/S} ch_{g}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})Td(\overline{T u_{g}})Td^{−1}(\overline{Y_{g,0}})g_{Y_{g,0}}
+ ∫_{Y_{g}/S} ch_{g}(∧^{j}\overline{F}^{∨} ⊗ \overline{η} |_{Y_{g}})Td^{−1}(\overline{N}_{Y_{g}×ℙ^{1}/Y_{g,0}})T̃d(\overline{T f_{g}}, \overline{T u_{g}} |_{Y_{g,0}})
+ ∫_{Y_{g}/S} ch_{g}(∧^{j}F^{∨} ⊗ η |_{Y_{g}})R(N_{Y_{g}×ℙ^{1}/Y_{g,0}})Td(T f_{g}).

Denote by \overline{Ω}(W (i_{g})) (resp. \overline{Ω}(−X_{g})) the middle (resp. sub) term of the induced short exact sequence of equivariant standard complexes. According to Lemma 3.12, we have

c̃h_{g}(\overline{Ξ}_{∇} |_{X}, h^{H}) − c̃h_{g}(\overline{Ω}(W (i_{g})), h^{H}) + c̃h_{g}(\overline{Ω}(−X_{g}), h^{H})

= ∑(−1)^{j}c̃h_{g}(\overline{χ}_{j}) − ∑(−1)^{j}c̃h_{g}(\overline{ε}_{j})
= ∑(−1)^{j}T_{g}(ω^{Y_{g}}, h^{∧^{j}F^{∨}⊗η|_{Yg}}) − ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})

+ ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{O(−Y_{g,0})⊗∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})

+ ∫_{Y_{g}×ℙ^{1}/S} ∑(−1)^{j}ch_{g}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})Td(\overline{T u_{g}})Td^{−1}(\overline{Y_{g,0}})g_{Y_{g,0}}

+ ∫_{Y_{g}/S} ∑(−1)^{j}ch_{g}(∧^{j}\overline{F}^{∨} ⊗ \overline{η} |_{Y_{g}})Td^{−1}(\overline{N}_{Y_{g}×ℙ^{1}/Y_{g,0}})T̃d(\overline{T f_{g}}, \overline{T u_{g}} |_{Y_{g,0}})

+ ∫_{Y_{g}/S} ∑(−1)^{j}ch_{g}(∧^{j}F^{∨} ⊗ η |_{Y_{g}})R(N_{Y_{g}×ℙ^{1}/Y_{g,0}})Td(T f_{g})

− ∑(−1)^{j}T_{g}(ω^{X_{g}}, h^{ξ_{j}|_{Xg}}) + ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{ξ̃_{j}|_{W (ig)}})

− ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{O(−X_{g})⊗ξ̃_{j}|_{W (ig)}})

− ∫_{W (i_{g})/S} ∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}})Td^{−1}(\overline{X_{g}})g_{X_{g}}

− ∫_{X_{g}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})T̃d(\overline{T h_{g}}, \overline{T l_{g}^{0}} |_{X_{g}})

− ∫_{X_{g}/S} ∑(−1)^{j}ch_{g}(ξ_{j})R(N_{W (i_{g})/X_{g}})Td(T h_{g}).  (1)

Similarly, we consider the tensor products of \overline{Ξ}_{∇} |_{W (i_{g})} with the following three Koszul resolutions

0 → \overline{O}(−P_{g}^{0}) → \overline{O}_{W (i_{g})} → i_{P_{g}^{0}}_{∗}\overline{O}_{P_{g}^{0}} → 0,

0 → \overline{O}(−X̃_{g}) → \overline{O}_{W (i_{g})} → iX̃_{g}_{∗}\overline{O}X̃_{g} → 0,

and

0 → \overline{O}(−X̃_{g}) ⊗ \overline{O}(−P_{g}^{0}) → \overline{O}(−X̃_{g})⊕\overline{O}(−P_{g}^{0}) → \overline{O}_{W (i_{g})}
→ i_{X̃_{g}∩P_{g}^{0}}_{∗}\overline{O}_{X̃_{g}∩P_{g}^{0}} → 0.

We shall still denote by \overline{χ.} (resp. \overline{ε.}) the exact sequences consisting of homologies (resp. elements) in the induced exact sequences of equivariant standard complexes. For the first one, denote by g_{P_{g}^{0}} the Euler-Green current associated to P_{g}^{0} and write \overline{ξ.}^{∞} for the restriction \overline{Ξ}_{∇} |_{P}. Moreover, denote by \overline{Ω}(−P_{g}^{0}) the sub term of the induced short exact sequence of equivariant standard complexes and denote by b_{g} the composition of the inclusion P_{g}^{0} ↪ W (i_{g}) with the morphism l_{g}^{0}. According to Lemma 3.12, we have

c̃h_{g}(\overline{Ξ}_{∇} |_{P_{g}^{0}}, h^{H}) − c̃h_{g}(\overline{Ω}(W (i_{g})), h^{H}) + c̃h_{g}(\overline{Ω}(−P_{g}^{0}), h^{H})
= ∑(−1)^{j}c̃h_{g}(\overline{χ}_{j}) − ∑(−1)^{j}c̃h_{g}(\overline{ε}_{j})
= ∑(−1)^{j}T_{g}(ω^{Y_{g}}, h^{∧^{j}F_{∞}^{∨}⊗η|_{Yg}}) − ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})
+ ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{O(−Y_{g,∞})⊗∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})
+ ∫_{Y_{g}×ℙ^{1}/S} ∑(−1)^{j}ch_{g}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})Td(\overline{T u_{g}})Td^{−1}(\overline{Y_{g,∞}})g_{Y_{g,∞}}
+ ∫_{Y_{g}/S}\{∑(−1)^{j}ch_{g}(∧^{j}\overline{F} _{∞}^{∨} ⊗ \overline{η} |_{Y_{g}})Td^{−1}(\overline{N}_{Y_{g}×ℙ^{1}/Y_{g,∞}})
· T̃d(\overline{T f_{g}}, \overline{T u_{g}} |_{Y_{g,∞}})\}
+ ∫_{Y_{g}/S} ∑(−1)^{j}ch_{g}(∧^{j}F_{∞}^{∨} ⊗ η |_{Y_{g}})R(N_{Y_{g}×ℙ^{1}/Y_{g,∞}})Td(T f_{g})
− ∑(−1)^{j}T_{g}(ω^{P_{g}^{0}}, h^{ξ_{j}^{∞}|_{P^{0}_{g}}}) + ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{ξ̃_{j}|_{W (ig)}})
− ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{O(−P_{g}^{0})⊗ξ̃_{j}|_{W (ig)}})
− ∫_{W (i_{g})/S} ∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}})Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}}
− ∫_{P_{g}^{0}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j}^{∞})Td^{−1}(\overline{N}_{W (i_{g})/P_{g}^{0}})T̃d(\overline{T b_{g}}, \overline{T l_{g}^{0}} |_{P_{g}^{0}})
− ∫_{P_{g}^{0}/S} ∑(−1)^{j}ch_{g}(ξ_{j}^{∞})R(N_{W (i_{g})/P_{g}^{0}})Td(T b_{g})  (2)

where \overline{F}_{∞} is the non-zero degree part of the hermitian normal bundle \overline{N}_{∞} associated to i_{∞}.

For the second one, denote by gX̃_{g} the Euler-Green current associated to X̃_{g}

and denote by \overline{Ω}(−X̃_{g}) the sub term of the induced short exact sequence of equivariant standard complexes. Since the restriction of \overline{Ξ} to the component X̃ is equivariantly and orthogonally split, we know that c̃h_{g}(\overline{Ξ} |X̃_{g}, h^{H}) is equal to

0 and the summation ∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j}) vanishes on X̃_{g}. Using again Lemma 3.12, we obtain

− c̃h_{g}(\overline{Ω}(W (i_{g})), h^{H}) + c̃h_{g}(\overline{Ω}(−X̃_{g}), h^{H})
= ∑(−1)^{j}c̃h_{g}(\overline{χ}_{j}) − ∑(−1)^{j}c̃h_{g}(\overline{ε}_{j})
= − ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})
+ ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{j_{g}^{0}^{∗}O(−X̃_{g})⊗∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})
− ∫_{Y_{g}×ℙ^{1}/S}\{∑(−1)^{j}ch_{g}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})Td(\overline{T u_{g}})
· c̃h(j_{g}^{0}^{∗}\overline{O}(−X̃_{g}), \overline{O}_{Y_{g}×ℙ^{1}})\}
+ ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{ξ̃_{j}|_{W (ig)}})
− ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{O(−X̃_{g})⊗ξ̃_{j}|_{W (ig)}})
− ∫_{W (i_{g})/S} ∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}})Td^{−1}(\overline{X̃_{g}})gX̃_{g}.  (3)

Here the element c̃h(j_{g}^{0}^{∗}\overline{O}(−X̃_{g}), \overline{O}_{Y_{g}×ℙ^{1}}) is the equivariant secondary characteristic class of the following short exact sequence

0 → 0 → j_{g}^{0}^{∗}\overline{O}(−X̃_{g}) → \overline{O}_{Y_{g}×ℙ^{1}} → 0.

We now consider the last one. This is also a Koszul resolution because X̃_{g} and P_{g}^{0} intersect transversally. By [6, Theorem 3.20], the Euler-Green current associated to X̃_{g} ∩ P_{g}^{0} is the current c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g} + δX̃_{g}g_{P_{g}^{0}}. Then, by using

the isometry \overline{O}(X_{g}) ≅ \overline{O}(P_{g}^{0}) ⊗ \overline{O}(X̃_{g}) and Corollary 3.13, we get

− c̃h_{g}(\overline{Ω}(W (i_{g})), h^{H}) + c̃h_{g}(\overline{Ω}(−X̃_{g}), h^{H})
+ c̃h_{g}(\overline{Ω}(−P_{g}^{0}), h^{H}) − c̃h_{g}(\overline{Ω}(−X_{g}), h^{H})
= ∑(−1)^{j}c̃h_{g}(\overline{χ}_{j}) − ∑(−1)^{j}c̃h_{g}(\overline{ε}_{j})
= − ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})
+ ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{j_{g}^{0}^{∗}O(−X̃_{g})⊗∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})
+ ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{O(−Y_{g,∞})⊗∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})

− ∑(−1)^{j}T_{g}(ω^{Y_{g}×ℙ^{1}}, h^{O(−Y_{g,0})⊗∧^{j} F̃^{∨}⊗p_{Yg}^{∗} η|_{Yg}})

− ∫_{Y_{g}×ℙ^{1}/S} ∑(−1)^{j}ch_{g}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})Td(\overline{T u_{g}})c̃h(\overline{Θ})

+ ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{ξ̃_{j}|_{W (ig)}})

− ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{O(−X̃_{g})⊗ξ̃_{j}|_{W (ig)}})

− ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{O(−P_{g}^{0})⊗ξ̃_{j}|_{W (ig)}})

+ ∑(−1)^{j}T_{g}(ω^{W (i_{g})}, h^{O(−X_{g})⊗ξ̃_{j}|_{W (ig)}})

− ∫_{W (i_{g})/S}\{∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}})Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})

· [c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g} + δX̃_{g}g_{P_{g}^{0}}]\}.

(4)

Here the element c̃h(\overline{Θ}) is the equivariant secondary characteristic class of the following short exact sequence

\overline{Θ} : 0 → \overline{O}(−Y_{g,0}) → j_{g}^{0}^{∗}\overline{O}(−X̃_{g}) ⊕ \overline{O}(−Y_{g,∞}) → \overline{O}_{Y_{g}×ℙ^{1}} → 0.

Since s_{0} : Y × {0} → Y × ℙ^{1} and s_{∞} : Y × {∞} → Y × ℙ^{1} are sections of smooth morphism, the normal sequences

0 → \overline{T f_{g}} → \overline{T u_{g}} |_{Y_{g,0}}→ \overline{N}_{Y_{g}×ℙ^{1}/Y_{g,0}} → 0

and

0 → \overline{T f_{g}} → \overline{T u_{g}} |_{Y_{g,∞}}→ \overline{N}_{Y_{g}×ℙ^{1}/Y_{g,∞}} → 0

are orthogonally split so that T̃d(\overline{T f_{g}}, \overline{T u_{g}} |_{Y_{g,0}}) and T̃d(\overline{T f_{g}}, \overline{T u_{g}} |_{Y_{g,∞}}) are both equal to 0. Moreover, the normal bundles N_{Y_{g}×ℙ^{1}/Y_{g,0}} and N_{Y_{g}×ℙ^{1}/Y_{g,∞}} are isomorphic to trivial bundles so that R(N_{Y_{g}×ℙ^{1}/Y_{g,0}}) and R(N_{Y_{g}×ℙ^{1}/Y_{g,∞}}) are both equal to 0. Furthermore, we may drop all the terms where an integral is taken over X̃_{g} because ∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j}) vanishes on X̃_{g}. Now, we compute (1)−(2)−(3)+(4) which is

c̃h_{g}(\overline{Ξ}_{∇} |_{X}, h^{H}) − c̃h_{g}(\overline{Ξ}_{∇} |_{P_{g}^{0}}, h^{H}) + ∑(−1)^{j}T_{g}(ω^{X_{g}}, h^{ξ_{j}|_{Xg}})
− ∑(−1)^{j}T_{g}(ω^{P_{g}^{0}}, h^{ξ_{j}^{∞}|_{P^{0}_{g}}}) − ∑(−1)^{j}T_{g}(ω^{Y_{g}}, h^{∧^{j}F^{∨}⊗η|_{Yg}})
+ ∑(−1)^{j}T_{g}(ω^{Y_{g}}, h^{∧^{j}F_{∞}^{∨}⊗η|_{Yg}})
= ∫_{Y_{g}×ℙ^{1}/S}\{∑(−1)^{j}ch_{g}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})Td(\overline{T u_{g}}) · [Td^{−1}(\overline{Y_{g,0}})g_{Y_{g,0}}

− Td^{−1}(\overline{Y_{g,∞}})g_{Y_{g,∞}} + c̃h(j_{g}^{0}^{∗}\overline{O}(−X̃_{g}), \overline{O}_{Y_{g}×ℙ^{1}}) − c̃h_{g}(\overline{Θ})]\} − ∫_{W (i_{g})/S} ∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}})Td^{−1}(\overline{X_{g}})g_{X_{g}}

− ∫_{X_{g}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})T̃d(\overline{T h_{g}}, \overline{T l_{g}^{0}} |_{X_{g}})

− ∫_{X_{g}/S} ∑(−1)^{j}ch_{g}(ξ_{j})R(N_{W (i_{g})/X_{g}})Td(T h_{g})

+ ∫_{W (i_{g})/S} ∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}})Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}}

+ ∫_{P_{g}^{0}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j}^{∞})Td^{−1}(\overline{N}_{W (i_{g})/P_{g}^{0}})T̃d(\overline{T b_{g}}, \overline{T l_{g}^{0}} |_{P_{g}^{0}})

+ ∫_{P_{g}^{0}/S} ∑(−1)^{j}ch_{g}(ξ_{j}^{∞})R(N_{W (i_{g})/P_{g}^{0}})Td(T b_{g})

+ ∫_{W (i_{g})/S} ∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}})Td^{−1}(\overline{X̃_{g}})gX̃_{g}

− ∫_{W (i_{g})/S} ∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}})Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g}.

Denote by i_{X} (resp. i_{P}) the inclusion from X to W (i) (resp. P to W (i)). We may use the Atiyah-Segal-Singer type formula for immersions and the projection formula in cohomology to compute

i_{X}_{g}_{∗}( ∑(−1)^{j}ch_{g}(ξ_{j})R(N_{W (i_{g})/X_{g}})Td(T h_{g}))
=i_{X}_{g}_{∗}(R(N_{W (i_{g})/X_{g}})Td(T h_{g})i_{g}_{∗}(Td_{g}^{−1}(N_{X/Y})ch_{g}(η)))
=(i_{X}_{g} ∘ i_{g})_{∗}(R(N_{W (i_{g})/X_{g}})Td(T h_{g})Td_{g}^{−1}(N_{X/Y})ch_{g}(η)).

Note that the restriction of N_{W (i_{g})/X_{g}} to Y_{g} is trivial so that the last expression vanishes. An entirely analogous reasoning implies that

i_{P}_{g}_{∗}( ∑(−1)^{j}ch_{g}(ξ_{j}^{∞})R(N_{W (i_{g})/P_{g}^{0}})Td(T b_{g})) = 0.

Thus, we are left with the equality

c̃h_{g}(\overline{Ξ}_{∇} |_{X}, h^{H}) − c̃h_{g}(\overline{Ξ}_{∇} |_{P_{g}^{0}}, h^{H}) + ∑(−1)^{j}T_{g}(ω^{X_{g}}, h^{ξ_{j}|_{Xg}})

− ∑(−1)^{j}T_{g}(ω^{P_{g}^{0}}, h^{ξ_{j}^{∞}|_{P^{0}_{g}}}) − ∑(−1)^{j}T_{g}(ω^{Y_{g}}, h^{∧^{j}F^{∨}⊗η|_{Yg}})

+ ∑(−1)^{j}T_{g}(ω^{Y_{g}}, h^{∧^{j}F_{∞}^{∨}⊗η|_{Yg}})
= ∫_{Y_{g}×ℙ^{1}/S}\{∑(−1)^{j}ch_{g}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})Td(\overline{T u_{g}}) · [Td^{−1}(\overline{Y_{g,0}})g_{Y_{g,0}}

− Td^{−1}(\overline{Y_{g,∞}})g_{Y_{g,∞}} + c̃h(j_{g}^{0}^{∗}\overline{O}(−X̃_{g}), \overline{O_{Y_{g}×ℙ^{1}}}) − c̃h_{g}(\overline{Θ})]\}

− ∫_{W (i_{g})/S}\{∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}}) · [Td^{−1}(\overline{X_{g}})g_{X_{g}} − Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}}

− Td^{−1}(\overline{X̃_{g}})gX̃_{g} + Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g}]\}

− ∫_{X_{g}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})T̃d(\overline{T h_{g}}, \overline{T l_{g}^{0}} |_{X_{g}})
+ ∫_{P_{g}^{0}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j}^{∞})Td^{−1}(\overline{N}_{W (i_{g})/P_{g}^{0}})T̃d(\overline{T b_{g}}, \overline{T l_{g}^{0}} |_{P_{g}^{0}}).

Using the differential equation which T_{g}(\overline{ξ̃.}) satisfies, we compute

− ∫_{W (i_{g})/S}\{∑(−1)^{j}ch_{g}(\overline{ξ̃}_{j})Td(\overline{T l_{g}^{0}}) · [Td^{−1}(\overline{X_{g}})g_{X_{g}} − Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}}
− Td^{−1}(\overline{X̃_{g}})gX̃_{g} + Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g}]\}
= ∫_{W (i_{g})/S}\{Td(\overline{T l_{g}^{0}})T_{g}(\overline{ξ̃.}) · [Td^{−1}(\overline{X_{g}})δ_{X_{g}} − Td^{−1}(\overline{P_{g}^{0}})δ_{P_{g}^{0}}
− Td^{−1}(\overline{X̃_{g}})δX̃_{g} + Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))δX̃_{g}]\}
− ∫_{W (i_{g})/S}\{Td(\overline{T l_{g}^{0}})ch_{g}(p_{Y}^{∗} \overline{η})Td_{g}^{−1}(\overline{N}_{W/Y ×ℙ^{1}})δ_{Y_{g}×ℙ^{1}} · [Td^{−1}(\overline{X_{g}})g_{X_{g}}

− Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}} − Td^{−1}(\overline{X̃_{g}})gX̃_{g} + Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g}]\}.

(5)

Here we have used the equation

Td^{−1}(\overline{X_{g}})c_{1}(\overline{O}(X_{g}))−Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0})) − Td^{−1}(\overline{X̃_{g}})c_{1}(\overline{O}(X̃_{g}))
+Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))c_{1}(\overline{O}(X̃_{g})) = 0 (6)

which is [21, (23)].

Again using the fact that \overline{ξ̃.} is equivariantly and orthogonally split on X̃, the first integral in the right-hand side of (5) is equal to

∫_{X_{g}/S} Td(\overline{T l_{g}^{0}})T_{g}(\overline{ξ.})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})
− ∫_{P_{g}^{0}/S} Td(\overline{T l_{g}^{0}})T_{g}(\overline{ξ.}^{∞})Td^{−1}(\overline{N}_{W (i_{g})/P_{g}^{0}}).

According to the normal sequence 0 → \overline{T h_{g}} → \overline{T l_{g}^{0}} |_{X_{g}}→ \overline{N}_{W (i_{g})/X_{g}} → 0, we may write

Td(\overline{T l_{g}^{0}}) = Td(\overline{T h_{g}})Td(\overline{N}_{W (i_{g})/X_{g}}) − dd^{c}T̃d(\overline{T h_{g}}, \overline{T l_{g}^{0}} |_{X_{g}}).

So we get

∫_{X_{g}/S} Td(\overline{T l_{g}^{0}})T_{g}(\overline{ξ.})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})
= ∫_{X_{g}/S} Td(\overline{T h_{g}})T_{g}(\overline{ξ.})
− ∫_{X_{g}/S} T̃d(\overline{T h_{g}}, \overline{T l_{g}^{0}} |_{X_{g}})δ_{Y_{g}}ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})
+ ∫_{X_{g}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})T̃d(\overline{T h_{g}}, \overline{T l_{g}^{0}} |_{X_{g}}).

Similarly we have

∫_{P_{g}^{0}/S} Td(\overline{T l_{g}^{0}})T_{g}(\overline{ξ.}^{∞})Td^{−1}(\overline{N}_{W (i_{g})/P_{g}^{0}})
= ∫_{P_{g}^{0}/S} Td(\overline{T b_{g}})T_{g}(\overline{ξ.}^{∞})
− ∫_{P_{g}^{0}/S} T̃d(\overline{T b_{g}}, \overline{T l_{g}^{0}} |_{P_{g}^{0}})δ_{Y_{g}}ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N}_{∞})Td^{−1}(\overline{N}_{W (i_{g})/P_{g}^{0}})
+ ∫_{P_{g}^{0}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j}^{∞})Td^{−1}(\overline{N}_{W (i_{g})/P_{g}^{0}})T̃d(\overline{T b_{g}}, \overline{T l_{g}^{0}} |_{P_{g}^{0}}).

Note that the normal sequence of \overline{T h_{g}} in \overline{T l_{g}^{0}} (resp. \overline{T b_{g}} in \overline{T l_{g}^{0}}) is orthogonally split on Y_{g} × \{0\} (resp. Y_{g} × \{∞\}), then T̃d(\overline{T h_{g}}, \overline{T l_{g}^{0}} |_{X_{g}})δ_{Y_{g}} and T̃d(\overline{T b_{g}}, \overline{T l_{g}^{0}} |_{P_{g}^{0}})δ_{Y_{g}} are both equal to 0. Combining these computations above we get

∫_{X_{g}/S} Td(\overline{T l_{g}^{0}})T_{g}(\overline{ξ.})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})
− ∫_{P_{g}^{0}/S} Td(\overline{T l_{g}^{0}})T_{g}(\overline{ξ.}^{∞})Td^{−1}(\overline{N}_{W (i_{g})/P_{g}^{0}})
= ∫_{X_{g}/S} Td(\overline{T h_{g}})T_{g}(\overline{ξ.})
+ ∫_{X_{g}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j})Td^{−1}(\overline{N}_{W (i_{g})/X_{g}})T̃d(\overline{T h_{g}}, \overline{T l_{g}^{0}} |_{X_{g}})
− ∫_{P_{g}^{0}/S} Td(\overline{T b_{g}})T_{g}(\overline{ξ.}^{∞})
− ∫_{P_{g}^{0}/S} ∑(−1)^{j}ch_{g}(\overline{ξ}_{j}^{∞})Td^{−1}(\overline{N}_{W (i_{g})/P_{g}^{0}})T̃d(\overline{T b_{g}}, \overline{T l_{g}^{0}} |_{P_{g}^{0}}).

(7)

We now compute the second integral in the right-hand side of (5). According to the normal sequence

0 → \overline{T u_{g}} → \overline{T l_{g}^{0}} |_{Y_{g}×ℙ^{1}}→ \overline{N}_{W (i_{g})/Y_{g}×ℙ^{1}} → 0,

we may write

Td(\overline{T l_{g}^{0}}) = Td(\overline{T u_{g}})Td(\overline{N}_{W (i_{g})/Y_{g}×ℙ^{1}}) − dd^{c}T̃d(\overline{T u_{g}}, \overline{T l_{g}^{0}} |_{Y_{g}×ℙ^{1}}).

Hence

− ∫_{W (i_{g})/S}\{Td(\overline{T l_{g}^{0}})ch_{g}(p_{Y}^{∗} \overline{η})Td_{g}^{−1}(\overline{N}_{W/Y ×ℙ^{1}})δ_{Y_{g}×ℙ^{1}} · [Td^{−1}(\overline{X_{g}})g_{X_{g}}

− Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}} − Td^{−1}(\overline{X̃_{g}})gX̃_{g} + Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g}]\}

= − ∫_{Y_{g}×ℙ^{1}/S}\{Td(\overline{T u_{g}})ch_{g}(p_{Y}^{∗} \overline{η})Td_{g}^{−1}(\overline{F̃}) · j_{g}^{0}^{∗}[Td^{−1}(\overline{X_{g}})g_{X_{g}}

− Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}} − Td^{−1}(\overline{X̃_{g}})gX̃_{g} + Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g}]\}

+ ∫_{Y_{g}×ℙ^{1}/S}\{T̃d(\overline{T u_{g}}, \overline{T l_{g}^{0}} |_{Y_{g}×ℙ^{1}})ch_{g}(p_{Y}^{∗} \overline{η})Td_{g}^{−1}(\overline{N}_{W/Y ×ℙ^{1}})
· [Td^{−1}(\overline{X_{g}})(δ_{X_{g}} − c_{1}(\overline{O}(X_{g}))) − Td^{−1}(\overline{P_{g}^{0}})(δ_{P_{g}^{0}} − c_{1}(\overline{O}(P_{g}^{0})))
− Td^{−1}(\overline{X̃_{g}})(δX̃_{g} − c_{1}(\overline{O}(X̃_{g})))
+ Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))(δX̃_{g} − c_{1}(\overline{O}(X̃_{g})))]\}.

By our choices of the metrics, we have Td_{g}^{−1}(\overline{N}_{W/Y ×ℙ^{1}}) |_{Y_{g,0}}= Td_{g}^{−1}(\overline{N}), Td(\overline{X_{g}}) |_{Y_{g,0}}= 1 and Td_{g}^{−1}(\overline{N}_{W/Y ×ℙ^{1}}) |_{Y_{g,∞}}= Td_{g}^{−1}(\overline{N}_{∞}), Td(\overline{P_{g}^{0}}) |_{Y_{g,∞}}= 1. Furthermore, by replacing all tangent bundles by relative tangent bundles, on can carry through the proof given in [21, P. 378-379] to show that

T̃d(\overline{T u_{g}}, \overline{T l_{g}^{0}} |_{Y_{g}×ℙ^{1}}) |_{Y_{g,0}}= T̃d(\overline{T f_{g}}, \overline{T h_{g}} |_{Y_{g}})

and

T̃d(\overline{T u_{g}}, \overline{T l_{g}^{0}} |_{Y_{g}×ℙ^{1}}) |_{Y_{g,∞}}= T̃d(\overline{T f_{g}}, \overline{T b_{g}} |_{Y_{g}}).

So combining with the equation (6), we get

− ∫_{W (i_{g})/S}\{Td(\overline{T l_{g}^{0}})ch_{g}(p_{Y}^{∗} \overline{η})Td_{g}^{−1}(\overline{N}_{W/Y ×ℙ^{1}})δ_{Y_{g}×ℙ^{1}} · [Td^{−1}(\overline{X_{g}})g_{X_{g}}

− Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}} − Td^{−1}(\overline{X̃_{g}})gX̃_{g} + Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g}]\}

= − ∫_{Y_{g}×ℙ^{1}/S}\{Td(\overline{T u_{g}})ch_{g}(p_{Y}^{∗} \overline{η})Td_{g}^{−1}(\overline{F̃}) · j_{g}^{0}^{∗}[Td^{−1}(\overline{X_{g}})g_{X_{g}}

− Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}} − Td^{−1}(\overline{X̃_{g}})gX̃_{g} + Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g}]\} 1082 Shun Tang

+ ∫_{Y_{g}/S} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})T̃d(\overline{T f_{g}}, \overline{T h_{g}} |_{Y_{g}})
− ∫_{Y_{g}/S} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N}_{∞})T̃d(\overline{T f_{g}}, \overline{T b_{g}} |_{Y_{g}}).

(8)

At last, using the fact that the intersections in the deformation diagram are transversal and the fact that j_{g}^{0}(Y_{g} × ℙ^{1}) has no intersection with X̃_{g}, we can compute

∫_{Y_{g}×ℙ^{1}/S}\{∑(−1)^{j}ch_{g}(∧^{j}\overline{F̃}^{∨} ⊗ p_{Y_{g}}^{∗} \overline{η} |_{Y_{g}})Td(\overline{T u_{g}}) · [Td^{−1}(\overline{Y_{g,0}})g_{Y_{g,0}}

− Td^{−1}(\overline{Y_{g,∞}})g_{Y_{g,∞}} + c̃h(j_{g}^{0}^{∗}\overline{O}(−X̃_{g}), \overline{O}_{Y_{g}×ℙ^{1}}) − c̃h_{g}(\overline{Θ})]\}
= ∫_{Y_{g}×ℙ^{1}/S}\{Td(\overline{T u_{g}})ch_{g}(p_{Y}^{∗} \overline{η})Td_{g}^{−1}(\overline{F̃}) · j_{g}^{0}^{∗}[Td^{−1}(\overline{X_{g}})g_{X_{g}} − Td^{−1}(\overline{P_{g}^{0}})g_{P_{g}^{0}}
− Td^{−1}(\overline{X̃_{g}})gX̃_{g} + Td^{−1}(\overline{X̃_{g}})Td^{−1}(\overline{P_{g}^{0}})c_{1}(\overline{O}(P_{g}^{0}))gX̃_{g}]\}.

(9)

Gathering (5), (7), (8) and (9) we finally get c̃h_{g}(\overline{Ξ}_{∇} |_{X}, h^{H}) − c̃h_{g}(\overline{Ξ}_{∇} |_{P_{g}^{0}}, h^{H}) + ∑(−1)^{j}T_{g}(ω^{X_{g}}, h^{ξ_{j}|_{Xg}})

− ∑(−1)^{j}T_{g}(ω^{P_{g}^{0}}, h^{ξ_{j}^{∞}|_{P^{0}_{g}}}) − ∑(−1)^{j}T_{g}(ω^{Y_{g}}, h^{∧^{j}F^{∨}⊗η|_{Yg}})
+ ∑(−1)^{j}T_{g}(ω^{Y_{g}}, h^{∧^{j}F_{∞}^{∨}⊗η|_{Yg}})
= ∫_{X_{g}/S} Td(\overline{T h_{g}})T_{g}(\overline{ξ.}) − ∫_{P_{g}^{0}/S} Td(\overline{T b_{g}})T_{g}(\overline{ξ.}^{∞})
+ ∫_{Y_{g}/S} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})T̃d(\overline{T f_{g}}, \overline{T h_{g}} |_{Y_{g}})
− ∫_{Y_{g}/S} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N}_{∞})T̃d(\overline{T f_{g}}, \overline{T b_{g}} |_{Y_{g}}).  (10)

On the other hand, by definition, we have

δ(\overline{Ξ} |_{P}) :=c̃h_{g}(\overline{ξ.}^{∞}, \overline{η}) − ∑_{k}(−1)^{k}T_{g}(ω^{Y_{g}}, h^{∧^{k}F_{∞}^{∨}⊗η|_{Yg}})

+ ∑_{k}(−1)^{k}T_{g}(ω^{P_{g}}, h^{ξ_{k}^{∞}|_{Pg}})

− ∫_{Y_{g}/S} Td(T f_{g})Td_{g}^{−1}(F)ch_{g}(η)R(N_{g})

− ∫_{P_{g}/S} T_{g}(\overline{ξ.}^{∞})Td(\overline{T b′_{g}})

Documenta Mathematica 15 (2010) 1049–1108

− ∫_{Y_{g}/S} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N}_{∞})T̃d(\overline{T f_{g}}, \overline{T b′_{g}} |_{Y_{g}})

where b′ : P → S is the composition of the inclusion P ↪ W (i) and the morphism l. Note that P_{g}^{0} is an open and closed submanifold of P_{g} and \overline{ξ.}^{∞} is orthogonally split on the other components since they all belong to X̃_{g}, then we can rewrite δ(\overline{Ξ} |_{P}) as

δ(\overline{Ξ} |_{P}) =c̃h_{g}(\overline{Ξ}_{∇} |_{P_{g}^{0}}, h^{H}) − ∑_{k}(−1)^{k}T_{g}(ω^{Y_{g}}, h^{∧^{k}F_{∞}^{∨}⊗η|_{Yg}})

+ ∑_{k}(−1)^{k}T_{g}(ω^{P_{g}^{0}}, h^{ξ_{k}^{∞}|_{P^{0}_{g}}})

− ∫_{Y_{g}/S} Td(T f_{g})Td_{g}^{−1}(F)ch_{g}(η)R(N_{g})

− ∫_{P_{g}^{0}/S} T_{g}(\overline{ξ.}^{∞})Td(\overline{T b_{g}})

− ∫_{Y_{g}/S} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N}_{∞})T̃d(\overline{T f_{g}}, \overline{T b_{g}} |_{Y_{g}}).

Comparing with the definition of δ(\overline{Ξ} |_{X}), the equality (10) implies that

δ(\overline{Ξ} |_{X}) − δ(\overline{Ξ} |_{P}) = 0

which completes the proof of this deformation theorem.

Now we consider the zero section imbedding i_{∞} : Y → P = ℙ(N_{∞} ⊕O_{Y}). Here we use the fact that N_{∞} is isomorphic to N_{X/Y}, we caution the reader that this is not necessarily an isometry since \overline{N}_{∞} carries the quotient metric induced by the Kähler metric on P but N_{X/Y} carries the quotient metric induced by the Kähler metric on X. We recall that on P we have a tautological exact sequence

0 → O(−1) → π_{P}^{∗} (N_{∞} ⊕ O_{Y}) → Q → 0.

The equivariant section σ : O_{P} → π_{P}^{∗} (N_{∞} ⊕ O_{Y}) → Q induces the following Koszul resolution

0 → ∧^{rkQ}Q^{∨} → ⋯ → Q^{∨} → O_{P} → i_{∞}_{∗}O_{Y} → 0.

Since σ is equivariant, the image of O_{P_{g}} under σ |_{P_{g}} is contained in Q_{g}. This means that σ |_{P_{g}} induces a Koszul resolution on P_{g} of the following form

0 → ∧^{rkQ_{g}}Q_{g}^{∨} → ⋯ → Q_{g}^{∨} → O_{P_{g}} → i_{∞,g}_{∗}O_{Y_{g}} → 0.

Proposition 3.19. Let \overline{κ} := \overline{κ}(\overline{η}, \overline{N}_{∞}) be a hermitian Koszul resolution on P defined in Definition 3.3. Then for n ≫ 0, we have δ(\overline{κ}(n)) = 0.

Proof. Denote the non-zero degree part of Q |_{P_{g}} by Q_{⊥}, then we have the following isometry

∧^{i}\overline{Q}^{∨} |_{P_{g}}= ∧^{i}(\overline{Q}_{g}^{∨} ⊕ \overline{Q}_{⊥}^{∨}) ≅ ⨁_{t+s=i}(∧^{t}\overline{Q}_{g}^{∨} ⊗ ∧^{s}\overline{Q}_{⊥}^{∨}).

Consider the following complex of equivariant hermitian vector bundles on P_{g}

0 → ∧^{rkQ_{g}} \overline{Q}_{g}^{∨} ⊗ (∧^{k}\overline{Q}_{⊥}^{∨} ⊗ π_{P_{g}}^{∗} \overline{η} |_{Y_{g}}) → ⋯ → \overline{Q}_{g}^{∨} ⊗ (∧^{k}\overline{Q}_{⊥}^{∨} ⊗ π_{P_{g}}^{∗} \overline{η} |_{Y_{g}})

→ ∧^{k}\overline{Q}_{⊥}^{∨} ⊗ π_{P_{g}}^{∗} \overline{η} |_{Y_{g}}→ 0

which provides a resolution of i_{∞,g}_{∗}(∧^{k}\overline{F} _{∞}^{∨} ⊗ \overline{η} |_{Y_{g}}) where F_{∞}, as before, is the non-zero degree part of the normal bundle N_{∞} associated to i_{∞}. We denote this resolution by \overline{κ}^{(k)}, then according to the arguments given before this proposition we have a decomposition of complexes \overline{κ}_{∇} |_{P_{g}} ≅ ⨁_{k≥0} \overline{κ}_{∇}^{(k)}[−k]

where \overline{κ}_{∇}^{(k)}[−k] is obtained from \overline{κ}_{∇}^{(k)} by shifting degree. Replacing \overline{κ} by \overline{κ}(n) for big enough n, we may assume that all elements in \overline{κ} and \overline{κ}^{(k)} are acyclic. Therefore, by Bisumt-Ma’s immersion formula we have the following equality

c̃h_{g}(b′_{g}_{∗}\overline{κ}^{(k)}) =T_{g}(ω^{Y_{g}}, h^{∧^{k}F_{∞}^{∨}⊗η|_{Yg}}) − ∑_{i=0}^{rkQ_{g}}(−1)^{i}T_{g}(ω^{P_{g}}, h^{∧^{i}Q_{g}^{∨}⊗∧^{k}Q_{⊥}^{∨}⊗π_{Pg}^{∗} η|_{Yg}})

+ ∫_{Y_{g}/S} ch_{g}(∧^{k}F_{∞}^{∨} ⊗ η |_{Y_{g}})R(N_{∞,g})Td(T f_{g})
+ ∫_{P_{g}/S} Td(\overline{T b′_{g}})T_{g}(\overline{κ}^{(k)})
+ ∫_{Y_{g}/S} ch_{g}(∧^{k}\overline{F} _{∞}^{∨} ⊗ \overline{η} |_{Y_{g}})Td^{−1}(\overline{N}_{∞,g})T̃d(\overline{T f_{g}}, \overline{T b′_{g}} |_{Y_{g}}).

It is easily seen from the decomposition \overline{κ}_{∇} |_{P_{g}}= ⨁_{k≥0} \overline{κ}_{∇}^{(k)}[−k] that the secondary characteristic class c̃h_{g}(\overline{κ}) appearing in the definition of δ(\overline{κ}) is exactly ∑(−1)^{k}c̃h_{g}(b′_{g}_{∗}\overline{κ}^{(k)}). So taking the alternating sum of both two sides of the equality above and using the fact that equivariant analytic torsion form is additive for direct sum of acyclic bundles, we know that to prove δ(\overline{κ}) = 0, we are left to show that ∑(−1)^{k}T_{g}(\overline{κ}^{(k)}) is equal to T_{g}(\overline{κ}). In fact, by using [21, Lemma 3.15], we have modulo Im∂ + Im\overline{∂}

∑(−1)^{k}T_{g}(\overline{κ}^{(k)}) = ∑(−1)^{k}ch_{g}(∧^{k}\overline{Q}_{⊥}^{∨})ch_{g}(π_{P_{g}}^{∗} \overline{η} |_{Y_{g}})T_{g}(\overline{∧^{·}Q_{g}^{∨}})
= Td_{g}^{−1}(\overline{Q}_{⊥})ch_{g}(π_{P_{g}}^{∗} \overline{η} |_{Y_{g}})T_{g}(\overline{∧^{·}Q_{g}^{∨}})
= Td_{g}^{−1}(\overline{Q})ch_{g}(π_{P_{g}}^{∗} \overline{η} |_{Y_{g}})T_{g}(\overline{∧^{·}Q_{g}^{∨}})Td(\overline{Q}_{g})
= ch_{g}(π_{P_{g}}^{∗} \overline{η} |_{Y_{g}})T_{g}(\overline{∧^{·}Q^{∨}})
= T_{g}(\overline{κ}).

So we are done.

It’s now ready to finish the proof of the vanishing theorem for equivariant closed immersions. Let \overline{η} be an equivariant hermitian vector bundle on Y, assume that

\overline{Ψ} : 0 → \overline{ξ}_{m} → ⋯ → \overline{ξ}_{1} → \overline{ξ}_{0} → i_{∗}\overline{η} → 0

is a resolution of i_{∗}\overline{η} by equivariant hermitian vector bundles on X which satisfies Bismut assumption (A). We need to prove that for n ≫ 0, δ(\overline{Ψ}(n)) = 0.

Proof. (of Theorem 3.1) We first construct a resolution of p_{Y}^{∗} \overline{η} on W (i) as

\overline{Ξ} : 0 → \overline{ξ̃}_{m} → ⋯ → \overline{ξ̃}_{0} → \overline{ξ̃}_{0} → j_{∗}p_{Y}^{∗} (\overline{η}) → 0

which satisfies the condition (i) and (ii) in Definition 3.16. Then the restriction of \overline{Ξ} to X (resp. P) provides a resolution of i_{∗}\overline{η} (resp. i_{∞}_{∗}\overline{η}). Over X, we can find a third resolution \overline{Φ} of i_{∗}\overline{η} which dominates \overline{Ψ} and \overline{Ξ} |_{X}. Namely we get short exact sequences of exact sequences

0 → \overline{Ker}(n) → \overline{Φ}(n) → \overline{Ψ}(n) → 0

and

0 → \overline{Ker}′(n) → \overline{Φ}(n) → \overline{Ξ}(n) |_{X}→ 0.

Then after omitting i_{∗}\overline{η} their restrictions to X_{g} become two exact sequences of complexes. Since n ≫ 0 we may assume that all elements and homologies in the induced double complexes are acyclic, so that by taking direct images we get two exact sequences of equivariant standard complexes on S. These two short exact sequences of equivariant standard complexes clearly satisfy the assumptions in Lemma 3.12. Therefore, using Lemma 3.12, Bismut-Ma’s immersion formula and the double complex formula of equivariant Bott-Chern singular currents (cf. Theorem 2.18), we obtain that

c̃h_{g}(\overline{Ψ}(n)) − c̃h_{g}(\overline{Φ}(n)) + c̃h_{g}(\overline{Ker}(n))
+ T_{g}(ω^{X_{g}}, h^{Ψ(n)_{∇}}) − T_{g}(ω^{X_{g}}, h^{Φ(n)_{∇}}) + T_{g}(ω^{X_{g}}, h^{Ker(n)_{∇}})
= ∫_{X_{g}/S}[T_{g}(\overline{Ψ}(n)_{∇}) − T_{g}(\overline{Φ}(n)_{∇}) + T_{g}(\overline{Ker}(n)_{∇})] · Td(\overline{T h_{g}})

which implies that

δ(\overline{Φ}(n)) = δ(\overline{Ψ}(n)) + δ(\overline{Ker}(n)).

By applying Bismut-Ma’s immersion formula to the case where the immersion is the identity map and \overline{η} is equal to the zero bundle, we get δ(\overline{Ker}(n)) = 0 so that δ(\overline{Φ}(n)) = δ(\overline{Ψ}(n)). Similarly, we have δ(\overline{Φ}(n)) = δ(\overline{Ξ}(n) |_{X}) and hence δ(\overline{Ψ}(n)) = δ(\overline{Ξ}(n) |_{X}). An entirely analogous reasoning implies that δ(\overline{κ}(n)) = δ(\overline{Ξ}(n) |_{P}). Then the vanishing of δ(\overline{Ψ}(n)) follows from Theorem 3.18 and Proposition 3.19.

4 Equivariant arithmetic Grothendieck groups with fixed wave front sets

By an arithmetic ring D we understand a regular, excellent, Noetherian integral ring, together with a finite set S of embeddings D ↪ ℂ, which is invariant under a conjugate-linear involution F_{∞} (cf. [15, Def. 3.1.1]). Denote by μ_{n} the diagonalisable group scheme over D associated to ℤ/nℤ. A μ_{n}-equivariant arithmetic scheme over D is a Noetherian scheme of finite type, endowed with a μ_{n}-projective action over D (cf. [21, Section 2]). Let X be a μ_{n}-equivariant arithmetic scheme whose generic fibre is smooth, then X(ℂ), the set of complex points of the variety ∐_{σ∈S} X ×_{D} ℂ, is a disjoint union of projective manifolds. This manifold admits an action of the group of complex n-th roots of unity and an anti-holomorphic involution induced by F_{∞} which is still denoted by F_{∞}. It was shown in [31, Prop. 3.1] that if X is regular, then the fixed point subscheme X_{μ_{n}} is also regular. Fix a primitive n-th root of unity ζ_{n} and denote its corresponding holomorphic automorphism on X(ℂ) by g, by GAGA principle we have a natural isomorphism X_{μ_{n}}(ℂ) ≅ X(ℂ)_{g}.

Definition 4.1. An equivariant hermitian sheaf (resp. vector bundle) \overline{E} on X is a coherent sheaf (resp. vector bundle) E on X, assumed locally free on X(ℂ), endowed with a μ_{n}-action which lifts the action of μ_{n} on X and a hermitian metric h on the associated bundle E_{ℂ}, which is invariant under F_{∞} and g.

Remark 4.2. Let \overline{E} be an equivariant hermitian sheaf (resp. vector bundle) on X, the restriction of \overline{E} to the fixed point subscheme X_{μ_{n}} has a natural ℤ/nℤ-grading structure \overline{E} |_{X_{μn}} ≅ ⊕_{k∈ℤ/nℤ}\overline{E}_{k}. We shall often write \overline{E}_{μ_{n}} for \overline{E}_{0}. It is clear that the associated bundle of \overline{E}_{μ_{n}} over X(ℂ) is exactly equal to \overline{E}_{g}.

Over a complex manifold M, we may consider the current space which is the continuous dual of the space of smooth complex valued differential forms (cf. [27, Chapter IX]). The wave front set WF(ω) of a current ω over M is a closed conical subset of the cotangent bundle T_{ℝ}^{∗}M_{0} := T_{ℝ}^{∗}M\{0}. This conical subset measures the singularities of ω, actually the projection of WF(ω) in M is equal to the singular locus of the support of ω. It also allows us to define certain products and pull-backs of currents. We refer to [19, Chapter VIII] for the definition and various properties of wave front set. Now let X be a μ_{n}-equivariant arithmetic scheme with smooth generic fibre and let S be a conical subset of T_{ℝ}^{∗}X(ℂ)_{g,0}, denote by D^{p,p}(X(ℂ)_{g}, S) the set of currents ω of type (p, p) on X(ℂ)_{g} which satisfy F_{∞}^{∗}ω = (−1)^{p}ω and whose wave front sets are contained in S, we shall write Ũ(X_{μ_{n}}, S) for the current class

Ũ(X(ℂ)_{g}, S) := ⨁_{p≥0}(D^{p,p}(X(ℂ)_{g}, S)/(Im∂ + Im\overline{∂})).

Let \overline{E} be an equivariant hermitian sheaf or vector bundle on X. Following the same notations and definitions as in [21, Section 3], we write ch_{g}(\overline{E}) for the equivariant Chern character form ch_{g}((E_{ℂ}, h)) associated to the hermitian holomorphic vector bundle (E_{ℂ}, h) on X(ℂ). Similarly, we have the equivariant Todd form Td_{g}(\overline{E}). Furthermore, let \overline{ε} : 0 → \overline{E}′ → \overline{E} → \overline{E}′′ → 0 be an exact sequence of equivariant hermitian sheaves or vector bundles on X, we can associate to it an equivariant Bott-Chern secondary characteristic class c̃h_{g}(\overline{ε}) ∈ Ũ(X_{μ_{n}}, ∅) which satisfies the differential equation

dd^{c}c̃h_{g}(\overline{ε}) = ch_{g}(\overline{E}′) − ch_{g}(\overline{E}) + ch_{g}(\overline{E}′′).

Definition 4.3. Let X be a μ_{n}-equivariant arithmetic scheme with smooth generic fibre and let S be a conical subset of T_{ℝ}^{∗}X(ℂ)_{g,0}, we define the equivariant arithmetic Grothendieck group Ĝ_{0}(X, μ_{n}, S) (resp. K̂_{0}(X, μ_{n}, S)) with respect to X and S as the free abelian group generated by the elements of Ũ(X_{μ_{n}}, S) and by the equivariant isometry classes of equivariant hermitian sheaves (resp. vector bundles) on X, together with the relations (i). for every exact sequence \overline{ε} as above, c̃h_{g}(\overline{ε}) = \overline{E}′ − \overline{E} + \overline{E}′′; (ii). if α ∈ Ũ(X_{μ_{n}}, S) is the sum of two elements α′ and α′′ in Ũ(X_{μ_{n}}, S), then the equality α = α′ + α′′ holds in Ĝ_{0}(X, μ_{n}, S) (resp. K̂_{0}(X, μ_{n}, S)).

Remark 4.4. (i). When S′ ⊂ S, [30, Theorem 3.9 (ii)] implies that the natural map from Ũ(X_{μ_{n}}, S′) to Ũ(X_{μ_{n}}, S) is injective. So the first generating relation in Definition 4.3 does make sense. (ii). When X is regular, one can carry out the proof of [21, Proposition 4.2] to show that the natural morphism from K̂_{0}(X, μ_{n}, S) to Ĝ_{0}(X, μ_{n}, S) is an isomorphism. (iii). The definition of the equivariant arithmetic Grothendieck group implies that there are exact sequences

\xymatrix{Ũ(X_{μ_{n}}, S) \ar[r]^{a} & Ĝ_{0}(X, μ_{n}, S) \ar[r]^{π} & G_{0}(X, μ_{n}) \ar[r] & 0}

and

\xymatrix{Ũ(X_{μ_{n}}, S) \ar[r]^{a} & K̂_{0}(X, μ_{n}, S) \ar[r]^{π} & K_{0}(X, μ_{n}) \ar[r] & 0}

where a is the natural map which sends α ∈ Ũ(X_{μ_{n}}, S) to the class of α in Ĝ_{0}(X, μ_{n}, S) (resp. K̂_{0}(X, μ_{n}, S)) and π is the forgetful map. Here the group G_{0}(X, μ_{n}) is the Grothendieck group of μ_{n}-equivariant coherent sheaves which are locally free on X(ℂ), by a theorem of Quillen (cf. [26, Thm. 3 Cor. 1]) we know that it is isomorphic to the ordinary Grothendieck group of μ_{n}-equivariant coherent sheaves.

In [29, Section 3], we have introduced the ring structure of K̂_{0}(X, μ_{n}, ∅). Since we may have a well-defined product of two currents if their wave front sets have no intersection, and the wave front set is invariant under the operation of multiplying a smooth current, we know that the Grothendieck group K̂_{0}(X, μ_{n}, S) has a K̂_{0}(X, μ_{n}, ∅)-module structure. The same thing goes to Ĝ_{0}(X, μ_{n}, S).

Furthermore, consider the isomorphism R(μ_{n}) ≅ K_{0}(D)[T]/(1 − T^{n}). Let \overline{I} be the μ_{n}-equivariant hermitian D-module whose term of degree 1 is D endowed with the trivial metric and whose other terms are 0. Then we may make K̂_{0}(D, μ_{n}, ∅) an R(μ_{n})-algebra under the ring morphism which sends T to \overline{I}. By doing pull-backs, we may endow every arithmetic Grothendieck group we defined before with an R(μ_{n})-module structure. Notice finally that there is a well-defined map from Ĝ_{0}(X, μ_{n}, ∅) (resp. K̂_{0}(X, μ_{n}, ∅)) to the space of complex closed differential forms, which is defined by the formula ch_{g}(\overline{E} + α) := ch_{g}(\overline{E}) + dd^{c}α where \overline{E} is an equivariant hermitian sheaf (resp. vector bundle) and α ∈ Ũ(X_{μ_{n}}, ∅). Now we investigate the wave front set of a current after doing push-forward. Let f be a holomorphic map of compact complex manifolds, we may define a push-forward f_{∗} on current space which is the dual map of the pull-back of smooth forms. When f is smooth, the push-forward f_{∗} extends the integration of smooth forms over the fibre. Assume that we are given a smooth morphism f : U → V of compact complex manifolds, then f_{∗} induces a current K over the product space V × U defined as

K(α ⊗ β) = (f_{∗}β)(α)

where α and β are smooth forms over V and U respectively. Define

M = {(v, u) ∈ V × U | f(u) = v}

which is a submanifold in V × U. From the fact that f_{∗}β is just the integration of smooth forms over the fibre, it is easily seen that the current K ∈ D^{∗}(V ×U) is exactly the object dS_{M} in [19, Theorem 8.1.5]. Then by that theorem, the wave front set of K is equal to

WF(K) = {(v, u, ξ, −f^{∗}(ξ)) ∈ T_{ℝ}^{∗}V × T_{ℝ}^{∗}U | f(u) = v, ξ ≠ 0}.

Let S be a conical subset of T_{ℝ}^{∗}U_{0}, we fix some notations as follows.

WF(K)_{V} ={(v, ξ) ∈ T_{ℝ}^{∗}V_{0} | ∃u ∈ U, (v, u, ξ, 0) ∈ WF(K)}
WF′(K)_{U} ={(u, η) ∈ T_{ℝ}^{∗}U_{0} | ∃v ∈ V, (v, u, 0, −η) ∈ WF(K)}
WF′(K)_{V} ∘ S ={(v, ξ) ∈ T_{ℝ}^{∗}V_{0} | ∃(u, η) ∈ S, (v, u, ξ, −η) ∈ WF(K)}.

Theorem 4.5. Let notations and assumptions be as above. Assume that ω is a current over U whose wave front set is contained in S with S ∩ WF′(K)_{U} = ∅, then the wave front set of f_{∗}ω is contained in

S′ := WF(K)_{V} ∪ WF′(K) ∘ S.

Proof. This follows from [19, Theorem 8.2.12 and 8.2.13].

Remark 4.6. (i) In our situation, the condition S ∩ WF′(K)_{U} = ∅ is always satisfied because by definition we have WF′(K)_{U} = ∅.

(ii). In our situation, S′ is always equal to WF′(K) ∘ S because WF(K)_{V} = ∅. (iii). If S is the empty set, then S′ is also empty. This is compatible with the push-forward of smooth forms. (iv). Assume that the restriction of f to a closed submanifold W is also smooth. Denote by N_{U/W} the normal bundle of W in U. If S = N_{U/W,ℝ}^{∨} \ {0}, then S′ = ∅.

We now turn to the arithmetic case. Let X, Y be two μ_{n}-equivariant arithmetic schemes with smooth generic fibres, and let f : X → Y be an equivariant morphism over D which is smooth on the complex numbers. Fix a μ_{n}(ℂ)- invariant Kähler metric on X(ℂ) so that we get a Kähler fibration with respect to the holomorphic submersion f_{ℂ} : X(ℂ) → Y (ℂ). Let \overline{E} be an f-acyclic μ_{n}-equivariant hermitian sheaf on X, we know that the direct image f_{∗}E is locally free on Y (ℂ) and it can be endowed with a natural equivariant structure and the L^{2}-metric. Let Ĝ_{0}^{ac}(X, μ_{n}, S) be the group generated by f-acyclic equivariant hermitian sheaves on X and the elements of Ũ(X_{μ_{n}}, S), with the same relations as in Definition 4.3. A theorem of Quillen (cf. [26, Cor.3 P. 111]) for the algebraic analogs of these groups implies that the natural map Ĝ_{0}^{ac}(X, μ_{n}, S) → Ĝ_{0}(X, μ_{n}, S) is an isomorphism. So the following definition does make sense.

Definition 4.7. Let notations and assumptions be as above. The push-forward morphism f_{∗} : Ĝ_{0}(X, μ_{n}, S) → Ĝ_{0}(Y, μ_{n}, S′) is defined in the following way. (i). For every f-acyclic μ_{n}-equivariant hermitian sheaf \overline{E} on X, f_{∗}\overline{E} = (f_{∗}E, f_{∗}h^{E}) − T_{g}(ω^{X}, h^{E}). (ii). For every element α ∈ Ũ(X_{μ_{n}}, S), f_{∗}α = ∫_{X_{g}/Y_{g}} Td_{g}(T f, h^{T f})α ∈

Ũ(Y_{μ_{n}}, S′).

Remark 4.8. If Y is regular, by Remark 4.4 (ii) we know that K̂_{0}(Y, μ_{n}, S′) is naturally isomorphic to Ĝ_{0}(Y, μ_{n}, S′) so that (f_{∗}E, f_{∗}h^{E}) admits a finite equivariant hermitian resolution; if the morphism f is flat and Y is reduced, then (f_{∗}E, f_{∗}h^{E}) is locally free when E is so. Therefore in both two cases above, one can also define a reasonable push-forward morphism f_{∗} : K̂_{0}(X, μ_{n}, S) → K̂_{0}(Y, μ_{n}, S′).

Theorem 4.9. The push-forward morphism f_{∗} is a well-defined group homomorphism.

Proof. The argument is the same as in the proof of [29, Theorem 6.2].

Lemma 4.10. (Projection formula) For any elements y ∈ K̂_{0}(Y, μ_{n}, ∅) and x ∈ Ĝ_{0}(X, μ_{n}, S), the identity f_{∗}(f^{∗}y · x) = y · f_{∗}x holds in Ĝ_{0}(Y, μ_{n}, S′).

Proof. Assume that y = \overline{E} is an equivariant hermitian vector bundle and x = \overline{F} is an f-acyclic equivariant hermitian sheaf, then f^{∗}y · x = f^{∗}\overline{E} ⊗ \overline{F}. By projection formula for direct images and the definition of L^{2}-metric, we know that f_{∗}(f^{∗}\overline{E} ⊗ \overline{F}) is isometric to \overline{E} ⊗ f_{∗}\overline{F}. Moreover, concerning the analytic torsion form, we have T_{g}(ω^{X}, h^{f^{∗}E⊗F}) = ch_{g}(\overline{E})T_{g}(ω^{X}, h^{F}). So the projection formula f_{∗}(f^{∗}y · x) = y · f_{∗}x holds in this case. Assume that y = \overline{E} is an equivariant hermitian vector bundle and x = α is represented by some singular current. We write f_{g}^{∗} and f_{g}_{∗} for the pull-back and push-forward of currents respectively, then

f_{∗}(f^{∗}y · x) =f_{∗}(f_{g}^{∗}ch_{g}(\overline{E})α) = f_{g}_{∗}(f_{g}^{∗}ch_{g}(\overline{E})αTd_{g}(\overline{T f}))
=ch_{g}(\overline{E})f_{g}_{∗}(αTd_{g}(\overline{T f}))
=ch_{g}(\overline{E}) ∫_{X_{g}/Y_{g}} αTd_{g}(\overline{T f}) = y · f_{∗}x.

Here we have used an extension of projection formula of smooth forms p_{∗}(p^{∗}α_{1}∧ α_{2}) = α_{1}∧p_{∗}α_{2} (cf. [14, Prop. IX p. 303]) to the case where the second variable α_{2} is replaced by a singular current. The fact that this extension is valid follows from the definition of p_{∗} and the definition of the product of smooth form and singular current. Assume that y = β is represented by some smooth form and x = \overline{E} is an f-acyclic hermitian sheaf, then

f_{∗}(f^{∗}y · x) =f_{∗}(f_{g}^{∗}(β)ch_{g}(\overline{F})) = f_{g}_{∗}(f_{g}^{∗}(β)ch_{g}(\overline{F})Td_{g}(\overline{T f}))
=βf_{g}_{∗}(ch_{g}(\overline{E})Td_{g}(\overline{T f}))
=β ∫_{X_{g}/Y_{g}} ch_{g}(\overline{E})Td_{g}(\overline{T f}) = β(ch_{g}(\overline{f_{∗}F}) − dd^{c}T_{g}(ω^{X}, h^{F}))

which is exactly y · f_{∗}x. Finally, assume that y = β is represented by some smooth form and x = α is represented by some singular current, then

f_{∗}(f^{∗}y · x) =f_{∗}(f_{g}^{∗}(β)dd^{c}α) = f_{g}_{∗}(f_{g}^{∗}(β)dd^{c}αTd_{g}(\overline{T f}))
=βdd^{c}f_{g}_{∗}(αTd_{g}(\overline{T f}))

which is also equal to y · f_{∗}x. Since f_{∗} and f^{∗} are both group homomorphisms, we may conclude the projection formula by linear extension.

Remark 4.11. Lemma 4.10 implies that f_{∗} is a homomorphism of R(μ_{n})- modules, and hence it induces a push-forward morphism after taking localization.

To end this section, we recall an important lemma which will be used frequently in our later arguments.

Lemma 4.12. ([21, Lemma 4.5]) Let X be a regular μ_{n}-equivariant arithmetic scheme and let \overline{E} be an equivariant hermitian vector bundle on X_{μ_{n}} such that \overline{E}_{μ_{n}} = 0. Then the element λ_{−1}(\overline{E}) is invertible in K̂_{0}(X_{μ_{n}}, μ_{n}, S)_{ρ}.

5 Arithmetic concentration theorem

In this section, we shall prove the arithmetic concentration theorem which is an analog of Thomason’s result in Arakelov geometry. Let X be a μ_{n}- equivariant arithmetic scheme with smooth generic fibre, we consider a special closed immersion i : X_{μ_{n}} ↪ X where X_{μ_{n}} is the fixed point subscheme of X. We shall first construct a well-defined group homomorphism i_{∗} between equivariant arithmetic G_{0}-groups as in the algebraic case. To construct i_{∗}, some analytic datum, which is the equivariant Bott-Chern singular current, should be involved. Precisely speaking, let \overline{η} be a μ_{n}-equivariant hermitian sheaf on X_{μ_{n}} and let \overline{ξ}. be a bounded complex of μ_{n}-equivariant hermitian sheaves which provides a resolution of i_{∗}\overline{η} on X. Such a resolution always exists since the generic fibre of X is supposed to be smooth. Then we may have an equivariant Bott-Chern singular current T_{g}(\overline{ξ}.) ∈ Ũ(X_{μ_{n}}). Note that on the complex numbers the 0-degree part of the normal bundle N := N_{X/X_{g}} vanishes (cf. [21, Prop. 2.12]) so that the wave front set of T_{g}(\overline{ξ}.) is the empty set. This fact means that the following definition does make sense.

Definition 5.1. Let notations and assumptions be as above. The embedding morphism

i_{∗} : Ĝ_{0}(X_{μ_{n}}, μ_{n}, S) → Ĝ_{0}(X, μ_{n}, S)

is defined in the following way. (i). For every μ_{n}-equivariant hermitian sheaf \overline{η} on X_{μ_{n}}, suppose that \overline{ξ}. is a resolution of i_{∗}\overline{η} on X whose metrics satisfy Bismut assumption (A), i_{∗}[\overline{η}] = ∑_{k}(−1)^{k}[\overline{ξ}_{k}] + T_{g}(\overline{ξ.}). (ii). For every α ∈ Ũ(X_{μ_{n}}, S), i_{∗}α = αTd_{g}^{−1}(\overline{N}).

Theorem 5.2. The embedding morphism i_{∗} is a well-defined group homomorphism.

Proof. The argument is the same as in the proof of [29, Theorem 5.2].

Lemma 5.3. (Projection formula) For any elements x ∈ K̂_{0}(X, μ_{n}, ∅) and y ∈ Ĝ_{0}(X_{μ_{n}}, μ_{n}, S), the identity i_{∗}(i^{∗}x · y) = x · i_{∗}y holds in Ĝ_{0}(X, μ_{n}, S).

Proof. Assume that x = \overline{E} is an equivariant hermitian vector bundle and y = \overline{F} is an equivariant hermitian sheaf. Let \overline{ξ.} be a resolution of i_{∗}\overline{F} on X, then \overline{E}⊗\overline{ξ.} provides a resolution of i_{∗}(i^{∗}\overline{E} ⊗ \overline{F}). By definition we have

i_{∗}(i^{∗}x · y) = ∑(−1)^{k}[\overline{ξ}_{k} ⊗ \overline{E}] + ch_{g}(\overline{E})T_{g}(\overline{ξ.})

which is exactly x · i_{∗}y. Assume that x = α is represented by some smooth form and y = \overline{F} is an equivariant hermitian sheaf. Again let \overline{ξ.} be a resolution of i_{∗}\overline{F} on X, then

i_{∗}(i^{∗}x · y) = αTd_{g}^{−1}(\overline{N}_{X/X_{g}})ch_{g}(\overline{F}) = α[dd^{c}T_{g}(\overline{ξ.}) + ∑(−1)^{k}ch_{g}(\overline{ξ}_{k})]

which is exactly x · i_{∗}y. Now assume that x = \overline{E} is an equivariant hermitian vector bundle and y = α is represented by some singular current, then

i_{∗}(i^{∗}x · y) = i_{∗}(ch_{g}(\overline{E})α) = ch_{g}(\overline{E})αTd_{g}^{−1}(\overline{N}_{X/X_{g}})

which is exactly x · i_{∗}y. Finally, if x is represented by some smooth form and y is represented by some singular current then their product is well-defined and i_{∗}(i^{∗}x · y) is obviously equal to x · i_{∗}y. Note that i_{∗} and i^{∗} are group homomorphisms, so we may conclude the projection formula from its correctness on generators. This completes the proof.

Remark 5.4. Lemma 5.3 implies that i_{∗} is even a homomorphism of R(μ_{n})- modules so that it induces a homomorphism between arithmetic G_{0}-groups after taking localization.

With Remark 5.4, we may formulate the arithmetic concentration theorem as follows.

Theorem 5.5. The embedding morphism i_{∗} : K̂_{0}(X_{μ_{n}}, μ_{n}, S)_{ρ} → K̂_{0}(X, μ_{n}, S)_{ρ} is an isomorphism if X is regular. In this case, the inverse morphism of i_{∗} is given by λ_{−1}^{−1}(\overline{N}_{X/X_{μn}}^{∨}) · i^{∗} where N_{X/X_{μn}} is the normal bundle of i(X_{μ_{n}}) in X.

Before we give the proof of this concentration theorem, we recall a crucial lemma as follows.

Lemma 5.6. Let \overline{η} be an equivariant hermitian vector bundle on X_{μ_{n}}. Assume that \overline{ξ.} is an equivariant hermitian resolution of i_{∗}\overline{η} on X whose metrics satisfy Bismut assumption (A). Then the equality

λ_{−1}(\overline{N}_{X/X_{μn}}^{∨}) · \overline{η} − ∑_{j}(−1)^{j}i^{∗}(\overline{ξ}_{j}) = T_{g}(\overline{ξ}.)

holds in the group K̂_{0}(X_{μ_{n}}, μ_{n}, S).

Proof. This is [29, Lemma 5.13].

Proof. (of Theorem 5.5) Denote by U the complement of X_{μ_{n}} in X, then j : U ↪ X is a μ_{n}-equivariant open subscheme of X whose fixed point set is empty. We consider the following double complex

\xymatrix{Ũ(X_{μ_{n}}, S)_{ρ} \ar[r]^{i_{∗}} \ar[d]^{a} & Ũ(X_{μ_{n}}, S)_{ρ} \ar[r]^{j^{∗}} \ar[d]^{a} & Ũ(U_{μ_{n}}, ∅)_{ρ} \ar[r] \ar[d]^{a} & 0 \\ K̂_{0}(X_{μ_{n}}, μ_{n}, S)_{ρ} \ar[r]^{i_{∗}} \ar[d]^{π} & K̂_{0}(X, μ_{n}, S)_{ρ} \ar[r]^{j^{∗}} \ar[d]^{π} & K̂_{0}(U, μ_{n}, ∅)_{ρ} \ar[r] \ar[d]^{π} & 0 \\ K_{0}(X_{μ_{n}}, μ_{n})_{ρ} \ar[r]^{i_{∗}} \ar[d] & K_{0}(X, μ_{n})_{ρ} \ar[r]^{j^{∗}} \ar[d] & K_{0}(U, μ_{n})_{ρ} \ar[r] \ar[d] & 0 \\ 0 & 0 & 0}

whose first and second columns are both exact sequences according to Remark 4.4 (iii). For the third column, K_{0}(U, μ_{n})_{ρ} is equal to 0 by [31, (2.1.3)], Ũ(U_{μ_{n}}, ∅)_{ρ} is also equal to 0 since U_{μ_{n}} is empty. Then from Remark 4.4 (iii) we know that K̂_{0}(U, μ_{n}, ∅)_{ρ} is equal to 0. We claim that i_{∗} : K̂_{0}(X_{μ_{n}}, μ_{n}, S)_{ρ} → K̂_{0}(X, μ_{n}, S)_{ρ} is surjective. Indeed, for any element x ∈ K̂_{0}(X, μ_{n}, S)_{ρ} we may find an element y ∈ K̂_{0}(X_{μ_{n}}, μ_{n}, S)_{ρ} such that i_{∗}π(y) = π(x) because the third line is exact. This means x − i_{∗}(y) is in the kernel of π, so there exists an element α ∈ Ũ(X_{μ_{n}}, S)_{ρ} such that α = x − i_{∗}(y) in K̂_{0}(X, μ_{n}, S)_{ρ}. Set β = αTd_{g}(\overline{N}), we get i_{∗}(y + β) = i_{∗}(y) + α = x in K̂_{0}(X, μ_{n}, S)_{ρ}. Hence, i_{∗} is surjective. We now prove that the embedding morphism i_{∗} : K̂_{0}(X_{μ_{n}}, μ_{n}, S)_{ρ} → K̂_{0}(X, μ_{n}, S)_{ρ} is really an isomorphism by constructing its inverse morphism. Let ω be an element in Ũ(X_{μ_{n}}, S), by definition we have

λ_{−1}^{−1}(\overline{N}_{X/X_{μn}}^{∨}) · i^{∗}i_{∗}(ω) =λ_{−1}^{−1}(\overline{N}_{X/X_{μn}}^{∨}) · ωTd_{g}^{−1}(\overline{N}_{X/X_{g}})
=ch_{g}(λ_{−1}^{−1}(\overline{N}_{X/X_{g}}^{∨}))ωTd_{g}^{−1}(\overline{N}_{X/X_{g}})
=ω.

Let \overline{η} be an equivariant hermitian vector bundle on X_{μ_{n}} and assume that \overline{ξ.} is an equivariant hermitian resolution of i_{∗}\overline{η} on X whose metrics satisfy Bismut assumption (A), then by the definition of the embedding morphism i_{∗} and Lemma 5.6 we have

λ_{−1}^{−1}(\overline{N}_{X/X_{μn}}^{∨}) · i^{∗}i_{∗}(\overline{η}) =λ_{−1}^{−1}(\overline{N}_{X/X_{μn}}^{∨}) · i^{∗}(∑_{k}(−1)^{k}\overline{ξ}_{k} + T_{g}(\overline{ξ.})) = \overline{η}.

So the inverse morphism of i_{∗} is of the form λ_{−1}^{−1}(\overline{N}_{X/X_{μn}}^{∨}) · i^{∗} and we are done.

6 A Lefschetz fixed point formula for singular arithmetic schemes with smooth generic fibres

6.1 The statement

We formulate in this subsection the statement of our main theorem, a singular Lefschetz fixed point formula for equivariant arithmetic schemes with smooth generic fibres. Its proof will be given in next two subsections. Let f : X → Y be a μ_{n}-equivariant morphism between two arithmetic schemes with smooth generic fibres, which is smooth on the complex numbers. This morphism f is automatically projective and hence proper, according to the definition of equivariant arithmetic scheme. Suppose that f factors through some regular equivariant arithmetic scheme Z. More precisely, our assumption is that there exist an equivariant closed immersion i : X ↪ Z and an equivariant morphism h : Z → Y such that f = h ∘ i and h is also smooth on the complex numbers. Moreover, we shall assume that the μ_{n}-action on Y is trivial. Let η be an equivariant coherent sheaf on X, then there exists a bounded complex of equivariant vector bundles which provides a resolution of i_{∗}η on Z because Z is regular. Since any two equivariant resolution of i_{∗}η can be dominated by a third one, the symbol Tor_{O_{Z}}^{k} (i_{∗}η, O_{Z_{μn}}) does make sense. We choose arbitrary μ_{n}-invariant Kähler forms ω^{Z} and ω^{X} on Z(ℂ) and X(ℂ) respectively, the Kähler form ω^{X} is not necessarily the Kähler form induced by ω^{Z}. The Kähler form on X(ℂ) induced by ω^{Z} will be denoted by ω_{X}^{Z}. Denote by N the normal bundle of i_{ℂ}(X(ℂ)) in Z(ℂ), we endow it with the quotient metric provided that T X(ℂ) carries the Kähler metric corresponding to ω_{X}^{Z}. Let \overline{F} be the non-zero degree part of \overline{N}, then by [13, Exp. VII, Lem. 2.4 and Prop. 2.5] for any equivariant hermitian sheaf \overline{η} on X there exists a canonical isomorphism on X_{g}

Tor_{O_{Z}}^{k} (i_{∗}η, O_{Z_{μn}})_{ℂ} ≅ ∧^{k}F^{∨} ⊗ η_{ℂ} |_{X_{g}}

which is equivariant. This means we may endow Tor_{O_{Z}}^{k} (i_{∗}η, O_{Z_{μn}})_{ℂ} with a hermitian metric induced by the metrics on F and η so that it becomes an equivariant hermitian sheaf on X_{μ_{n}}. Moreover, we know that the hermitian vector bundle \overline{F} fits the following exact sequence

(\overline{F}, ω^{X}) : 0 → \overline{N}_{X/X_{g}} → \overline{N}_{Z/Z_{g}} → \overline{F} → 0

where N_{Z/Z_{g}} admits the quotient metric associated to ω^{Z} and N_{X/X_{g}} admits the quotient metric associated to ω^{X}. Similarly, we shall denote by (\overline{F}, ω_{X}^{Z}) the hermitian exact sequence \overline{F} whose metric on N_{X/X_{g}} is induced by ω_{X}^{Z}. The push-forward homomorphism from the arithmetic G_{0}-group Ĝ_{0}(X, μ_{n}, ∅) to Ĝ_{0}(Y, μ_{n}, ∅) with respect to the Kähler form ω^{X} is denoted by f_{∗} as usual. The push-forward homomorphism from Ĝ_{0}(X_{μ_{n}}, μ_{n}, ∅) to Ĝ_{0}(Y, μ_{n}, ∅) with respect to the Kähler form ω_{X}^{Z} will be denoted by f_{μ_{n}}^{Z}_{∗}.

Write T̃d(T f_{g}, ω^{X}, ω_{X}^{Z}) for the secondary characteristic class of the exact sequence

\xymatrix{0 \ar[r] & (T f_{g}, ω^{X}) \ar[r]^{Id} & (T f_{g}, ω_{X}^{Z}) \ar[r] & 0 \ar[r] & 0}

where the middle term carries the metric induced by ω_{X}^{Z} and the sub term carries the metric induced by ω^{X}. Then the singular Lefschetz fixed point formula for equivariant arithmetic schemes with smooth generic fibres can be formulated as follows.

Theorem 6.1. Let notations and assumptions be as above. Then for any equivariant hermitian sheaf \overline{η} on X, the equality

f_{∗}(\overline{η}) =f_{μ_{n}}^{Z}_{∗}(i_{μ_{n}}^{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨})) · ∑_{k}(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}))

+ ∫_{X_{g}/Y} Td(T f_{g}, ω^{X})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω^{X})Td_{g}^{−1}(\overline{F})
− ∫_{X_{g}/Y} Td_{g}(T f)ch_{g}(η)R_{g}(N_{X/X_{g}})
+ ∫_{X_{g}/Y} T̃d(T f_{g}, ω^{X}, ω_{X}^{Z})ch_{g}(\overline{η})Td_{g}(\overline{N}_{Z/Z_{g}})Td_{g}^{−1}(\overline{F})

holds in the group Ĝ_{0}(Y, μ_{n}, ∅)_{ρ}.

Remark 6.2. This arithmetic Lefschetz fixed point formula was inspired by [31, Théorème 3.5].

6.2 Equivariant arithmetic G_{0}-theoretic vanishing theorem

The central actor in the proof of Theorem 6.1 is the following vanishing theorem in equivariant arithmetic G_{0}-theory, which can be viewed as a translation of Theorem 3.1.

Theorem 6.3. Let notations and assumptions be as in last subsection. Let \overline{η} be an equivariant hermitian sheaf on X, and let

\overline{Ψ} : 0 → \overline{ξ}_{m} → ⋯ → \overline{ξ}_{1} → \overline{ξ}_{0} → i_{∗}\overline{η} → 0

be a resolution of i_{∗}\overline{η} by equivariant hermitian vector bundles on Z. Denote by h_{μ_{n}}_{∗} the push-forward homomorphism from K̂_{0}(Z_{μ_{n}}, μ_{n}, N_{g,ℝ}^{∨} \ {0}) to Ĝ_{0}(Y, μ_{n}, ∅) with respect to the Kähler form ω^{Z}. Then the formula

f_{μ_{n}}^{Z}_{∗}(∑(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}})) − h_{μ_{n}}_{∗}(∑(−1)^{k}(\overline{ξ}_{k} |_{Z_{μn}}))
= ∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td(\overline{T h_{g}}) + ∫_{X_{g}/Y} Td(T f_{g})Td_{g}^{−1}(F)ch_{g}(η)R(N_{g})
+ ∫_{X_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})

holds in Ĝ_{0}(Y, μ_{n}, ∅).

Proof. Following the same arguments given in the proof of Lemma 3.9, we may show that the deformation to the normal cone W (i) admits an equivariant hermitian very ample invertible sheaf \overline{L} which is relative to the morphism l : W (i) → Y. By Theorem 3.1 and the fact that L is very ample, we conclude that there exists an integer k_{0} > 0 such that for n ≥ k_{0}, L^{⊗n} is l-acyclic and δ(\overline{Ψ}(n)_{ℂ}) = 0. Then l factors through an equivariant projective space bundle ℙ(E) where E is locally free of rank r + 1 on Y and l_{∗}L^{⊗k_{0}} is an equivariant quotient of E. Denote by p : ℙ(E) → Y the canonical projection. On P := ℙ(E), we have a canonical exact sequence

H : 0 → O_{P} → p^{∗}(E^{∨})(1) → ⋯ → p^{∗}(∧^{r+1}E^{∨})(r + 1) → 0.

Restricting this sequence to Z, we obtain an exact sequence of exact sequences

0 → Ψ → Ψ ⊗ h^{∗}(E^{∨})(1) → ⋯ → Ψ ⊗ h^{∗}(∧^{r+1}E^{∨})(r + 1) → 0.

Endow E with any μ_{n}(ℂ)-invariant hermitian metric. We claim that the assumption that Theorem 6.3 holds for \overline{Ψ} ⊗ h^{∗}(∧^{n}\overline{E}^{∨})(n) with n ≥ 1 implies that it holds for \overline{Ψ}. In fact, since H is an exact sequence of flat modules, for any k ≥ 0 we have the following exact sequence on X_{μ_{n}}

0 → Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}) →Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}) ⊗ f_{μ_{n}}^{∗}(\overline{E}^{∨})(1) → ⋯
→Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}) ⊗ f_{μ_{n}}^{∗}(∧^{r+1}\overline{E}^{∨})(r + 1) → 0.

We compute

f_{μ_{n}}^{Z}_{∗}(Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}))
=f_{μ_{n}}^{Z}_{∗}(− ∑_{j=1}^{r+1}(−1)^{j}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}) ⊗ f_{μ_{n}}^{∗}(∧^{j}\overline{E}^{∨})(j))

+ ∫_{X_{g}/Y} Td(T f_{g}, ω_{X}^{Z})ch_{g}(∧^{k}\overline{F}^{∨})ch_{g}(\overline{η})(−1)^{r+1}c̃h_{g}(\overline{H})

and

∑_{k=0}^{m}(−1)^{k}h_{μ_{n}}_{∗}(\overline{ξ}_{k} |_{Z_{μn}})

= ∑_{k=0}^{m}(−1)^{k}h_{μ_{n}}_{∗}(− ∑_{j=1}^{r+1}(−1)^{j}\overline{ξ}_{k} |_{Z_{μn}} ⊗h_{μ_{n}}^{∗}(∧^{j}\overline{E}^{∨})(j))

+ ∑_{k=0}^{m}(−1)^{k} ∫_{Z_{g}/Y} Td(\overline{T h_{g}})ch_{g}(\overline{ξ}_{k})(−1)^{r+1}c̃h_{g}(\overline{H}).

Singular Fixed Point Formula of Lefschetz Type 1097

Moreover, we have ∫_{X_{g}/Y} Td(T f_{g})ch_{g}(Tor_{O_{Z}}^{k} (i_{∗}η, O_{Z_{μn}}))R(N_{g})

= ∫_{X_{g}/Y} − ∑_{j=1}^{r+1}(−1)^{j}ch_{g}(Tor_{O_{Z}}^{k} (i_{∗}η, O_{Z_{μn}}) ⊗ f_{μ_{n}}^{∗}(∧^{j}E^{∨})(j))R(N_{g})Td(T f_{g})

and

∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td(\overline{T h_{g}})

= ∫_{Z_{g}/Y} Td(\overline{T h_{g}})\{δ_{X_{g}}Td_{g}^{−1}(\overline{N})ch_{g}(\overline{η})(−1)^{r+1}c̃h_{g}(\overline{H})

− ∑_{k=0}^{m}(−1)^{k}ch_{g}(\overline{ξ}_{k})(−1)^{r+1}c̃h_{g}(\overline{H}) − ∑_{j=1}^{r+1}(−1)^{j}T_{g}(\overline{ξ.})ch_{g}(h_{μ_{n}}^{∗}(∧^{j}\overline{E}^{∨})(j))\}

by the double complex formula of equivariant Bott-Chern singular currents. At last, we also have

∫_{X_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})

= ∫_{X_{g}/Y} \{dd^{c}(−1)^{r+1}c̃h_{g}(\overline{H})ch_{g}(\overline{η}) − ∑_{j=1}^{r+1}(−1)^{j}ch_{g}(\overline{η} ⊗ f^{∗}(∧^{j}\overline{E}^{∨})(j))\}
· Td_{g}^{−1}(\overline{N})T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})
= − ∫_{X_{g}/Y} (−1)^{r+1}c̃h_{g}(\overline{H})ch_{g}(\overline{η}) · \{Td_{g}^{−1}(\overline{N})Td(\overline{T h_{g}})
− Td(T f_{g}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F})\}

− ∫_{X_{g}/Y} ∑_{j=1}^{r+1}(−1)^{j}ch_{g}(\overline{η} ⊗ f^{∗}(∧^{j}\overline{E}^{∨})(j))Td_{g}^{−1}(\overline{N})T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}}).

Gathering all these computations above and using our assumption, we get f_{μ_{n}}^{Z}_{∗}(∑(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}})) − h_{μ_{n}}_{∗}(∑(−1)^{k}\overline{ξ}_{k} |_{Z_{μn}})

− ∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td(\overline{T h_{g}}) − ∫_{X_{g}/Y} Td(T f_{g})Td_{g}^{−1}(F)ch_{g}(η)R(N_{g})
− ∫_{X_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})
= ∫_{X_{g}/Y} Td(T f_{g}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F})ch_{g}(\overline{η})(−1)^{r+1}c̃h_{g}(\overline{H})

Documenta Mathematica 15 (2010) 1049–1108

− ∑_{k=0}^{m}(−1)^{k} ∫_{Z_{g}/Y} Td(\overline{T h_{g}})ch_{g}(\overline{ξ}_{k})(−1)^{r+1}c̃h_{g}(\overline{H})

− ∫_{X_{g}/Y} Td(\overline{T h_{g}})Td_{g}^{−1}(\overline{N})ch_{g}(\overline{η})(−1)^{r+1}c̃h_{g}(\overline{H})

+ ∑_{k=0}^{m}(−1)^{k} ∫_{Z_{g}/Y} Td(\overline{T h_{g}})ch_{g}(\overline{ξ}_{k})(−1)^{r+1}c̃h_{g}(\overline{H})

+ ∫_{X_{g}/Y} (−1)^{r+1}c̃h_{g}(\overline{H})ch_{g}(\overline{η})\{Td_{g}^{−1}(\overline{N})Td(\overline{T h_{g}})

− Td(T f_{g}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F})\}

which vanishes. This ends the proof of our claim. By the construction of the projective space bundle P, we have already known that δ(\overline{Ψ}(n)_{ℂ}) vanishes from n = 1 to n = r + 1. Moreover, according to the projection formula of higher direct images, the operation of tensoring with the element l^{∗}(∧^{n}\overline{E}^{∨}) doesn’t change the property of l-acyclicity. Hence we also have δ(\overline{Ψ} ⊗ h^{∗}(∧^{n}\overline{E}^{∨})(n)_{ℂ}) = 0. By the generating relations and the definition of push-forward morphisms of arithmetic G_{0}-groups, this is equivalent to say that Theorem 6.3 holds for \overline{Ψ} ⊗ h^{∗}(∧^{n}\overline{E}^{∨})(n). Therefore the equality in the statement of this theorem follows from our claim before.

Corollary 6.4. Let notations and assumptions be as in Theorem 6.3, and let x be any element in K̂_{0}(Z, μ_{n}, ∅)_{ρ}. Then the formula

f_{μ_{n}}^{Z}_{∗}(i^{∗}x |_{X_{μn}} · ∑(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}))
− h_{μ_{n}}_{∗}(x |_{Z_{μn}} · ∑(−1)^{k}(\overline{ξ}_{k} |_{Z_{μn}}))
= ∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td(\overline{T h_{g}})ch_{g}(x)
+ ∫_{X_{g}/Y} Td(T f_{g})Td_{g}^{−1}(F)ch_{g}(η)R(N_{g})ch_{g}(i^{∗}x)
+ ∫_{X_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})ch_{g}(i^{∗}x)T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})

holds in Ĝ_{0}(Y, μ_{n}, ∅)_{ρ}.

Proof. If x = \overline{E} is an equivariant hermitian vector bundle on Z, then \overline{ξ.} ⊗ \overline{E} provides a resolution of i_{∗}(\overline{η}⊗i^{∗}\overline{E}). Hence the formula follows from Theorem 6.3 in this case. If x = α is represented by some smooth form, then

f_{μ_{n}}^{Z}_{∗}(i^{∗}x |_{X_{μn}} · ∑(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}))

= ∫_{Z_{g}/Y} Td(T f_{g}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F})ch_{g}(\overline{η})δ_{X_{g}}α

and

h_{μ_{n}}_{∗}(x |_{Z_{μn}} · ∑(−1)^{k}(\overline{ξ}_{k} |_{Z_{μn}})) = ∫_{Z_{g}/Y} Td(\overline{T h_{g}})α ∑(−1)^{k}ch_{g}(\overline{ξ}_{k}).

Moreover, by the definition of ch_{g}(x) we have

∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td(\overline{T h_{g}})ch_{g}(x) = ∫_{Z_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})δ_{X_{g}}Td(\overline{T h_{g}})α
− ∫_{Z_{g}/Y} ∑(−1)^{k}ch_{g}(\overline{ξ}_{k})Td(\overline{T h_{g}})α

and

∫_{X_{g}/Y} Td(T f_{g})Td_{g}^{−1}(F)ch_{g}(η)R(N_{g})ch_{g}(i^{∗}x) = 0.

Finally, using the definition of T̃d we compute

∫_{X_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})ch_{g}(i^{∗}x)T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})
= ∫_{Z_{g}/Y} Td(T f_{g}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F})ch_{g}(\overline{η})δ_{X_{g}}α
− ∫_{Z_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})δ_{X_{g}}Td(\overline{T h_{g}})α.

Gathering all computations above, we know that the formula still holds for x which is represented by smooth form. Since both two sides are additive, we are done.

Corollary 6.5. Let notations and assumptions be as in Theorem 6.3, and let y be any element in K̂_{0}(Z_{μ_{n}}, μ_{n}, ∅)_{ρ}. Then the formula

f_{μ_{n}}^{Z}_{∗}(i_{μ_{n}}^{∗}y · ∑(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}})) − h_{μ_{n}}_{∗}(y · ∑(−1)^{k}(\overline{ξ}_{k} |_{Z_{μn}}))

= ∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td(\overline{T h_{g}})ch_{g}(y)
+ ∫_{X_{g}/Y} Td(T f_{g})Td_{g}^{−1}(F)ch_{g}(η)R(N_{g})ch_{g}(i_{μ_{n}}^{∗}y)
+ ∫_{X_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})ch_{g}(i_{μ_{n}}^{∗}y)T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})

holds in Ĝ_{0}(Y, μ_{n}, ∅)_{ρ}.

Proof. Provided Corollary 6.4, it is enough to prove that for any y ∈ K̂_{0}(Z_{μ_{n}}, μ_{n}, ∅)_{ρ} there exists an element x ∈ K̂_{0}(Z, μ_{n}, ∅)_{ρ} such that i_{Z}^{∗}x = y.

Here i_{Z} stands for the inclusion Z_{μ_{n}} ↪ Z. Actually, set x = i_{Z}_{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨})· y), we have

i_{Z}^{∗}x = i_{Z}^{∗}i_{Z}_{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨}) · y) = λ_{−1}(\overline{N}_{Z/Z_{μn}}^{∨}) · λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨}) · y = y.

This follows from our arithmetic concentration theorem.

6.3 Proof of the fixed point formula

In this subsection, we provide a complete proof of Theorem 6.1 the singular Lefschetz fixed point formula. Before that, we need to translate Bismut-Ma’s immersion formula to an arithmetic G_{0}-theoretic version. That’s the following.

Theorem 6.6. Let notations and assumptions be as in Section 6.1. Assume that \overline{η} is an equivariant hermitian sheaf on X and \overline{ξ.} is a bounded complex of equivariant hermitian vector bundles providing a resolution of i_{∗}\overline{η} on Z whose metrics satisfy Bismut assumption (A). Then the equality

f_{∗}^{Z}(\overline{η}) − ∑_{j=0}^{m}(−1)^{j}h_{∗}(\overline{ξ}_{j}) = ∫_{X_{g}/Y} ch_{g}(η)R_{g}(N)Td_{g}(T f)

+ ∫_{Z_{g}/Y} T_{g}(\overline{ξ}.)Td_{g}(\overline{T h})
+ ∫_{X_{g}/Y} ch_{g}(\overline{η})T̃d_{g}((T f, ω_{X}^{Z}), \overline{T h} |_{X})Td_{g}^{−1}(\overline{N})

holds in Ĝ_{0}(Y, μ_{n}, ∅).

Proof. We first suppose that η and ξ. are all acyclic, then the verification follows rather directly from the generating relations of arithmetic G_{0}-theory. In fact

f_{∗}^{Z}(\overline{η}) − ∑_{j=0}^{m}(−1)^{j}h_{∗}(\overline{ξ}_{j}) =\overline{f_{∗}η} − T_{g}(ω_{X}^{Z}, h^{η}) − (∑_{j=0}^{m}(−1)^{j}(\overline{h_{∗}ξ_{j}} − T_{g}(ω^{Z}, h^{ξ_{j}})))

=c̃h_{g}(h_{∗}\overline{Ξ}) − T_{g}(ω_{X}^{Z}, h^{η}) + ∑_{j=0}^{m}(−1)^{j}T_{g}(ω^{Z}, h^{ξ_{j}}).

And the right-hand side of the last equality is exactly the left-hand side of Bismut-Ma’s immersion formula. We emphasize again that to simplify the right-hand side of Bismut-Ma’s immersion formula, we have used an Atiyah-Segal-Singer type formula for immersion

i_{g}_{∗}(Td_{g}^{−1}(N)ch_{g}(x)) = ch_{g}(i_{∗}(x)).

To remove the condition of acyclicity, one can use the argument which is essentially the same as in the proof of Theorem 6.3. Since it doesn’t use any new techniques, we omit it here.

Definition 6.7. The inclusion i : X ↪ Z induces an embedding morphism

i_{∗} : Ĝ_{0}(X, μ_{n}, ∅) → K̂_{0}(Z, μ_{n}, N_{g,ℝ}^{∨} \ {0})

which is defined as follows. (i). For every μ_{n}-equivariant hermitian sheaf \overline{η} on X, suppose that \overline{ξ.} is a resolution of i_{∗}\overline{η} on Z whose metrics satisfy Bismut assumption (A), i_{∗}[\overline{η}] = ∑_{k}(−1)^{k}[\overline{ξ}_{k}] + T_{g}(\overline{ξ.}). (ii). For every α ∈ Ũ(X_{μ_{n}}, ∅), i_{∗}α = αTd_{g}^{−1}(\overline{N})δ_{X_{g}}.

Remark 6.8. Similar to Theorem 5.2 and Lemma 5.3, one can prove that the embedding morphism is a well-defined homomorphism of R(μ_{n})-modules.

Proof. (of Theorem 6.1) We first prove that this fixed point formula holds when ω^{X} is equal to ω_{X}^{Z}, namely the Kähler metric on X(ℂ) is induced by the Kähler metric on Z(ℂ). By Theorem 6.6 and Definition 6.7, we have the following equality

f_{∗}^{Z}(\overline{η}) =h_{∗}i_{∗}(\overline{η}) + ∫_{X_{g}/Y} ch_{g}(η)R_{g}(N)Td_{g}(T f)
+ ∫_{X_{g}/Y} ch_{g}(\overline{η})T̃d_{g}((T f, ω_{X}^{Z}), \overline{T h} |_{X})Td_{g}^{−1}(\overline{N})

which holds in Ĝ_{0}(Y, μ_{n}, ∅). Now we claim that for any element y ∈ K̂_{0}(Z_{μ_{n}}, μ_{n}, N_{g,ℝ}^{∨} \ {0})_{ρ}, we have

h_{μ_{n}}_{∗}(y) − h_{∗}i_{Z}_{∗}(y) = h_{μ_{n}}_{∗}(y · R_{g}(N_{Z/Z_{μn}})).

Since all morphisms are homomorphisms of R(μ_{n})-modules, we can only consider the generators of K̂_{0}(Z_{μ_{n}}, μ_{n}, N_{g,ℝ}^{∨} \ {0}). Indeed, by applying Theorem 6.6 to the closed immersion i_{Z}, for any equivariant hermitian vector bundle \overline{E} on Z_{μ_{n}} we have

h_{μ_{n}}_{∗}(\overline{E}) − h_{∗}i_{Z}_{∗}(\overline{E}) = ∫_{Z_{g}/Y} ch_{g}(E)R_{g}(N_{Z/Z_{μn}})Td_{g}(T h_{g})
=h_{μ_{n}}_{∗}(ch_{g}(E)R_{g}(N_{Z/Z_{μn}})).

The first equality holds because the exact sequence

0 → \overline{T h_{g}} → \overline{T h} |_{Z_{g}}→ \overline{N}_{Z/Z_{g}} → 0

is orthogonally split on Z_{g} so that T̃d_{g}(\overline{T h_{g}}, \overline{T h} |_{Z_{g}}) = 0. The second equality follows from [21, Lemma 7.3] and the fact that ch_{g}(E)R_{g}(N_{Z/Z_{μn}}) is dd^{c}-closed. On the other hand, let α be an element in Ũ(Z_{μ_{n}}, N_{g,ℝ}^{∨} \ {0}), we have

h_{μ_{n}}_{∗}(α) − h_{∗}i_{Z}_{∗}(α) = ∫_{Z_{g}/Y} αTd_{g}(\overline{T h_{g}}) − ∫_{Z_{g}/Y} αTd_{g}^{−1}(\overline{N}_{Z/Z_{μn}})Td_{g}(\overline{T h})
= ∫_{Z_{g}/Y} αTd_{g}^{−1}(\overline{N}_{Z/Z_{μn}})dd^{c}T̃d_{g}(\overline{T h_{g}}, \overline{T h} |_{Z_{g}}) = 0.

Combing these two computations, we get our claim by linear extension. Now using arithmetic concentration theorem, we compute

h_{∗}i_{∗}(\overline{η}) =h_{∗}i_{Z}_{∗}i_{Z} _{∗}^{−1}i_{∗}(\overline{η})
=h_{μ_{n}}_{∗}(i_{Z} _{∗}^{−1}i_{∗}(\overline{η}) · (1 − R_{g}(N_{Z/Z_{μn}})))
=h_{μ_{n}}_{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨}) · i_{Z}^{∗}i_{∗}(\overline{η}) · (1 − R_{g}(N_{Z/Z_{μn}})))
=h_{μ_{n}}_{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨}) · \{∑_{k}(−1)^{k}(\overline{ξ}_{k} |_{Z_{μn}})

+ T_{g}(\overline{ξ.})\} · (1 − R_{g}(N_{Z/Z_{μn}})))

=h_{μ_{n}}_{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨}) · ∑_{k}(−1)^{k}(\overline{ξ}_{k} |_{Z_{μn}}))

+ h_{μ_{n}}_{∗}(Td_{g}(\overline{N}_{Z/Z_{μn}})T_{g}(\overline{ξ.}))

− h_{μ_{n}}_{∗}(Td_{g}(N_{Z/Z_{μn}}) ∑_{k}(−1)^{k}ch_{g}(ξ_{k})R_{g}(N_{Z/Z_{μn}})).

According to Corollary 6.5, by setting y = λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨}), we compute

h_{μ_{n}}_{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨}) · ∑_{k}(−1)^{k}(\overline{ξ}_{k} |_{Z_{μn}}))

=f_{μ_{n}}^{Z}_{∗}(i_{μ_{n}}^{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨})) · ∑(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}))

− ∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td(\overline{T h_{g}})Td_{g}(\overline{N}_{Z/Z_{g}})

− ∫_{X_{g}/Y} Td(T f_{g})Td_{g}^{−1}(F)ch_{g}(η)R(N_{g})Td_{g}(N_{Z/Z_{g}})

− ∫_{X_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})Td_{g}(\overline{N}_{Z/Z_{g}})T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})

=f_{μ_{n}}^{Z}_{∗}(i_{μ_{n}}^{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨})) · ∑(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}))

− ∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td_{g}(\overline{T h}) − ∫_{X_{g}/Y} Td_{g}(T f)ch_{g}(η)R(N_{g})

− ∫_{X_{g}/Y} ch_{g}(\overline{η})Td_{g}^{−1}(\overline{N})Td_{g}(\overline{N}_{Z/Z_{g}})T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}}).

Here we have used various relations of character forms or characteristic classes

Documenta Mathematica 15 (2010) 1049–1108

Singular Fixed Point Formula of Lefschetz Type 1103

arising from the following double complex

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & (T f_{g}, ω_{X}^{Z}) \ar[r] \ar[d] & \overline{T h_{g}} \ar[d] \ar[r] & _{\overline{N}_{g}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & (T f, ω_{X}^{Z}) \ar[r] \ar[d] & \overline{T h} \ar[d] \ar[r] & \overline{N} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & (N_{X/X_{g}}, ω_{X}^{Z}) \ar[r] \ar[d] & _{\overline{N}_{Z/Z_{g}}} \ar[r] \ar[d] & \overline{F} \ar[d] \ar[r] & 0 \\  & 0 & 0 & 0}

whose columns are all orthogonally split. Also, for this double complex, one may use Example 2.3 (iv) to compute that

T̃d_{g}((T f, ω_{X}^{Z}), \overline{T h} |_{X}) =T̃d_{g}(\overline{F}, ω_{X}^{Z})Td(\overline{N}_{g})Td(T f_{g}, ω_{X}^{Z})
+ T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})Td_{g}(\overline{N}_{Z/Z_{g}}).  (11)

We deduce from (11) that

∫_{X_{g}/Y} ch_{g}(\overline{η})T̃d_{g}((T f, ω_{X}^{Z}), \overline{T h} |_{X})Td_{g}^{−1}(\overline{N})

= ∫_{X_{g}/Y} ch_{g}(\overline{η})T̃d((T f_{g}, ω_{X}^{Z}), \overline{T h_{g}} |_{X_{g}})Td_{g}^{−1}(\overline{N})Td_{g}(\overline{N}_{Z/Z_{g}})
+ ∫_{X_{g}/Y} ch_{g}(\overline{η})T̃d_{g}(\overline{F})Td_{g}^{−1}(\overline{F})Td(T f_{g}, ω_{X}^{Z}).  (12)

Moreover, we have

h_{μ_{n}}_{∗}(Td_{g}(\overline{N}_{Z/Z_{μn}})T_{g}(\overline{ξ.})) = ∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td(\overline{T h_{g}})Td_{g}(\overline{N}_{Z/Z_{g}})
= ∫_{Z_{g}/Y} T_{g}(\overline{ξ.})Td_{g}(\overline{T h})  (13)

and

h_{μ_{n}}_{∗}(Td_{g}(N_{Z/Z_{μn}}) ∑_{k}(−1)^{k}ch_{g}(ξ_{k})R_{g}(N_{Z/Z_{μn}}))

= ∫_{Z_{g}/Y} Td_{g}(N_{Z/Z_{μn}})δ_{X_{g}}ch_{g}(η)Td_{g}^{−1}(N)R_{g}(N_{Z/Z_{g}})Td(T h_{g})

Documenta Mathematica 15 (2010) 1049–1108

= ∫_{X_{g}/Y} Td_{g}(N_{X/X_{g}})Td_{g}(F)ch_{g}(η)Td_{g}^{−1}(N)
· [R_{g}(N_{X/X_{g}}) + R_{g}(N) − R(N_{g})]Td(T f_{g})Td(N_{g})
= ∫_{X_{g}/Y} Td_{g}(T f)ch_{g}(η)[R_{g}(N_{X/X_{g}}) + R_{g}(N) − R(N_{g})].  (14)

Gathering (12), (13) and (14) we finally get

f_{∗}^{Z}(\overline{η}) =f_{μ_{n}}^{Z}_{∗}(i_{μ_{n}}^{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨})) · ∑_{k}(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}))

+ ∫_{X_{g}/Y} Td(T f_{g}, ω_{X}^{Z})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F})
− ∫_{X_{g}/Y} Td_{g}(T f)ch_{g}(η)R_{g}(N_{X/X_{g}})

which completes the proof of Theorem 6.1 in the case where the Kähler metric on X(ℂ) is induced by the Kähler metric on Z(ℂ).

In general, in analogy with the notation T̃d(T f_{g}, ω^{X}, ω_{X}^{Z}), we write T̃d_{g}(N_{X/X_{g}}, ω^{X}, ω_{X}^{Z}) for the secondary characteristic class of the exact sequence

\xymatrix{0 \ar[r] & N_{X/X_{g}} \ar[r]^{Id} & N_{X/X_{g}} \ar[r] & 0 \ar[r] & 0}

where the middle term carries the metric induced by ω_{X}^{Z} and the sub term carries the metric induced by ω^{X}. Similarly, we have the notation T̃d_{g}(T f, ω^{X}, ω_{X}^{Z}). Then by applying the argument in the proof of (11) to the double complex

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & (T f_{g}, ω^{X}) \ar[r] \ar[d] & (T f_{g}, ω_{X}^{Z}) \ar[d] \ar[r] & 0 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & (T f, ω^{X}) \ar[r] \ar[d] & (T f, ω_{X}^{Z}) \ar[d] \ar[r] & 0 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & (N_{X/X_{g}}, ω^{X}) \ar[r] \ar[d] & (N_{X/X_{g}}, ω_{X}^{Z}) \ar[r] \ar[d] & 0 \ar[d] \ar[r] & 0 \\  & 0 & 0 & 0}

Singular Fixed Point Formula of Lefschetz Type 1105

We get

T̃d_{g}(T f, ω^{X}, ω_{X}^{Z}) =T̃d(T f_{g}, ω^{X}, ω_{X}^{Z})Td_{g}(N_{X/X_{g}}, ω_{X}^{Z})
+ T̃d_{g}(N_{X/X_{g}}, ω^{X}, ω_{X}^{Z})Td(T f_{g}, ω^{X}).

Moreover, by [30, Proposition 2.8], we obtain that

T̃d_{g}(\overline{F}, ω^{X}) = T̃d_{g}(\overline{F}, ω_{X}^{Z}) + T̃d_{g}(N_{X/X_{g}}, ω^{X}, ω_{X}^{Z})Td_{g}(\overline{F}).

With these two comparison formulae, we can compute

∫_{X_{g}/Y} Td(T f_{g}, ω^{X})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω^{X})Td_{g}^{−1}(\overline{F})

− ∫_{X_{g}/Y} Td(T f_{g}, ω_{X}^{Z})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F})

= ∫_{X_{g}/Y} Td(T f_{g}, ω^{X})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω^{X})Td_{g}^{−1}(\overline{F})

− ∫_{X_{g}/Y} Td(T f_{g}, ω^{X})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F}) + ∫_{X_{g}/Y} Td(T f_{g}, ω^{X})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F}) − ∫_{X_{g}/Y} Td(T f_{g}, ω_{X}^{Z})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω_{X}^{Z})Td_{g}^{−1}(\overline{F})

= ∫_{X_{g}/Y} T̃d(T f_{g}, ω^{X}, ω_{X}^{Z})ch_{g}(\overline{η})

· [Td_{g}(N_{X/X_{g}}, ω_{X}^{Z})Td_{g}(\overline{F}) − Td_{g}(\overline{N}_{Z/Z_{g}})]Td_{g}^{−1}(\overline{F}) + ∫_{X_{g}/Y} Td(T f_{g}, ω^{X})ch_{g}(\overline{η})T̃d_{g}(N_{X/X_{g}}, ω^{X}, ω_{X}^{Z})

= ∫_{X_{g}/Y} ch_{g}(\overline{η})T̃d_{g}(T f, ω^{X}, ω_{X}^{Z})

− ∫_{X_{g}/Y} T̃d(T f_{g}, ω^{X}, ω_{X}^{Z})ch_{g}(\overline{η})Td_{g}(\overline{N}_{Z/Z_{g}})Td_{g}^{−1}(\overline{F}).

At last, using [21, Lemma 7.3], we get the equality f_{∗}(\overline{η}) − f_{∗}^{Z}(\overline{η}) = ∫_{X_{g}/Y} ch_{g}(\overline{η})T̃d_{g}(T f, ω^{X}, ω_{X}^{Z}).

Together with the fact that the other two terms have nothing to do with the

Documenta Mathematica 15 (2010) 1049–1108

choice of the metric ω^{X}, we finally obtain that

f_{∗}(\overline{η}) =f_{μ_{n}}^{Z}_{∗}(i_{μ_{n}}^{∗}(λ_{−1}^{−1}(\overline{N}_{Z/Z_{μn}}^{∨})) · ∑_{k}(−1)^{k}Tor_{O_{Z}}^{k} (i_{∗}\overline{η}, \overline{O}_{Z_{μn}}))

+ ∫_{X_{g}/Y} Td(T f_{g}, ω^{X})ch_{g}(\overline{η})T̃d_{g}(\overline{F}, ω^{X})Td_{g}^{−1}(\overline{F})
− ∫_{X_{g}/Y} Td_{g}(T f)ch_{g}(η)R_{g}(N_{X/X_{g}})
+ ∫_{X_{g}/Y} T̃d(T f_{g}, ω^{X}, ω_{X}^{Z})ch_{g}(\overline{η})Td_{g}(\overline{N}_{Z/Z_{g}})Td_{g}^{−1}(\overline{F})

which ends the proof of Theorem 6.1.

Remark 6.9. Let Y be an affine arithmetic scheme Spec(D), and choose ω^{X} to be the induced Kähler form ω_{X}^{Z}. Then the formula in Theorem 6.1 is the content of [25, Conjecture 5.1].

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Shun Tang

Département de Mathématiques Bâtiment 425 Université Paris-Sud 11 91405 Orsay France shun.tang@math.u-psud.fr