Lobachevskii Journal of Mathematics Vol.4, 1999, 47–70 \begin{matrix}c \\ ◯ \end{matrix}A.L. Onishchik

A.L. Onishchik ON THE CLASSIFICATION OF COMPLEX ANALYTIC SUPERMANIFOLDS

(submitted by B.N.Shapukov)

Abstract. We consider the problem of classification of complex analytic supermanifolds with a given reduction M. As is well known, any such supermanifold is a deformation of its retract, i.e., of a supermanifold (M, O) whose structure sheaf O is the Grassmann algebra over the sheaf of holomorphic sections of a holomorphic vector bundle E → M. Thus, the problem is reduced to the following two classification problems: of holomorphic vector bundles over M and of supermanifolds with a given retract (M, O). We are dealing here with the second problem. By a well-known theorem of Green [9], it can be reduced to the calculation of the 1-cohomology set of a certain sheaf of automorphisms of O. We construct a non-linear resolution of this sheaf giving rise to a non-linear cochain complex whose 1-cohomology is the desired one. For a compact manifold M, we apply Hodge theory to construct a finite-dimensional affine algebraic variety which can serve as a moduli variety for our classification problem; it is analogous to the Kuranishi family of complex structures on a compact manifold (see [6, 7]).

Introduction

In this paper, we apply the machinery of non-abelian cohomology to the problem of classification of complex analytic supermanifolds with a given retract. Namely, we associate to any holomorphic vector bundle E over a complex manifold M a non-abelian cochain complex in the sense of [8, 12] whose 1-cohomology describes the set of all complex analytic supermanifolds such that the corresponding split supermanifold is (M, ⋀ E), where E is the sheaf of holomorphic sections of E. This complex is actually of a type considered by Nijenhuis and Richardson [7] in connection with the deformation theory of algebras, i.e., it is related to a differential graded Lie superalgebra. The corresponding differential Lie superalgebra was introduced in [9];

Work supported in part by the Russian Foundation for Fundamental Research (Grant 98-01-00329).

1991 Mathematical Subject Classification.Primary 18G50, 32C11, 32C35, 58A50. Key words and phrases.Cochain complex, 1-cohomology set, resolution, complex su-

permanifold, split complex supermanifold, retract, tangent sheaf.

it gives the cohomology of M with values in the tangent sheaf of the split supermanifold (M, ⋀ E). For a compact manifold M, our complex gives rise to a finite-dimensional affine algebraic variety which can serve as a moduli variety for our classification problem; it is analogous to the Kuranishi family of complex structures on a compact manifold (see [6]). A characteristic feature of the theory is its global character: it deals with classification of structures rather than with their deformations.

The constructions mentioned above are presented in Sections 2, 3 of this paper. The necessary facts of non-abelian homological algebra can be found in Section 1. All manifolds are assumed to be paracompact.

The author would like to thank the Department of Mathematics of the University Leuven (Belgium) for hospitality in January – March 1998 when the main results of this paper were obtained. The work was completed in November 1998 during the stay in the Humboldt University Berlin, where the author was invited by Sonderforschungsbereich 288 “Differentialgeometrie und Quantenphysik”. The present text is essentially a part of the preprint [13] which contains a more detailed exposition of non-abelian cochain complexes.

1. Preliminaries

1.1. Non-abelian cohomology. First we give some definitions and fix our notation.

A set A with a fixed point e ∈ A will be called a set with distinguished point and will be denoted by (A, e). E.g., a group G will be usually regarded as the set with distinguished point (G, e), where e ∈ G is the unit element. If (A, e) and (B, e) are two sets with distinguished points, then a homomorphism φ : (A, e) → (B, e) is a mapping φ : A → B such that φ(e) = e. An isomorphism is a bijective homomorphism. The kernel and the image of a homomorphism φ : (A, e) → (B, e) are the subsets with distinguished points Ker φ = (φ^{−1}(e), e) ⊂ (A, e) and Im φ = (φ(A), e) ⊂ (B, e). One defines in a usual way an exact sequence of sets with distinguished points.

Let G and Γ be certain groups. We say that G is a Γ-group if an action σ of Γ on G by automorphisms is given. A mapping φ : Γ → G is called a crossed homomorphism if

φ(ab) = φ(a)(σ(a)(φ(b)), a, b ∈ Γ.

Clearly, φ is a homomorphism of of sets with distinguished points, and Ker φ is a subgroup of Γ.

A crossed homomorphism may be interpreted in other terms. Let us consider the group of bijective transformations of the group G normalizing the group l(G) of left translations l_{g}, g ∈ G. This group Hol G is called the holomorph of G. One verifies easily that

Hol G = l(G) ⋊ Aut G.  (1.1)

Any crossed homomorphism φ : Γ → G allows to define an action ρ of Γ on G given by the following homomorphism of the group Γ into Hol G:

ρ(a) = l_{φ(a)} ∘ σ(a), a ∈ Γ.  (1.2)

Clearly, ρ coincides with σ on Ker φ, and Im φ is the orbit ρ(Γ)(e). Conversely, suppose two groups Γ, G and a homomorphism Γ → Hol G be given. By (1.1), we have the decomposition (1.2), where φ : Γ → G and σ : Γ → Aut G are mappings uniquely determined by ρ. One checks easily that σ is a homomorphism, giving a structure of Γ-group on G, and that φ is a crossed homomorphism of Γ into the Γ-group G.

Mutatis mutandi, one extends the above definitions onto sheaves of sets and of groups on a topological space X. A sheaf will be always understood as an ‘espace étalé’ over X. The constant sheaf of sets associated with a set S is, by definition, the direct product X × S.

Now we recall some definitions and simple facts from [8], Section 1 (see also [12]).

Suppose that we are given a triple K = {K^{0}, K^{1}, K^{2}}, where K^{0}, K^{1} are groups, while K^{2} is a set with distinguished point, two actions ρ : K^{0} → Hol K^{1} and σ_{2} : K^{0} → Aut K^{2}, and a homomorphism of sets with distinguished points δ_{1} : K^{1} → K^{2}, such that

δ_{1} ∘ ρ(a) = σ_{2}(a) ∘ δ_{1}, a ∈ K^{0}.  (1.3)

Then we say that we have a (non-abelian) cochain complex. Elements of K^{p} are called p-cochains. The stabilizer of the point e ∈ K^{1} under the action ρ is a subgroup of K^{0} called the group of 0-cocycles or the 0-cohomology group

H^{0}(K) = Z^{0}(K).

We also define the set of 1-cocycles Z^{1}(K) = Ker δ_{1}. By (1.3), it is invariant under ρ, and hence we can define the 1-cohomology set

H^{1}(K) = Z^{1}(K)/ρ.

This set is usually endowed with the distinguished point ε = ρ(K^{0})(e).

Due to (1.2), ρ(a) = l_{δ_{0}(a)} ∘ σ_{1}(a), where σ_{1} : K^{0} → Aut K^{1} is an action of K^{0} by automorphisms of the group K^{1} and δ_{0} : K^{0} → K^{1} a crossed homomorphism. Clearly, H^{0}(K) = Z^{0}(K) = Ker δ_{0}.

A classical example is the Čech cochain complex with values in a sheaf of groups. Let F be a sheaf of groups on a topological space X and 𝔘 = (U_{i}) an open cover of X. Denote by C^{p} = C^{p}(𝔘, F), p ≥ 0, the group of p-cochains of 𝔘 with values in F, i.e.,

C^{p}(𝔘, F) = ∏_{U_{i_{0}} ∩...∩U_{ip} ≠∅} Γ(U_{i_{0}} ∩ … ∩ U_{i_{p}}, F).

Consider the triple C = C(𝔘, F) = (C^{0}, C^{1}, C^{2}). To make it a cochain complex, we set

(ρ(a)(b))_{ij} = a_{i}bija_{j}^{−1}, a ∈ C^{0}, b ∈ C^{1},
(σ_{2}(a)(c))_{ijk} = a_{i}c_{ijk}a_{i}^{−1}, a ∈ C^{0}, c ∈ C^{2},
δ_{1}(b)_{ijk} = b_{ij}b_{jk}b_{ik}^{−1}, b ∈ C^{1}.

Then

(σ_{1}(a)(b))_{ij} = a_{j}bija_{j}^{−1}, a ∈ C^{0}, b ∈ C^{1},
δ_{0}(a)_{ij} = a_{i}a_{j}^{−1}, a ∈ C^{0}.

The sets of cocycles are denoted by Z^{p}(𝔘, F), and the cohomology sets by H^{p}(𝔘, F), p = 0, 1. The group H^{0}(𝔘, F) coincides with the group of global sections Γ(X, F) and does not depend of 𝔘. This is not true for the 1cohomology. Passing to the limit over all open covers 𝔘 of X, we get the 1-cohomology set of X with values in F denoted by H^{1}(X, F). Note that the natural mapping H^{1}(𝔘, F) → H^{1}(X, F) is injective for any open cover 𝔘.

Let S be a sheaf of sets in X and let us denote by Aut S the sheaf of germs of automorphisms of this sheaf. Its sections over an open set U ⊂ X are, by definition, bijective continuous mappings S|U → S|U leaving each stalk invariant. Clearly, Aut S is a sheaf of groups. In what follows, we will often use this notation in a more restrictive sense, having in mind that automorhisms should preserve certain algebraic structure given in the sheaf S (it may be a sheaf of groups, a sheaf of superalgebras etc.).

Let 𝔘 = (U_{i}) be an open cover of X, and fix a cocycle z ∈ Z^{1}(𝔘, Aut S). Then we can twist S with the help of z getting a new sheaf S^{z} on X. This means that we glue together any two sheaves S|U_{i} and S|U_{j} over U_{i} ∩ U_{j} ≠ ∅ identifying a^{i} ∈ (S|U_{i})_{x} and a^{j} ∈ (S|U_{j})_{x}, x ∈ U_{i} ∩ U_{j}, under the condition a^{i} = z_{ij}(a^{j}). If S is a sheaf of groups and we use automorphisms of a sheaf of groups, then, clearly, S^{z} is a sheaf of groups, too, etc. The twisted sheaf S^{z} depends, up to isomorphism, only on the cohomology class ζ ∈ H^{1}(X, Aut S) determined by z. E.g., if z is cohomologous to e, i.e., z_{ij} = c_{i}c_{j}^{−1} for a 0cochain c ∈ C^{0}(𝔘, Aut S), then a^{i} = z_{ij}(a^{j}) implies c_{i}^{−1}(a^{i}) = c_{j}^{−1}(a^{j}). Thus, the correspondence (a^{i}) ↦ (c_{i}^{−1}(a^{i})) is an isomorphism of S^{z} onto S.

In a natural way, one defines homomorphisms of cochain complexes and the corresponding homomorphisms of cohomology groups and sets (see [8], Section 1.3 and [12], Section 1.2). A classical example is as follows. Let F and G be two sheaves of groups on X and f : G → F a homomorphism of sheaves of groups. Then, for any open cover 𝔘 of X, the homomorphism of complexes f : C(𝔘, G) → C(𝔘, F) is defined in an obvious way. The latter induces cohomology homomorphisms f_{p}^{∗} : H^{p}(𝔘, G) → H^{p}(𝔘, F), and, after passing to the limit, homomorphisms f_{p}^{∗} : H^{p}(X, G) → H^{p}(X, F), p = 0, 1.

As in the classical homological algebra, there are exact cohomology sequences associated with a subcomplex or with a short exact sequence of complexes. We will need the following result related to the non-abelian Čech cohomology (see [12], Section 1.4).

Theorem 1.1. Let the exact sequence of sheaves of groups

e → G →^{i} F →^{g} Q → e.  (1.4)

be given. Then the following sequence of groups and sets with distinguished points is exact:

\begin{matrix}e → H^{0}(X, G) →^{i_{0}^{∗}} H^{0}(X, F) →^{g_{0}^{∗}} H^{0}(X, Q) →^{δ_{0}^{∗}} H^{1}(X, G) →^{i_{1}^{∗}} \\ →^{i_{1}^{∗}} H^{1}(X, F) →^{g_{1}^{∗}} H^{1}(X, Q). \end{matrix}

Moreover, there is a natural action ρ^{∗} of the group H^{0}(X, Q) on H^{1}(X, G) such that the corresponding orbit mapping coincides with δ_{0}^{∗}. For any two γ, γ′ ∈ H^{1}(X, G), we have i_{1}^{∗}(γ) = i_{1}^{∗}(γ′) if and only if γ, γ′ lie on the same orbit of ρ^{∗}.

Consider the special case when (1.4) splits, i.e., there exists a homomorphism of sheaves of groups j : Q → F such that g ∘ j = id.

Corollary. If the sequence (1.4) splits by a homomorphism j : Q → F, then there is the following exact sequence of sets with dustinguished points:

e → H^{1}(X, G)/(σ_{1}^{∗}∘j) →^{\overline{i_{1}^{∗}}} H^{1}(X, F) →^{g_{1}^{∗}} H^{1}(X, Q) → e,

where \overline{i_{1}^{∗}} is injective and induced by i_{1}^{∗}, while σ_{1}^{∗} is the action of the group H^{0}(X, F) induced by its action on G by conjugations.

The definition of a cochain complex extends readily to sheaves on a fixed topological space X. In particular, one can define a sheaf of cochain complexes K on X and get its cohomology sheaves H^{p}(K), p = 0, 1, that are sheaves on X as well. Clearly, with any cochain sheaf K = {K^{0}, K^{1}, K^{2}} one associates the cochain complex of its global sections Γ(X, K) = {Γ(X, K^{0}), Γ(X, K^{1}), Γ(X, K^{2})}.

A sheaf of cochain complexes R on X is called a resolution of a sheaf of groups F if H^{0}(R) = F and H^{1}(R) = ε (the trivial sheaf of sets). This means that the following sequence of sheaves is exact:

e → F →^{i} R^{0} →^{δ_{0}} R^{1} →^{δ_{1}} R^{2},

where i is the natural inclusion.

The cochain complex of global sections Γ(X, R) has, in general, non-trivial 1-cohomology. Applying the technics of exact cohomology sequences, one gets the following result establishing a relation between this cohomology set and H^{1}(X, F) (see [8], Section 2.3).

Theorem 1.2. Suppose that a resolution R of a sheaf of groups F on X is given. Then the following sequence of sets with distinguished points is exact:

e → H^{1}(Γ(X, R)) →^{μ} H^{1}(X, F) →^{i_{1}^{∗}} H^{1}(X, R^{0}),

where μ is injective.

It is useful to describe μ explicitly. Take z ∈ Z^{1}(Γ(X, R)) = {c ∈ Γ(X, R) | δ_{1}(c) = e}. There exists an open cover 𝔘 = (U_{i}) of X such that z = δ_{0}(a_{i}), where a_{i} ∈ Γ(U_{i}, R^{0}), for any i. Define b ∈ Z^{1}(𝔘, R^{0}) by b_{ij} = a_{i}^{−1}a_{j}. Then δ_{0}(b_{ij}) = e and hence b ∈ Z^{1}(𝔘, F). Then μ maps the cohomology class of z onto that of b.

Let G be another sheaf of groups on X and S its resolution. Suppose that we have a homomorphism of sheaves of complexes φ = {φ_{0}, φ_{1}, φ_{2}} of S into R. This means that we have the commutative diagram

e ⟶ G ⟶^{i} S^{0} ⟶^{δ_{0}} S^{1} ⟶^{δ_{1}} S^{2}
φ↓  ↓^{φ_{0}}  ↓^{φ_{1}}  ↓^{φ_{2}}
e ⟶ F ⟶_{i} R^{0} ⟶_{δ_{0}} R^{1} ⟶_{δ_{1}} R^{2}.

Then we have the corresponding homomorphism φ : Γ(X, S) → Γ(X, R) of the complexes of global sections. It is useful to mention the following result ([8], Section 2.3):

Theorem 1.3. Let φ : S → R be a homomorphism of resolutions of sheaves of groups G and F, respectively, inducing the homomorphism of sheaves φ : G → F. Then the following diagram is commutative:

H^{1}(Γ(X, S)) ⟶^{μ} H^{1}(X, G) ⟶^{i_{1}^{∗}} H^{1}(X, S^{0})
φ^{∗}↓  ↓^{φ_{1}^{∗}}  ↓^{φ_{0}^{∗}}
H^{1}(Γ(X, R)) ⟶_{μ} H^{1}(X, F) ⟶_{i_{1}^{∗}} H^{1}(X, R^{0}).

1.2. Supermanifolds and vector bundles. Let M be a complex analytic manifold of complex dimension n. We will regard it as the ringed space (M, F), where F is the sheaf of holomorphic functions on M. We want to classify, up to isomorphisms leaving any point of M fixed, all complex analytic supermanifolds of dimension n|m with reduction (M, F). This can be easily made in terms of non-abelian cohomology.

Let us denote by ξ_{1}, …, ξ_{m} the standard basis of ℂ^{m} and form the Grassmann algebra A_{m} = ⋀_{ℂ}(ξ_{1}, …, ξ_{m}) = ⋀_{ℂ}(ℂ^{m}). We may regard it either as an algebra graded by the exterior powers A_{m}^{p} = ⋀_{ℂ}^{p}(ℂ^{m}), or as a superalgebra (ℤ_{2}-graded algebra). The same is true for the sheaf of algebras F_{A_{m}} = F ⊗ A_{m}. Consider the sheaf of groups Aut F_{A_{m}} formed by germs of automorphisms of this sheaf of superalgebras. Take an open cover 𝔘 = (U_{i}) of M and a cocycle z ∈ Z^{1}(𝔘, Aut F_{A_{m}}). The twisted sheaf O = F_{A_{m}}^{z} is a sheaf of ℂ-superalgebras on M that can be regarded as the structure sheaf of a complex supermanifold (M, O) of dimension n|m. Denote by J the subsheaf of ideals of O generated by the subsheaf O¯1 of odd elements. One verifies easily that O/J ≃ F, i.e., the reduction (the underlying manifold) of (M, O) is (M, F). Up to isomorphisms leaving any point of M fixed, this supermanifold depends on the cohomology class ζ ∈ H^{1}(M, Aut F_{A_{m}}) only. Clearly, each supermanifold of dimension n|m with reduction (M, F) can be got in this way, using its transition functions. Thus, H^{1}(M, Aut F_{A_{m}}) can be interpreted as the set of all supermanifolds of dimension n|m with reduction (M, F), regarded up to isomorphisms leaving any point of M fixed.

Note that we have the injection of the sheaf of groups j : F_{GL_{m}(ℂ)} → Aut F_{A_{m}} which extends, in a natural way, any linear transformation of ℂ^{n} to an automorphisms of the algebra A_{m}. The image of j will be the subsheaf of automorphisms preserving the ℤ-grading. If z = j_{1}(z′), where z′ ∈ Z^{1}(𝔘, F_{GL_{m}(ℂ)}), then the twisting by z preserves the ℤ-grading, and hence the corresponding structure sheaf O will be a sheaf of ℤ-graded algebras. As usually, we shall interprete H^{1}(M, F_{GL_{m}(ℂ)}) as the set of all holomorphic vector bundles of rank m over M regarded up to isomorphisms leaving any point of M fixed. One sees easily that O = ⋀_{F} E, where E = (F_{ℂ)^{m}})^{z′} is the sheaf of holomorphic sections of the vector bundle E corresponding to the cocycle z′. The supermanifolds (M, O) of this special form (and those isomorphic to these supermanifolds) are called split. Thus, the set of split supermanifolds is the image of the mapping j_{1}^{∗} : H^{1}(M, F_{GL_{m}(ℂ))}) → H^{1}(M, Aut F_{A_{m}}).

Let (M, O) be a supermanifold of dimension n|m with reduction (M, F). Then we have the folowing filtration of O by powers of its ideal subsheaf J generated by O¯1:

O = J^{0} ⊃ J ⊃ J^{2} ⊃ …  (1.5)

It is finite, since J^{m+1} = 0. Clearly, O/J ≃ F, and E = J /J^{2} is a locally free sheaf of F-modules of rank m over (M, F).

In the case when (M, O) is split, i.e., O = ⋀_{F} E, where E is the sheaf of holomorphic sections of a holomorphic vector bundle, the filtration (1.5) is accociated with the grading, i.e.,

J^{p} = ⨁_{q≥p} ⋀_{F}^{q} E, p ≥ 0.

Now consider the sheaf Aut O of automorphisms of the sheaf of superalgebras O, and, for any p ≥ 0, its subsheaf

Aut_{(2p)}O = {a ∈ Aut O | a(f) − f ∈ J^{2p} for any f ∈ O}.  (1.6)

Then we get the following finite filtration:

Aut O = Aut_{(0)}O ⊃ Aut_{(2)}O ⊃ ….  (1.7)

Any automorphism preserves the filtration (1.5), and hence any Aut_{(2p)}O is a subsheaf of normal subgroups.

Consider again the simplest case when O = F_{A_{m}}. Then J /J^{2} = F_{ℂ^{m}}. Any automorphism of O leaves J and J^{2} invariant and hence induces an automorphism of F_{ℂ^{m}}, i.e., an element of F_{GL_{m}(ℂ)}. We got a homomorphism of sheaves of groups g : Aut O → F_{GL_{m}(ℂ)}. Clearly, g ∘ j = id. Also Ker g = Aut_{(2)}O. Thus, we have the exact sequence of sheaves of groups

e → Aut_{(2)}F_{A_{m}} →^{i} Aut F_{A_{m}} →^{g} F_{GL_{m}(ℂ)} → e  (1.8)

that splits by j : F_{GL_{m}(ℂ)} → Aut F_{A_{m}}.

The mappings

\begin{matrix}g_{1}^{∗} : H^{1}(M, Aut F_{A_{m}}) → H^{1}(M, F_{GL_{m}(ℂ)}), \\ j_{1}^{∗} : H^{1}(M, F_{GL_{m}(ℂ)}) → H^{1}(M, Aut F_{A_{m}}) \end{matrix}

have a nice interpretation in terms of supermanifolds and vector bundles. Namely, g_{1}^{∗} assigns to any supermanifold (M, O) with reduction (M, F) the holomorphic vector bundle E over M, whose sheaf of holomorphic sections is isomorphic to E = J /J^{2}, J being the ideal subsheaf of O described above. Now, j_{1}^{∗} assigns to any holomorphic vector bundle E the corresponding split supermanifold with the structure sheaf O = ⋀_{F} E. Since g_{1}^{∗} ∘ j_{1}^{∗} = id, g_{1}^{∗} is surjective. Now, j_{1}^{∗} ∘ g_{1}^{∗} assignes to any supermanifold (M, O) the split supermanifold (M, ⋀_{F} E), where E = J /J^{2}. The latter one is called the retract of (M, O). One checks easily that its structure sheaf is isomorphic to the sheaf of graded algebras associated with the filtration (1.5) of O.

We see that the classification problem for supermanifolds of dimension n|m and reduction (M, F) reduces to the folowing two problems:

(1) To classify holomorphic vector bundles of rank m over (M, F), i.e., to compute H^{1}(M, F_{GL_{m}(ℂ)}).

(2) For a fixed holomorphic vector bundle E of rank m over (M, F), to classify the supermanifolds whose retracts are isomorphic to (M, ⋀_{F} E), i.e., to describe the inverse image (g_{1}^{∗})^{−1}(ζ) of the corresponding class ζ ∈ H^{1}(M, F_{GL_{m}(ℂ)}).

In what follows, we will study the second problem. In cohomological terms, it is solved by the following result (see [3]).

Theorem 1.4. Let E be a holomorphic vector bundle over (M, F) and (M, O) the corresponding split supermanifold. Then there is a bijection between the set of all supermanifolds having (M, O) as their retract (considered up to isomorphisms leaving fixed any point of M) and the quotient H^{1}(M, Aut_{(2)}O)/ Aut E, where the action is induced by the action of Aut E on the sheaf Aut_{(2)}O by conjugations.

If ω ∈ H^{1}(M, Aut_{(2)}O) is determined by a cocycle w ∈ Z^{1}(𝔘, Aut_{(2)}O), then the orbit of ω corresponds to the sheaf O^{w}, where w is regarded as a 1-cocycle with values in Aut O.

Note that Theorem 1.4 is immediately implied by Corollary of Theorem 1.1. In fact, let z ∈ Z^{1}(𝔘, F_{GL_{m}(ℂ)}) be a cocycle representing E. Then we have the cocycle j_{1}(z) ∈ Z^{1}(𝔘, Aut F_{A_{m}}) which we will denote again by z. Let (M, O) be the split supermanifold corresponding to E. Then we may identify Aut O = Aut(F_{A_{m}})^{z} with (Aut F_{A_{m}})^{Int z}, and Aut E = Aut(F_{ℂ^{m}})^{z} with (F_{GL_{m}(ℂ)})^{Int z}, where Int z denotes the cocycle of inner automorphisms determined by z. Twisting (1.8), we get the following split exact sequence:

e → Aut_{(2)}O →^{i(z)} Aut O →^{g(z)} Aut E → e.  (1.9)

Then we have to apply Corollary of Theorem 1.1 to (1.9), taking into account that Ker g(z)_{1}^{∗} = Im i(z)_{1}^{∗} identifies with (g_{1}^{∗})^{−1}(ζ), where ζ ∈ H^{1}(M, F_{GL_{m}(ℂ)}) is the cohomology class of z (see [12], Section 1).

2. Resolutions of automorphism sheaves

2.1. Tangent sheaf and exponential mapping. We retain the notation of Section 1.2. For an arbitrary supermanifold (M, O), denote by T = Der O its tangent sheaf. This is a sheaf of Lie superalgebras with respect to the Lie bracket [ , ].

Here we suppose that (M, O) is the split supermanifold corresponding to a holomorphic vector bundle E over M, i.e., that O = ⋀_{F} E. Then O and T are sheaves of graded algebras, the ℤ-grading in T being given by

T = ⨁_{p≥−1} T_{p}, where T_{p} = {u ∈ T | u(O_{q}) ⊂ O_{q+p} for all q ∈ ℤ}.

We also note that T is a locally free graded sheaf of F-modules on the manifold (M, F). This means that it can be regarded as the sheaf of holomorphic sections of a certain graded holomorphic vector bundle over M (see [2]). We call it the supertangent bundle and denote by ST(M, O). Note that for any p ≥ −1 there exists a natural exact sequence of holomorphic vector bundles

0 → E^{∗} ⊗ ⋀^{p+1} E → ST_{p}(M, O) → T(M) ⊗ ⋀^{p} E → 0,  (2.1)

where T(M) is the tangent bundle of M.

Following [14], we shall apply the tangent sheaf for studying the sheaves of automorphisms Aut_{(2p)}O, p ≥ 1, defined by (1.6). Consider the even part of T ; this is the graded sheaf of Lie algebras

T¯0 = ⨁_{p≥0} T_{2p}.

This sheaf admits the natural finite filtration

T¯0 = T (0) ¯0 ⊃ T (2) ¯0 ⊃ …, where T (2p) ; ¯0 = ⨁_{q≥p} T_{2q}.  (2.2)

Define the exponential mapping exp : T (2) ¯0 → Aut_{(2)}O expressed by the usual exponential series. Since our filtrations are finite, this series will be actually a polynomial. One also defines the inverse mapping log = exp^{−1}. Thus, exp is an isomorphism of sheaves of sets (but, in general, not of groups). Clearly, exp maps each subsheaf T (2p) ; ¯0, p ≥ 1, onto Aut_{(2p)}O.

To get a homomorphism of sheaves of groups, we define the mapping λ_{2p} : Aut_{(2p)}O → T_{2p} assigning to any a ∈ Aut_{(2p)}O the (2p)-component (log a)_{2p} of log a. By the Campbell–Hausdorff formula, we have

log(ab) = log a + log b + \frac{1}{2}[log a, log b] + …,  (2.3)

and hence

(log(ab))_{2p} = (log a)_{2p} + (log b)_{2p}.

Thus, for any p ≥ 1, λ_{2p} is a homomorphism of sheaves of groups. Clearly, Ker λ_{2p} = Aut_{(2p+2)}O. Therefore we have the exact sequence of sheaves of groups

e → Aut_{(2p+2)}O → Aut_{(2p)}O →^{λ_{2p}} T_{2p} → 0.  (2.4)

Now we will consider similar constructions for differentiable functions and vector fields. Denote by F^{∞} the sheaf of differentiable complex-valued functions on M and by E^{∞} the sheaf of differentiable sections of E. Then the sheaf of algebras

O^{∞} = ⋀_{F^{∞}} E^{∞}

is the sheaf of differentiable sections of ⋀ E. The ringed space (M, O^{∞}) is the differentiable complex supermanifold corresponding to (M, O). We have

O^{∞} = F^{∞} ⊗ O.

Let T^{∞} = Der O^{∞} denote the tangent sheaf of (M, O^{∞}).

A derivation v ∈ T^{∞} will be called pseudoholomorphic if v(f̄) = 0 for all f ∈ F. Clearly, the subset PT of all pseudoholomorphic derivations is a subsheaf of graded Lie superalgebras and, moreover, of O^{∞}-submodules in T^{∞}. The sheaves of even derivations T ∞ ; ¯0 ⊃ PT ∞ ; ¯0 are sheaves of graded Lie algebras and admit filtrations by ideal subsheaves T ∞(2p) ; ¯0 and PT (2p) ; ¯0 similar to (2.2).

Clearly, we have the filtration of O^{∞} by the powers of the ideal subsheaf J^{∞} generated by odd elements, giving rise to the fitration of the automorphism sheaf Aut O^{∞} by subsheaves of normal subgroups

Aut O^{∞} = Aut_{(0)}O^{∞} ⊃ Aut_{(2)}O^{∞} ⊃ Aut_{(4)}O^{∞} ⊃ …,
where Aut_{(2p)}O^{∞} = {a ∈ Aut O^{∞} | a(f) − f ∈ (J^{∞})^{2p} for all f ∈ O^{∞}}.

Now, define the subsheaves

PAut O^{∞} = {a ∈ Aut O^{∞} | a(f̄) = f̄ for all f ∈ F},
PAut_{(2p)}O^{∞} = Aut_{(2p)}O^{∞} ∩ PAut O^{∞}.

Then

PAut O^{∞} = PAut_{(0)}O^{∞} ⊃ PAut_{(2)}O^{∞} ⊃ PAut_{(4)}O^{∞} ⊃ ….

We also have the exponential mapping

exp : T ∞(2) ; ¯0 → Aut_{(2)}O^{∞}.

It is an isomorphism of sheaves of sets and maps T ∞(2p) ; ¯0 onto Aut_{(2p)}O^{∞} and PT (2p) ; ¯0 onto PAut_{(2p)}O^{∞}, p ≥ 1. Using log = exp^{−1}, we get the following exact sequences similar to (2.4):

e → PAut_{(2p+2)}O^{∞} → PAut_{(2p)}O^{∞} →^{λ_{2p}} PT_{2p} → 0.  (2.5)

Since PT_{2p} are fine sheaves, their 1-cohomology is 0.

Proposition 2.1. For any holomorphic vector bundle E over a complex manifold M, we have H^{1}(M, PAut_{(2p)}O^{∞}) = ε for all p ≥ 1.

Proof. We use the induction on −p. For p ≥ m + 1 the assertion is true. Suppose it to be true for p + 1. By Theorem 1.1, (2.5) gives the exact sequence

H^{1}(M, PAut_{(2p+2)}O) ⟶ H^{1}(M, PAut_{(2p)}O) ⟶^{(λ_{2p})_{1}^{∗}} H^{1}(M, PT_{2p}).

It easily implies our assertion, since both extreme terms are trivial.

Remark. In the same way, one can prove that any differentiable supermanifold is split (see [1]).

2.2. Construction of a resolution. Let M = (M, F) be a complex manifold, E a holomorphic vector bundle over (M, F), and E the sheaf of holomorphic sections of E. We will study again the split supermanifold (M, O), where O = ⋀_{F} E. In what follows, we will omit the subscript F in the notation of exterior algebras or tensor products over this sheaf of algebras. The goal of this section is to construct resolutions of the non-abelian sheaves Aut_{(2p)}O, p ≥ 1.

First, we construct a fine resolution of the tangent sheaf T = Der O. Consider the classical bigraded sheaf Φ = ⨁_{p,q} Φ^{p,q} of smooth complexvalued differential forms on M. Since T is a locally free analytic sheaf, we can form the standard Dolbeault–Serre resolution of T that is the bigraded sheaf R of ST-valued smooth forms of type (0, ∗). More precisely, we set

R_{p,q} = Φ^{0,q} ⊗ T_{p},
R = ⨁_{p≥−1,q≥0} R_{p,q},
¯∂(φ ⊗ u) = (¯∂φ) ⊗ u, φ ∈ R_{0,q}, u ∈ T_{p}.

Then the sequence

0 → T →^{i} R_{∗,0} →^{¯∂} R_{∗,1} →^{¯∂} ….  (2.6)

is the desired resolution. However, it is convenient to write this resolution in a more complicated form, expressing ST-valued forms as derivations of the sheaf Φ̂ of smooth ⋀ E-valued forms.

Thus, we first form the standard Dolbeault–Serre resolution of O, setting

Φ̂^{p,q} = Φ^{0,q} ⊗ O_{p},
Φ̂ = ⨁_{p,q≥0} Φ̂^{p,q},
¯∂(φ ⊗ u) = (¯∂φ) ⊗ u, φ ∈ Φ^{0,q}, u ∈ O_{p}.

Then, regarding Φ̂ as a sheaf of graded algebras with respect to the total degree, consider the sheaf of graded Lie algebras T̂ = Der Φ̂. Denote

D̄ = ad ¯∂.

Clearly, D̄ is a derivation of degree 1 (and of bidegree (0, 1)) of T̂, and

D̄^{2} = \frac{1}{2}[D̄, D̄] = \frac{1}{2} ad[¯∂, ¯∂] = 0.

By definition, we have

D̄u = ¯∂ ∘ u − (−1)^{p(u)}u ∘ ¯∂, u ∈ S.

Set

S = {u ∈ T̂ | u(f̄) = u(df̄) = 0 for any f ∈ F}.

One sees readily that S is a subsheaf of bigraded subalgebras and of F^{∞}- submodules of T̂. Further, for any u ∈ S and any f ∈ O we have

(D̄u)(f̄) = (D̄u)(df̄) = 0,

and so D̄(S) ⊂ T̂.

As in Section 2.1, consider the sheaves

O^{∞} = F^{∞} ⊗ O = Φ̂^{∗,0},

T^{∞} = Der O^{∞} and the subsheaf PT of pseudoholomorphic derivations. Then we can identify PT with S_{∗,0}. To do this, note that Φ̂ is a locally free sheaf of O^{∞}-modules. Suppose that x_{1}, …, x_{n} and ξ_{1}, …, ξ_{m} are respectively even and odd local coordinates of (M, O). Then a local basis of this sheaf is formed by the sections dx̄_{i_{1}} … dx̄_{i_{q}} ⊗ 1 = dx̄_{i_{1}} … dx̄_{i_{q}}, i_{1} < … < i_{q}. For any u ∈ PT , we put

u( ∑_{i_{1}<...<i_{q}} f_{i_{1}...i_{q}}dx̄_{i_{1}} … dx̄_{i_{q}}) = ∑_{i_{1}<...<i_{q}} u(f_{i_{1}...i_{q}})dx̄_{i_{1}} … dx̄_{i_{q}},

where f_{i_{1}...i_{q}} ∈ O^{∞}.

On the other hand, we have a natural injection T → PT that extends any derivation germ u of O to a derivation germ ũ of O^{∞} satisfying ũ(f̄) = 0 for any f ∈ F. This injection is easily described in terms of the local coordinates introduced above. Namely, suppose that u = ∑_{i} a_{i} \frac{∂}{∂x_{i}} + ∑_{j} b_{j} \frac{∂}{∂ξ_{j}}. Then ũ is expressed by the same formula in the local coordinates x_{1}, …, x_{n}, x̄_{1}, …, x̄_{n}, ξ_{1}, …, ξ_{m} of (M, O^{∞}).

This allows to define a natural injection i : T → S_{∗,0}. Thus, we get the sequence

0 → T →^{i} S_{∗,0} →^{D̄} S_{∗,1} →^{D̄} ….  (2.7)

By [9], Theorem 1, this is a fine resolution of T isomorphic to (2.6). Moreover, i is a homomorphism of graded Lie algebra sheaves, and hence the natural bracket in S may be used to calculate the bracket in H^{∗}(M, T ) induced by the Lie bracket defined in T .

Now we go over to the construction of resolutions for certain sheaves of automorphisms of O. Consider the subsheaf PAut O^{∞} of Aut O^{∞} defined in Section 2.1. Any a ∈ PAut O^{∞} can be extended to an automorphism of Φ̂; the extension is denoted by the same symbol a and is given by

a( ∑_{i_{1}<...<i_{q}} f_{i_{1}...i_{q}}dx̄_{i_{1}} … dx̄_{i_{q}}) = ∑_{i_{1}<...<i_{q}} a(f_{i_{1}...i_{q}})dx̄_{i_{1}} … dx̄_{i_{q}},

where f_{i_{1}...i_{q}} ∈ O^{∞}. One checks easily that in this way we get an injection PAut O^{∞} → Aut Φ̂. Further, u ↦ aua^{−1} is, clearly, an automorphism of T̂ for any a ∈ Aut Φ̂. If a ∈ PAut O^{∞}, then this automorphism maps S onto itself. Therefore, we get an action Int : a ↦ Int a of the sheaf PAut O^{∞} on S by the automorphisms Int a(u) = aua^{−1}.

Now we define the mapping δ_{0} : PAut O^{∞} → T̂_{∗,1} by

δ_{0}(a) = ¯∂ − a¯∂a^{−1},

and the mapping δ_{1} : T̂_{∗,1} → T̂_{∗,2} by

δ_{1}(u) = D̄u − \frac{1}{2}[u, u] = −\frac{1}{2}[u − ¯∂, u − ¯∂].

Clearly, δ_{1}(0) = 0.

Proposition 2.2.

(1) The mapping δ_{0} is a crossed homomorphism of PAut O^{∞} into S_{∗,1}.

(2) The corresponding affine action of PAut O^{∞} on S_{∗,1} is given by

ρ(a)(u) = a(u − ¯∂)a^{−1} + ¯∂.

(3) The mapping δ_{1} maps S_{∗,1} into S_{∗,2} and satisfies

δ_{1}(ρ(a)(u)) = aδ_{1}(u)a^{−1}.

Proof. One checks easily that δ_{0}(a) ∈ S_{∗,1} whenever a ∈ PAut O^{∞}. Also

δ_{0}(ab) = ¯∂− ab¯∂b^{−1}a^{−1} = ¯∂− a¯∂a^{−1} + a(¯∂− b¯∂b^{−1})a^{−1} = δ_{0}(a) + aδ_{0}(b)a^{−1}.

Now, by definition,

ρ(a)(u) = aua^{−1} + δ_{0}(a) = a(u − ¯∂)a^{−1} + ¯∂.

Finally,

δ_{1}(ρ(a)(u)) = −\frac{1}{2}[a(u − ¯∂)a^{−1}, a(u − ¯∂)a^{−1}] = aδ_{1}(u)a^{−1}.

Let us introduce the formal series

E(x) = \frac{exp x − 1}{x} = 1 + \frac{1}{2!}x + \frac{1}{3!}x^{2} + ….

Proposition 2.3. For any v ∈ T ∞(2) ; ¯0 we have

δ_{0}(exp v) = E(ad v)(D̄v).

Proof. By definition, we have

δ_{0}(exp v) = ¯∂ − (exp v)¯∂(exp v)^{−1} = ¯∂ − (exp ad v)(¯∂)
= E(ad v)(−[v, ¯∂]) = E(ad v)(D̄v),

since v is an even derivation.

Proposition 2.4. Any a ∈ Aut_{(2)}O can be uniquely extended to an automorphism b ∈ PAut_{(2)}O^{∞} such that b¯∂b^{−1} = ¯∂. This gives us, for any p ≥ 1, a natural injective homomorphism j : Aut_{(2p)}O → PAut_{(2p)}O^{∞} whose image is (Ker δ_{0}) ∩ PAut_{(2p)}O^{∞}.

Proof. Any b ∈ PAut_{(2)}O^{∞} satisfying δ_{0}(b) = 0 maps O onto itself. Thus, we get the restriction homomorphism (Ker δ_{0}) ∩ PAut_{(2)}O^{∞} → Aut_{(2)}O. If b|O = id, then log b(f) = 0 for any f ∈ O. Since log b(f̄) = 0 for any f ∈ F, we have log b = 0, and b = id. Thus, b is uniquely determined by its restriction onto O. On the other hand, if a ∈ Aut_{(2)}O, then log a, as was shown above, can be uniquely prolonged to a derivation v ∈ PT (2) ¯0 satisfying [¯∂, v] = 0. Then (exp v)¯∂(exp v)^{−1} = (exp ad v)(¯∂) = ¯∂, and thus exp v is the desired extension of a onto O^{∞}.

Define

S_{q}^{(2p)} = ⨁_{k≥p} S_{2k,q}.

Theorem 2.1. For any p ≥ 1, the sequence

PAut_{(2p)}O^{∞} →^{δ_{0}} S_{1}^{(2p)} →^{δ_{1}} S_{2}^{(2p)}  (2.8)

is a resolution of the sheaf Aut_{(2p)}O.

Proof. We see from Proposition 2.2 that (2.8) gives us a sheaf of cochain complexes, if we define the action σ_{q} of K_{p}^{0} = PAut_{(2p)}O^{∞} on K_{p}^{q} = S_{q}^{(2p)}, q = 1, 2, as Int. By Proposition 2.4, the 0-cohomology of it is isomorphic to Aut_{(2p)}O. We have only to check that the 1-cohomology is trivial, i.e., that Ker δ_{1} ⊂ Im δ_{0} in (2.8).

Since this is true for large p, we can proceed by induction in −p. Suppose that the sequence (2.8) associated with p + 1 is exact. Take u ∈ S_{1}^{(2p)} such that δ_{1}(u) = 0. If u_{2p} ∈ S_{2p,1} is the 2p-component of u, then, clearly, the 2p-component of δ_{1}(u) is D̄u. Hence D̄u = 0. Since (2.7) is exact, there exists v ∈ S_{2p,0} such that D̄v = u. Using Proposition 2.3, we see that

u′ = ρ(exp(−v))(u) = δ_{0}(exp(−v)) + (exp(−v))u(exp v)
= −E(− ad v)(D̄v) + exp(− ad v)(u)

lies in S_{1}^{(2p+2)}. Clearly, δ_{1}(u′) = 0. By the inductive hypothesis, u′ = δ_{0}(a), where a ∈ PAut_{(2p+2)}O^{∞}. Then

u = ρ(exp v)(u′) = ρ(exp v)(δ_{0}(a)) = ρ(exp v)(ρ(a)(0))
= ρ((exp v)a)(0) = δ_{0}((exp v)a).

Clearly, (exp v)a ∈ PAut_{(2p)}O^{∞}, and the exactness is proved.

Now we consider the corresponding complexes of global sections. For any p ≥ 1, denote

K_{p}^{0} = PAut_{(2p)} O^{∞} = Γ(M, PAut_{(2p)}O^{∞}),
K_{p}^{q} = S_{q}^{(2p)} = Γ(M, S_{q}^{(2p)}), q = 1, 2,

and define, as above, the operators δ_{q} : K_{p}^{q} → K_{p}^{q+1}, q = 0, 1, and the actions σ_{q} = Int of K_{p}^{0} on K_{p}^{q}, q = 1, 2. Then, using Theorems 2.1, 1.2 and Proposition 2.1, we get the following result.

Theorem 2.2. For any p ≥ 1, we have an isomorphism of sets with distinguished points

μ_{p} : H^{1}(K_{p}) → H^{1}(M, Aut_{(2p)}O).

The mapping μ_{p} can be expressed quite explicitly. Take z ∈ Z^{1}(K_{p}). There exists an open cover 𝔘 = (U_{i}) of M such that z = δ_{0}(a_{i}), where a_{i} ∈ Γ(U_{i}, PAut_{(2p)}O^{∞}), for any i. Define b ∈ Z^{1}(𝔘, PAut_{(2p)}O^{∞}) by b_{ij} = a_{i}^{−1}a_{j}. One sees that b ∈ Z^{1}(𝔘, Aut_{(2p)}O). Then μ_{p} maps the cohomology class of z onto that of b.

Due to Theorem 1.4, it is important to interprete the natural action of Aut E on H^{1}(M, Aut_{(2p)}O) in terms of the complex K_{p}. Clearly, any holomorphic automorphism of E is smooth, and hence we have the natural injection of the sheaf Aut E into Aut E^{∞}. On the other hand, any element of Aut E^{∞} extends uniquely to an automorphism of O^{∞} leaving fixed any element of F^{∞}, and thus we get the inclusions Aut E ⊂ Aut E^{∞} ⊂ PAut O^{∞}. As we saw above, there is an action Int of the group Aut E = Aut E on the resolution (2.8) of Aut_{(2p)}O, p ≥ 1. After verifying that Int a, for any a ∈ Aut E, is an automorphism of this resolution, we get from Theorem 1.3 the following assertion.

Proposition 2.5. The action of Aut E on the sheaf Aut_{(2p)}O by conjugations induces an action on the 1-cohomology of this sheaf corresponding, via the isomorphism μ_{p} of Theorem 2.2, to the action of this group on H^{1}(K_{p}) induced by the action Int on the resolution (2.8).

Note that Aut E is non-trivial whenever m = rk E ≥ 1. In fact, to any c ∈ ℂ^{×} there corresponds the dilatation d_{c} of E by the scalar c.

Proposition 2.6. We have

(Int d_{c})(u) = c^{2q}u, c ∈ ℂ^{×}, u ∈ Γ(M, S_{2q,r}).

Proof. For any f ∈ O_{k}, φ ∈ Φ^{0,l} we have u(φ ⊗ f) ∈ Γ(M, S_{2q+k,r+l}). Therefore

((Int d_{c})(u))(φ ⊗ f) = (d_{c}ud_{−c})(φ ⊗ f) = d_{c}(u(c^{−k}φ ⊗ f))
= c^{−k}d_{c}(u(φ ⊗ f)) = c^{−k}c^{2q+k}u(φ ⊗ f) = (c^{2q}u)(φ ⊗ f).

Example 1. Suppose that E = T^{∗}(M) is the cotangent bundle of M. Then O = Ω is the sheaf of holomorphic forms. Therefore Φ̂^{p,q} = Φ^{0,q} ⊗Ω^{p} = Φ^{p,q}. Thus, Φ̂ is the sheaf Φ of all smooth complex-valued forms on M. Due to Frölicher–Nijenhuis, the derivations of this sheaf can be easily described in terms of vector-valued forms. In [11], a construction of non-split supermanifolds with retract (M, Ω) was given. Without going into details, we show, how to express this construction in terms of the complex K_{2}.

Let ω ∈ Γ(M, Φ^{1,1}) be a (1,1)-form satisfying dω = 0. Consider the derivation u = ω∂ of Φ. Clearly, u ∈ S_{2,1}. Moreover,

D̄u = [¯∂, ω∂] = (¯∂ω)∂ + ω ¯∂∂ = (¯∂ω)∂ + ω(¯∂∂ + ∂ ¯∂) = 0,
[u, u] = [ω∂, ω∂] = 2ω(∂ω)∂ = 0.

Thus, δ_{1}(u) = 0, and u determines a cohomology class ũ ∈ H^{1}(M, Aut_{(2)}Ω). One sees easily that this correspondence coincides with that established in [11]. If, in particular, M is a compact Kähler manifold of dimension ≥ 2, then the correspondence ω ↦ ũ gives rise to an injection H^{1,1}(M, ℂ) → H^{1}(M, Aut_{(2)}Ω) (see [11], Theorem 2.1).

3. Applications

3.1. (2p)-projections. Here we consider an application of the resolution constructed in Section 2.2.

Suppose that p ≥ 1. In Section 2.1, we have defined the homomorphism of sheaves of groups λ_{2p} : Aut_{(2p)}O → T_{2p}. Actually, (2.3) implies that

(log(ab))_{2q} = (log a)_{2q} + (log b)_{2q}

for p ≤ q ≤ 2p − 1, whenever a, b ∈ Aut_{(2p)}O. Therefore we can define a homomorphism of sheaves of groups λ̂_{2p} : Aut_{(2p)}O^{∞} → T_{2p}^{∞} by

λ̂_{2p}(a) = ∑_{q=p}^{2p−1}(log a)_{2q}.

Clearly, Ker λ̂_{2p} = Aut_{(4p)}O^{∞}. Also λ̂_{2} = λ_{2}. Further, Ker λ̂_{2p} maps PAut_{(2p)}O^{∞} onto ⨁_{q=p}^{2p−1} PT_{2q}. From the definitions one sees easily that the following diagram is commutative:

e → Aut_{(2p)}O ⟶^{j} PAut_{(2p)}O^{∞} ⟶^{δ_{0}}  S_{1}^{(2p)}  ⟶^{δ_{1}}  S_{2}^{(2p)}
λ̂_{2p}↓  ↓λ̂_{2p}  ↓^{π_{2p}}  ↓^{π_{2p}}
0 → ⨁_{q=p}^{2p−1} T_{2q} ⟶_{i} ⨁_{q=p}^{2p−1} S_{2q,0} ⟶_{D̄} ⨁_{q=p}^{2p−1} S_{2q,1} ⟶_{D̄} ⨁_{q=p}^{2p−1} S_{2q,2}.

Here π_{2p} is the natural projection of S_{r}^{(2p)} onto ⨁_{q=p}^{2p−1} S_{2q,r}, r = 1, 2. Thus, we get a homomorphism π_{2p} of a resolution of Aut_{(2p)}O onto a resolution of ⨁_{q=p}^{2p−1} T_{2q} inducing λ̂_{2p} on these sheaves. By Theorem 1.3, we may identify (λ̂_{2p})_{1}^{∗} with the mapping (π_{2p})_{1}^{∗} : H^{1}(K_{p}) → ⨁_{q=p}^{2p−1} H^{1}(S_{2q,∗}), where S_{2q,∗} is the complex of sections of the resolution (2.7) of T_{2q}. The following theorem was essentially proved in [14] using the Čech cochain complex.

Theorem 3.1. If H^{2}(M, T_{2q}) = 0 for all q ≥ 2p, then

(λ̂_{2p})_{1}^{∗} : H^{1}(M, Aut_{(2p)}O) → ⨁_{q=p}^{2p−1} H^{1}(M, T_{2q})

is surjective.

Proof. Take u_{2q} ∈ S_{2q,1}, q = p, …, 2p − 1, satisfying D̄u_{2q} = 0. We want to construct elements u_{2k} ∈ S_{2k,1}, k ≥ 2p, such that u = u_{2p} + … + u_{4p−1} + u_{4p} +… satisfies D̄u− \frac{1}{2}[u, u] = 0. It suffices to prove the following inductive assertion:

For any s ≥ 2p, there exist elements u_{2k} ∈ S_{2k,1}, 2p ≤ k ≤ s, such that v(s) = u_{2p} + … + u_{4p−1} + u_{4p} + … + u_{2s} satisfies

δ_{1}(v(s)) = D̄v(s) − \frac{1}{2}[v(s), v(s)] ∈ S_{2}^{(2s+2)} .  (3.1)

This is true for s = 2p. Suppose that v(s − 1) = u_{2p} + … + u_{4p−1} + u_{4p} + … + u_{2s−2} ist constructed. Denote

w = δ_{1}(v(s − 1)) = w_{2s} + w_{2s+2} + …,

where w_{2k} ∈ S_{2k,2}. Then

D̄w = −\frac{1}{2}D̄[v(s − 1), v(s − 1)] = −[D̄v(q − 1), v(s − 1)]

= −[w + \frac{1}{2}[v(s − 1), v(s − 1)], v(s − 1)] = −[w, v(s − 1)] ∈ S_{2}^{(2s+2)} .

In particular, D̄w_{2s} = 0. Since H^{2}(M, T_{2s}) = 0, we have w_{2s} = −D̄u_{2s}, where u_{2s} ∈ Γ(M, S_{2s,1}). Then define v(s) = v(s − 1) + u_{2s}. One checks easily that v(s) satisfies (3.1), and our assertion is proved by induction.

Corollary 1. If H^{1}(M, T_{2q}) = H^{2}(M, T_{2q}) = 0 for all q ≥ p + 1, then (λ_{2p})_{1}^{∗} : H^{1}(M, Aut_{(2p)}O) → H^{1}(M, T_{2p}) is an isomorphism.

Proof. By [10], Proposition 3.2, (λ_{2p})_{1}^{∗} is injective. Under our conditions, (λ_{2p})_{1}^{∗} = (λ̂_{2p})_{1}^{∗}, and hence it is surjective by Theorem 3.1.

The following simple special case was considered in [2]:

Corollary 2. If m = rk E ≤ 3, then (λ_{2})_{1}^{∗} : H^{1}(M, Aut_{(2)}O) → H^{1}(M, T_{2}) is an isomorphism.

Example 2. Let M be an irreducible compact simply connected Hermitian symmetric space of dimension ≥ 2 and let E = T^{∗}(M). Then the conditions of Corollary 1 of Theorem 3.1 are fulfilled for p = 1, and hence (λ_{2})_{1}^{∗} : H^{1}(M, Aut_{(2)}Ω) → H^{1}(M, T_{2}) is an isomorphism. This is proved in [10], Section 4, where the complete classification of non-split supermanifolds with retract (M, Ω) is given.

3.2. Application of Hodge theory. Under assumptions of Section 2.2, endow M and E with smooth Hermitian metrics. Then we get natural Hermitian metrics in the vector bundles E^{∗} ⊗⋀ E and T(M)⊗⋀ E. Using (2.1), one can introduce a Hermitian metric in ST(M) such that the subbundle of ST_{p}(M), p ≥ 0, orthogonal to E^{∗} ⊗ ⋀^{p+1} E, is projected isometrically onto T(M) ⊗ ⋀^{p} E. Suppose that M is compact. Then we can develop the standard Hodge theory in the complex (Γ(M, R), ¯∂) or in its subcomplex S^{(2)} = (Γ(M, S^{(2)}), D̄) (see [4]). Let D̄^{∗} denote the operator conjugate to D̄ and □ = [D̄, D̄^{∗}] the Beltrami–Laplace operator. Their bidegrees are (0, −1) and (0, 0), respectively. Then we have the orthogonal decomposition

S^{(2)} = H ⊕ D̄ S^{(2)} ⊕D̄^{∗} S^{(2)},  (3.2)

where H = Ker □ is the bigraded subspace of harmonic elements. Moreover,

id = H + □G = H + D̄D̄^{∗}G + D̄^{∗}D̄G,  (3.3)

where H is the projection onto H in (3.2) and G is the so-called Green operator commuting with D̄ and D̄^{∗}. Then

Ker D̄ = H ⊕ D̄ S^{(2)},
Ker D̄^{∗} = H ⊕ D̄^{∗} S^{(2)} .

(3.4)

In particular,

H_{2p,q} ≃ H^{q}(M, T_{2p}), p ≥ 1, q ≥ 0.
H_{∗,q} ≃ H^{q}(M, T^{(2)}), q ≥ 0.

Clearly, the subcomplexes S^{(2p)} = Γ(M, S^{(2p)}), p ≥ 1, are invariant with respect to D̄^{∗}, □ and G.

Let us denote L_{0} = D̄^{∗}(S_{1}^{(2)}) and L_{1} = Ker D̄^{∗} ∩ S_{1}^{(2)}. Then (3.2) and (3.4) imply

S_{1}^{(2)} = D̄(L_{0}) ⊕ L_{1}.  (3.5)

We are going now to establish a non-linear analogue of this decomposition. Define the mapping Q : L_{0} × L_{1} → S_{1}^{(2)} by

Q(v_{2} + v_{4} + … + v_{2r}, u) = ρ(exp v_{2r}) … ρ(exp v_{2})(u).

Here v_{2q} ∈ (L_{0})_{2q}, q = 1, …, r, u ∈ L_{1}, r = [m/2], where m = rk E. Note that S^{(2p)} = 0 for p ≥ r + 1.

Theorem 3.2.

(1) The mapping Q : L_{0} × L_{1} → S_{1}^{(2)} is bijective.

(2) Q defines a bijection L_{0} × (Z^{1}(K_{2}) ∩ L_{1}) → Z^{1}(K_{2}).

(3) The natural mapping of Z^{1}(K_{2})∩L_{1} into H^{1}(K_{2}) = H^{1}(M, Aut_{(2)}O) is surjective.

Proof. (1) Let us prove, for any p = 1, …, r, the following assertion: (A_{p}). Assume that

w = w_{2} + … + w_{2p−2} + w_{2p} + … ∈ S_{1}^{(2)}

is given, where w_{2q} ∈ S_{2q,1} and w_{2q} ∈ L_{1} for q = 1, …, p − 1. Then there exist v_{2p} ∈ (L_{0})_{2p} such that

ρ(exp v)(w) = w_{2} + … + w_{2p−2} + w̃_{2p} + w̃_{2p+2} …,

where w̃_{2q} ∈ S_{2q,1} and w̃_{2p} ∈ L_{1}.

If (A_{p}) is proved for p = 1, …, r, then for any w ∈ S_{1}^{(2)} we can find elements v_{2p} ∈ (L_{0})_{2p}, p = 1, …, r, such that ρ(exp v_{2r}) … ρ(exp v_{2})(w) ∈ L_{1}. This implies that Q is surjective.

To prove (A_{p}), we note that by (3.5) w_{2p} = −D̄v_{2p} + w̃_{2p}, where v_{2p} ∈ (L_{0})_{2p}, w̃_{2p} ∈ (L_{1})_{2p}. Using Proposition 2.3, we see that

ρ(exp v_{2p})(w) = δ_{0}(exp v_{2p}) + (exp v_{2p})w(exp v_{2p})^{−1}
= E(ad v_{2p})(D̄v_{2p}) + (exp ad v_{2p})(w)
= w_{2} + … + w_{2p−2} + w_{2p} − D̄v_{2p} + w′
= w_{2} + … + w_{2p−2} + w̃_{2p} + w′,

where w′ ∈ S_{1}^{(2p+2)}.

To prove the injectivity of Q, suppose that

ρ(exp v_{2r}) … ρ(exp v_{2})(u) = ρ(exp v′_{2r}) … ρ(exp v′_{2})(u′),

where v_{2q}, v′_{2q} ∈ (L_{0})_{2q}, u, u′ ∈ L_{1}. We have to prove that v_{2q} = v′_{2q} for all q. Assume that it is true for q = r, …, p + 1. Then

ρ(exp v_{2p})(w) = ρ(exp v′_{2p})(w′),

where

w = ρ(exp v_{2p−2}) … ρ(exp v_{2})(u), w′ = ρ(exp v′_{2p−2}) … ρ(exp v′_{2})(u′).

Write

w = w_{2} + … + w_{2p−2} + w_{2p} + …, w′ = w′_{2} + … + w′_{2p−2} + w′_{2p} + …,

where w_{2q}, w′_{2q} ∈ S_{2q,1}. Using Proposition 2.3, we see that

ρ(exp v_{2p})(w) = w_{2} + … + w_{2p−2} + w_{2p} + D̄v_{2p} + …,
ρ(exp v′_{2p})(w) = w′_{2} + … + w′_{2p−2} + w′_{2p} + D̄v′_{2p} + …,

where the dots denote elements lying in S_{1}^{(2p+2)}. Therefore w_{2p} + D̄v_{2p} = w′_{2p} + D̄v′_{2p}, whence D̄v_{2p} = D̄v′_{2p}, and v_{2p} = v′_{2p}, since v_{2p}, v′_{2p} ∈ L_{0}. Thus, our assertion is proved by induction.

(2) follows from (1), since Z^{1}(K_{2}) is invariant under ρ, and (3) follows from (2).

Theorem 3.2 implies that any element of H^{1}(M, Aut_{(2)}O) is represented by a cocycle from the subset K = Z^{1}(K_{2}) ∩ L_{1}. We will study now the structure of this algebraic subset of the Frechet space S_{1}^{(2)}.

Proposition 3.1. The subset K is connected.

Proof. For any u = ∑_{k≥1} u_{2k}, where u ∈ S_{2k,1}, consider the complex curve u(t) = ∑_{k≥1} t^{k}u_{2k}, t ∈ ℂ, connecting u with 0 = u(0). If u ∈ Z^{1}(K_{2}), then u(t) ∈ Z^{1}(K_{2}) for any t ∈ ℂ. This follows, e.g., from Proposition 2.6, since d_{c} is an automorphism of the complex K_{2} for any c ∈ ℂ^{×}. Therefore, u(t) ∈ K for any t ∈ ℂ whenever u ∈ K.

Consider now the algebraic subset of S_{1}^{(2)} given by

u − \frac{1}{2}D̄^{∗}G[u, u] = Hu.  (3.6)

Clearly, (3.3) implies that K ⊂ K_{0}. Also, K_{0} ⊂ L_{1}.

Theorem 3.3.

(1) There exists a polynomial mapping φ : H_{∗,1} → L_{1} such that φ(H_{∗,1}) = K_{0}, H ∘φ = id, φ∘H = id on K_{0}, φ(0) = 0, d_{0}φ = id. In particular, H : K_{0} → H_{∗,1} is a bijection.

(2) H determines a bijection of K onto the connected algebraic subset V of H_{∗,1} ≃ H^{1}(M, T^{(2)}) given by the equation

H[φ(h), φ(h)] = 0, h ∈ H_{∗,1}.  (3.7)

Proof. We are going to solve the equation (3.6), where h = Hu ∈ H_{∗,1} is a given harmonic element, while u is an unknown element of S_{1}^{(2)} (necessarily lying in L_{1}). Writing h = ∑_{k≥1} h_{2k}, u = ∑_{k≥1} u_{2k}, where h_{2k}, u_{2k} ∈ S_{2k,1}, we obtain the following equations:

u_{2} = h_{2},
u_{4} − \frac{1}{2}D̄^{∗}G[u_{2}, u_{2}] = h_{4},
...............................
u_{2k} − \frac{1}{2}D̄^{∗}G ∑_{s=1}^{k−1}[u_{2s}, u_{2(k−s)}] = h_{2k},
...............................

One proves by induction that they admit a unique solution u = φ(h), where φ : H_{∗,1} → L_{1} is a polynomial mapping. Thus,

φ(h) − \frac{1}{2}D̄^{∗}G[φ(h), φ(h)] = h  (3.8)

for any h ∈ H_{∗,1}. Clearly, φ satisfies the conditions of (1).

To prove (2), we have to characterize those h ∈ H_{∗,1}, for which D̄φ(h) = \frac{1}{2}[φ(h), φ(h)]. It follows from (3.3) and (3.8) that

\begin{matrix}D̄φ(h) − \frac{1}{2}[φ(h), φ(h)] \\ = & Hu + \frac{1}{2}(D̄D̄^{∗}G[φ(h), φ(h)] − [φ(h), φ(h)]) \\ = & − \frac{1}{2}(H[φ(h), φ(h)] + D̄^{∗}D̄G[φ(h), φ(h)]). \end{matrix}  (3.9)

If D̄φ(h) − \frac{1}{2}[φ(h), φ(h)] = 0, then, clearly,

H[φ(h), φ(h)] = D̄^{∗}D̄G[φ(h), φ(h)] = 0.

Thus, h ∈ V. Conversely, suppose that h ∈ V, i.e., H[φ(h), φ(h)] = 0. We prove that D̄^{∗}D̄G[φ(h), φ(h)] = 0 which implies that D̄φ(h)− \frac{1}{2}[φ(h), φ(h)] = 0. In fact, we see from (3.9) that

D̄^{∗}D̄G[φ(h), φ(h)] = 2D̄^{∗}G[D̄φ(h), φ(h)]
=D̄^{∗}G[[φ(h), φ(h)] − D̄^{∗}D̄G[φ(h), φ(h)], φ(h)]
= − D̄^{∗}G[D̄^{∗}D̄G[φ(h), φ(h)], φ(h)].

Denoting ψ(h) = D̄^{∗}D̄G[φ(h), φ(h)], we obtain

ψ(h) = −D̄^{∗}[ψ(h), ψ(h)].

Writing ψ(h) = ∑_{k≥1} ψ_{2k}(h), where ψ_{2k}(h) ∈ S_{2k,1}, we easily prove by induction that ψ_{2k}(h) = 0 for any k ≥ 1. Thus, ψ(h) = 0.

The connectedness of V follows from Proposition 3.1.

Corollary. Suppose that H^{2}(M, T^{(4)}) = 0, where T^{(4)} = ⨁_{p≥2} T_{2p}. Then K = K_{0}, and the mapping H : K → H_{∗,1} is a bijection.

Proof. Under our condition, the equation (3.7) is satisfied identically, whence V = H_{∗,1}.

Thus, K is bijectively projected onto the connected algebraic subvariety V of the graded vector space H_{∗,1} ≃ H^{1}(M, T^{(2)}) of finite dimension given by (3.7). This is an analogue of the Kuranishi family of complex structures on a compact manifold (see [6, 7]). By Theorems 3.2 and 3.3, we can regard V as a moduli variety for the cohomology set H^{1}(M, Aut_{(2)}O). Of course, this cohomology set is in general a non-trivial quotient of V.

As an example, consider the case n = dim M = 1. Then, clearly, Z^{1}(K_{2}) = Ker D̄ ∩ S_{1}^{(2)}, L_{1} = H_{∗,1}, and hence V = K_{0} = H_{∗,1}.

Theorem 3.4. Suppose that M is a compact Riemannian surface and m ≤ 5. Two cocycles u, u′ ∈ H_{∗,1} represent the same cohomology class in H^{1}(K_{2}) if and only if there exists v_{2} ∈ Γ(M, T_{2}) such that

\begin{aligned}u′_{2} & = u_{2}, \\ u′_{4} & = u_{4} + H[v_{2}, u_{2}]. \end{aligned}  (3.10)

Proof. Let v = v_{2} + v_{4} ∈ S_{0}^{(2)} satisfies u′ = ρ(v)(u). Using Proposition 2.3, we see that

\begin{aligned}u′_{2} & = u_{2} + D̄v_{2}, \\ u′_{4} & = u_{4} + D̄v_{4} + \frac{1}{2}[v_{2}, D̄v_{2}] + [v_{2}, u_{2}]. \end{aligned}  (3.11)

Clearly, (3.11) implies (3.10), where D̄v_{2} = 0. Conversely, if (3.10) holds for a certain v_{2} ∈ Γ(M, T_{2}), then we can find a v_{4} ∈ S_{4,0} such that u′_{4} = u_{4}+D̄v_{4}+[v_{2}, u_{2}]. Clearly, (3.11) will be satisfied, whence u′ = ρ(v_{2}+v_{4})(u).

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70 A.L. ONISHCHIK Yaroslavl University, Sovietskaya 14, 150 000 Yaroslavl, Russia

E-mail address: arkadiy@onishchik.msk.ru or onishch@univ.uniyar.ac.ru Received July 27, 1999