1 Introduction

The Batalin–Vilkovisky formalism (henceforth the “BV formalism”) was originally designed in the early 1980’s as a tool to deal with the perturbative quantization of gauge theories. Nowadays, its importance goes far beyond its original purpose: the BV formalism has grown into one of the foundational languages of contemporary theoretical and mathematical physics [7, 8, 10, 14, 21], with several applications also to pure mathematics [8, 23].

It was Albert Schwarz in [24] who first elucidated the geometric framework that lies at the basis of the BV formalism, by recognizing the crucial role played by supergeometry. The BV formalism builds upon the BRST formalism, that in turn introduced a new point of view on the so-called Faddeev–Popov procedure. In particular, the BRST formalism identifies the space of fields of a gauge theory with “functions” on a supermanifold , typically constructed as the split supermanifold generated by an action Lie algebroid associated to a Lie group action , so that is trivialized as . In this framework, ghost fields are not just byproducts of an integration procedure, as in the Faddeev–Popov procedure. Instead, they are conceptually understood as proper geometric data: odd sections of related to the generators of the Chevalley–Eilenberg algebra resolving the (infinitesimal) gauge symmetry \({\mathfrak {g}}.\) The BV formalism enhances the above “BRST package” by crucially allowing symplectic geometry to enter the picture: starting from , one constructs a canonically associated (odd) symplectic supermanifold as the total space supermanifold of the (parity-shifted) cotangent bundle . This is called BV space or BV supermanifold. Functions on are given by polynomial functions on the fibers of , that is . This means that if in the BRST setting the local \(n\,{|}\,m\) coordinates \((x^i\!, \theta ^\alpha )\) of are identified with the fields \(x^i\) and the ghosts \(\theta ^\alpha \) of the related physical theory, in the BV setting these get supplemented by another set of \(m\,{|}\,n\) coordinates \(( q_\alpha \,{|}\, p_i)\), accounting for the fiber directions of and identified with the so-called anti-fields \(p_i\) and anti-ghosts \(q_\alpha \). In a similar fashion as above, whereas the odd ghosts \(\theta _\alpha \) provided a homological resolution for the gauge symmetry, the new odd generators in —the anti-fields \(q_i\)’s—provide a homological (Koszul–Tate) resolution of the critical locus of the action , showing once again the vicinity of the formalism with a homological or, better, derived geometric point of view.

Generally speaking, the most interesting aspects of supergeometry are those which do not arise as a generalization of the ordinary commutative theory, but instead force us to revise our classical geometric intuition and to confront ourselves with unexpected new features. These new features are both of local and global nature. Locally, the geometry of forms on supermanifolds and the related integration theory present the most peculiar non-trivial novelties: the failure of a trivial generalization of Poincaré duality leads to the introduction of a new kind of forms, which are crucial for the purpose of a meaningful integration theory and for this reason are called integral forms. On the other hand, globally, complex supermanifolds can be non-split or non-projected: this means that they cannot be reconstructed from ordinary geometric data, but instead they are genuinely new geometric spaces living a life of their own.

In this paper we relate these two aspects, by starting from the geometry of forms arising from a BV supermanifold of the type of above. More precisely, the paper is organized as follows. In Sect. 2 we recall the main definitions of the supergeometric objects and constructions that we will use. We then address the global aspects related to the geometry of forms on BV supermanifolds in Sect. 3. In particular, we prove that 1-forms on -type supermanifolds are an extension of vector bundles defined on the base supermanifold, Theorem 3.2. For real supermanifolds we show in Theorem 3.4 that this extension is always split and the splitting corresponds to a reduction of the structural symplectic supergroup. More interestingly, building upon a projection to (the cohomology of) the base supermanifold of , Theorem 3.3, in the case of complex supermanifolds we show that the extension is split if and only if the super Atiyah class of the base supermanifold vanishes, corresponding to the existence of a holomorphic connection, Theorem 4.3. We then show how this condition is related to the characteristic non-split or non-projected geometry of complex and algebraic supermanifolds in Theorem 4.9 and we comment on future perspectives in this context. Several examples spanning different levels of sophistication are discussed in Sect. 5. From a local point of view, in Sect. 6, we focus on the geometry related to the symplectic nature of the supermanifold and we show that the associated deformed de Rham double complex—whose differentials are given by the odd symplectic form and the de Rham differential—naturally arises as a de-quantization of the de Rham/Spencer double complex associated to the base supermanifold. Following Ševera, we show in Theorem 6.13 that the related spectral sequence yields semidensities on the odd symplectic BV supermanifold and their differential in the form of a super BV Laplacian.

2 Main definitions: local and global data

In this section we recollect the definitions of the main geometric objects that will be used throughout the paper. For thorough introductions to the geometry of supermanifolds the reader is suggested to refer to the books [2, 18]. The recent [22] offers a detailed account of the geometry of forms on supermanifolds. We let be a smooth, analytic or algebraic supermanifold of dimension \(p\,{|}\,q\) with structure sheaf given by and we denote by its reduced space, which is an ordinary (commutative smooth, analytic or algebraic) manifold of dimension p with structure sheaf . We will denote by and the even, respectively, odd part of the structure sheaf with respect to its \(\mathbb {Z}_2\)-gradation, and likewise for other sheaves or vector bundles introduced in the following. We define be the tangent sheaf of . This is a locally-free sheaf of (left) -modules of rank \(p\,{|}\,q\): if we let U be an open set in the topological space underlying and \(x_a \,{:}{=}\,\,z_i \,{|}\, \theta _\alpha \) for \(i = 1, \ldots , p\) and \(\alpha = 1, \ldots , q\) be a system of local coordinates over U for , then

where is the structure sheaf of and the local generators \(\partial _{z_i}\)’s are even and the \(\partial _{\theta _\alpha }\)’s are odd. Given the tangent sheaf as defined above, we can immediately introduce two related sheaves. The first one is the cotangent sheaf , which is the dual of , i.e. . The second one is the parity shifted tangent sheaf , which is a locally-free sheaf of -module of rank \(q\,{|}\,p\). With reference to the above trivialization over U, the parity shifted tangent sheaf is locally generated as follows:

where we stress that the local generators \(\pi \partial _{\theta }\)’s are even and the \(\pi \partial _{z}\)’s are odd. Sections of the parity-shifted tangent sheaf are called \(\Pi \)-vector fields or vector fields for short. We call the dual of the parity-shifted tangent sheaf the sheaf of 1-forms on and we denote it as . This is a locally-free sheaf of (right) -modules of rank \(q\,{|}\,p\): over an open set U one has the trivialization

where the local generators \(d\theta \)’s are even and the dz’s are odd. The duality paring between and over U reads \(dx_a (\pi \partial _{x_b}) = \delta _{ab}\) for any a, b both even or odd. The name sheaf of 1-forms is in some sense justified by the fact that in supergeometry it is customary take the de Rham differential to be an odd morphism, so that one indeed has a sheaf morphism which satisfies the \(\mathbb {Z}_2\)-graded Leibniz rule. Application of the (super)symmetric power functor to the sheaf of 1-forms and to yields respectively the algebra of forms and \(\Pi \)-vector fields on . A section of is called a k-form and a section of is called a (k-)\(\Pi \)-polyfield or k-polyfield for short. In this context the de Rham differential lifts to the exterior derivative , which is an odd derivation of , i.e. it obeys the \(\mathbb {Z}_2\)-graded Leibniz rule in the form

$$\begin{aligned} d (\omega \eta ) = d \omega \eta + (-1)^{|\omega |} \omega \,d \eta , \end{aligned}$$

where \(\omega \) and \(\eta \) are two generic forms in and where we have denoted by \(|\omega |\) the \(\mathbb {Z}_2\)-degree, henceforth parity of \(\omega \). Further, it is easy to see that the exterior derivative is nilpotent: the pair defines a sheaf of dg-algebras, the de Rham complex of .

As it is well known, differential forms are not suitable for integration in a supergeometric setting [18]: this leads to the introduction of a second complex, which is “dual” to the de Rham complex. This is the so-called complex of integral forms, which are defined as sections of the tensor product of sheaves , where is the Berezinian sheaf of , see [18, 21, 22], which substitutes the notion of canonical sheaf of an ordinary manifold and whose sections are tensor densities. The interested reader can refer to [18] or the recent [5] and [22] for a construction ab initio of the differential for integral forms.

After this preliminary recollections of definitions, conventions and notations we introduce one of the main objects under study in this paper.

Definition 2.1

(BV supermanifold \(\textsf{M}\)) Let be a smooth, analytic or algebraic supermanifold of dimension \(p\,{|}\,q\) and let be its sheaf of 1-forms. We call the \(p+q \,{|}\, p + q\)-dimensional supermanifold defined as a ringed space by the pair , where the topological space is given by the total space of the vector bundle of rank \(p+q\) endowed with its canonical topology, and the structure sheaf \({\mathcal {O}}_{\tiny {{\textsf{M}}}}\) is defined as , where is taken with its -module structure.

Remark 2.2

Notice that the canonical topology on the total space of a vector bundle is defined locally via the product topology, and then glueing along the transition functions: the related quotient topology is the desired topology on the total space of the vector bundle.

Remark 2.3

The previous definition says that “functions” on \({\textsf{M}}\) are polynomial functions on the fibers , i.e. they are polyfields having shifted parity. Locally, on open sets of the kind for U an open set in , one has that

for even and odd fiber coordinates and \(\mathbb {K}\) the field of real or complex numbers. In the following subsection we will give an explicit local description via transition functions.

Remark 2.4

A notational remark is in order. Indeed, in the introduction of the paper we have denoted the supermanifold \({\textsf{M}}\) defined above in Definition 2.1 with , to distinguish it from the supermanifold arising in the context of the BRST formalism. In the following we will always consider the supermanifold .

2.1 Local description

It is worth to provide a local description of \({\textsf{M}}\) in terms of transition functions of its local coordinates. We let \((U, x_a)\) be a local chart for the \(p\,{|}\,q\)-dimensional supermanifold , where we stress that the index a spans both even and odd local coordinates. Then \((\pi ^{-1} (U), x_a, p_a)\) is a chart for \({\textsf{M}}\) with

$$\begin{aligned} p_a \,{:}{=}\,(-1)^{|x_a|}\, \partial _{dx_a}. \end{aligned}$$

The following is immediate.

Lemma 2.5

(Transition functions of \({{\textsf{M}}}\)) Let \((U, x_a)\) and \((V, z_b)\) be two charts on with \(U \cap V \ne \varnothing \), and let \((\pi ^{-1}(U), x_a, p_a)\) and \((\pi ^{-1} (V), z_b, q_b)\) be the corresponding open sets on \({\textsf{M}} \). Then the transition functions of  \({{\mathsf {{M}}}}\) read

Proof

The first ones are obvious, being the transition functions on . For the latter, it is enough to observe that from \(d z_b = dx_a (\partial _{x_a} z_b)\) it follows that

where we have made use of the definition of \(q_b\) in the last step. \(\square \)

3 The geometry of forms: split and non-split extensions

We now study the geometry of the cotangent sheaf of the supermanifold \({\textsf{M}}\). Note that this is a locally-free sheaf of -modules of rank \(p+q \,{|}\, p+q\). We can characterize its transition functions thanks to Lemma 2.5.

Lemma 3.1

(Transition functions of ) Let \({\textsf{M}} \) be defined as above and let \((dx_a, dp_a)\) and \((dz_b, dq_b)\) be two local bases of on the open sets \(\pi ^{-1} (U)\) and \(\pi ^{-1}(V)\) on \({\textsf{M}} \) with \(U \cap V \ne \varnothing \). Then the transition functions of read

$$\begin{aligned} dx_a&= dz_b \biggl ( \frac{\partial x_a}{\partial z_b} \biggr ), \end{aligned}$$
(3.1)
$$\begin{aligned} dp_a&= \biggl (\frac{\partial z_b}{\partial x_a } \biggr )\, dq_b + (-1)^{|x_a| + |z_b|} \,d \biggl ( \frac{\partial z_b}{\partial x_a} \biggr )\, q_b. \end{aligned}$$
(3.2)

Proof

The first ones are obvious. For the transition functions of the dp’s we observe that we have

$$\begin{aligned} d p_a = dz_b \biggl ( \frac{\partial p_a}{\partial z_b} \biggr ) + dq_b \biggl ( \frac{\partial p_a}{\partial q_b} \biggr ). \end{aligned}$$
(3.3)

The first summand reads

The second summands reads

\(\square \)

The previous lemma describes locally in terms of its transition functions, but it yields information also on its global geometry, as the following shows.

Theorem 3.2

( as extension of vector bundles) Let \({\textsf{M}} \) be defined as above. Then the canonical exact sequence

induces the isomorphism . In particular, is an extension of locally-free sheaves

(3.4)

Proof

We work in the same setting of Lemma 3.1. We first observe that the transformations of equation (3.1) identify the sections \((dx_a)\)’s as a local basis of (notice the slight abuse of notation). The first summand in the transformations given by equation (3.2) identifies the transformations of the parity-reversed dual of , as the \(dp_a\) have opposite parity with respect to the \(dx_a\). This is hence identified with . The second summand in (3.2) gives the off-diagonal term of the extension of with . \(\square \)

It follows from the previous theorem that in order to study the geometry of one needs to consider the cohomology group

(3.5)

which controls the splitting of the exact sequence (3.4), that will be called -extension in the rest of the paper. For ease of reading, we have deferred to the appendix a very concrete construction of the Ext-group related to an extension, which highlights the structure of the representatives in the above cohomology group in terms of the transition functions of the vector bundles involved. As we shall see, this concrete approach will play a significant role in what follows.

It is convenient to re-express this Ext-group appearing in (3.5) as a cohomology group computed on the supermanifold —and hence on —instead of \({\textsf{M}}\).

Theorem 3.3

(Projection to ) Let be a smooth, analytic or algebraic supermanifold, and let \({\mathsf {{M}}} \) be constructed as above with its projection map. Then one has the following natural isomorphism:

Proof

First, notice that

Since in the given hypotheses, is an affine morphism, then by Leray’s spectral sequence

Finally, by projection formula applied to \(R^i \pi _{*} \)—in the case \(i=0\) (see [13, p. 253]) we have

Further, since as -modules, this can be rewritten as

Finally, the linear dependence on p in the second summand of (3.2) shows that this extension class appears in degree one only in , i.e. in the summand of the above direct image, so that one finds

The conclusions follow observing that

and that , so that

\(\square \)

The above theorem can be applied to smooth real supermanifolds, as to show the existence of a reduction of the structure group of . To this end, following [18, Chapter 4, Section 10], we recall that the structure group of is given by the symplectic supergroup \(\Pi \text {Sp} (p+q\,{|}\,p+q)\), that can be understood as the stabilizer of the “metric” in given by the odd symplectic form \(\omega \)—whose related geometry will be discussed in Sect. 6, see Definition 6.6.

Theorem 3.4

(Splitting & reduction of symplectic supergroup) Let be a smooth supermanifold and let be the smooth supermanifold associated to as defined above. Then the following are true:

  1. (1)

    The -extension

    is split, i.e. non-canonically.

  2. (2)

    There exists a reduction of the structure group of as follows:

    where \(\Pi { {\text {Sp}}}(p+q\,{|}\,p+q)\) is the symplectic supergroup.

Proof

For the first point it is enough to observe that the existence of a smooth partition of unity in the smooth category leads to the exactness of the Čech cochain complex of any sheaf in degree \(i > 0\), which is therefore fine, thus soft and acyclic. Applying this to yields the conclusion, i.e.

The second point follows from the first one and Theorem 6.14 in Appendix, which generalizes to the \({\mathbb {Z}}_2\)-graded context. To this end it is enough to observe that the structure of the transition functions as in (A.4) follows from Lemma 3.1. \(\square \)

Remark 3.5

It is to be noted that the above splitting is non-canonical. It would be interesting to see if it is possible to classify or provide constraints on smooth supermanifolds such that the above reduction of the structure group is possible via a suitable choice of charts.

4 Connections and obstructions: the super Atiyah class

Theorem 3.4 proves the existence of a splitting for the -extension in the smooth category: this fact could have been easily inferred directly from equation (3.5). On the other hand, the “projection” result of Theorem 3.3 allows for a very nice interpretation when working in the complex analytic or algebraic category, where sheaves admit non-trivial higher cohomologies and the splitting of the -extension is far from obvious. Quite the contrary, we will see that in general the conditions under which the -extension splits are quite restrictive. To this end, in the following we restrict ourselves to work on complex supermanifolds in the holomorphic category: the reader shall see that everything holds true also in the algebraic category.

Definition 4.1

(Affine connection on ) Let be a complex supermanifold and let be the (holomorphic) tangent sheaf of An affine connection on is an even morphism of sheaves of \({\mathbb {C}}\)-vector spaces such that it satisfies the Leibniz rule

for any and , where is the even de Rham differential, see [18].

Obstructions to the existence of an affine connection on a complex supermanifold [2, 3, 11, 17] can be established in same fashion of the original Atiyah’s result [1] for ordinary complex manifolds. We spell out the main points of the construction following [2], which is very close to the original [1].

First, one defines the sheaf of 1-jets of . One starts introducing the sheaf of \({\mathbb {C}}\)-vector spaces given by

for U an open set of Notice that sections of are 1-forms valued in the tangent bundle. The sheaf can be endowed with the structure of sheaf of -modules as follows: let and . One defines the product

(4.1)

where is the even de Rham differential, see again [18]. One can then verify that the sequence of sheaves of -modules given by

(4.2)

where \( \alpha (\tau ) \,{:}{=}\,(0, \tau )\), and \(\beta ((X, \tau )) = X\), is exact. Notice that since is locally-free, then the sequence (4.2) is locally split, hence there exists a covering \(\{U_a \}_{a \in I}\) such that and are free and

which guarantees that is locally-free. On the other hand, due to the non-trivial -module structure of the sheaf of 1-jets of , the previous short exact sequence of locally-free sheaves of -modules (4.2)—henceforth 1-jets short exact sequence—does not necessarily split. Applying the functor , and taking the long exact sequence in cohomology one gets

(4.3)

We call this long exact sequence in cohomology the 1-jets long exact cohomology sequence. We can thus give the following definition.

Definition 4.2

(Super Atiyah class) Let be a complex supermanifold and let be its tangent sheaf. We define the Atiyah class of to be the image of the identity map via the 1-connecting homomorphism \(\delta \) in the 1-jets long exact cohomology sequence, i.e.

The following theorem is adapted from [1] to the super-setting, and it shows how the super Atiyah class is related to the existence of an affine connection on the complex supermanifold .

Theorem 4.3

(Pseudo-Atiyah) Let be a complex supermanifold and let be the tangent sheaf of , then:

  1. (1)

    the 1-jets short exact sequence (4.2) splits if and only if there exists an affine connection on ;

  2. (2)

    there exists an affine connection on if and only if   is trivial.

In particular, let \(\{U_i\}_{i \in I}\) be an open covering for and \(\{ g_{ij} \}_{i, j \in I}\) be the transition functions of   on the intersections \(U_i \cap U_j\), then the Atiyah class of   is represented by the Čech 1-cocycle

(4.4)

Proof

The first point is the crucial one. First, we let \(\nabla \) be an affine connection on and we define the morphism by \(j (X) \,{:}{=}\,(X, \nabla X)\). Notice that \(s_{\nabla }\) is a well-defined morphism of sheaves of -modules, as

by equation (4.1). By definition of the 1-jets short exact sequence (4.2), one has that the surjective morphism is given by \(\beta ((X, \tau )) = X\). Therefore , which implies that the affine connection \(\nabla \) determines a splitting \(s_{\nabla } \) of the 1-jets short exact sequence, i.e.

Vice versa, let the 1-jets short exact sequence (4.2) be split. Then there exists a morphism of sheaves of -modules such that . We let then be defined by \(p ((X, \tau )) = \tau \). Notice that p is \({\mathbb {C}}\)-linear, but not -linear. Let us then define It is immediate that \(\nabla ^{(s)}\) is \({\mathbb {C}}\)-linear. Finally, it satisfies the Leibniz rule, indeed

for any and It follows that \(\nabla ^{(s)}\) defines an affine connection.

The second point of the theorem depends on the first one. Let . Then, by definition . By exactness, it follows from the 1-jets long cohomology exact sequence (4.3)

that there exists an element such that , where \(\beta \) is the surjection in 1-jets short exact sequence (4.2), which therefore splits. By the previous point of the theorem, this is equivalent to the existence of an affine connection on .

Vice versa, let be such that it admits an affine connection. Then the 1-jets short exact sequence is split by the previous point of the theorem. This implies that there exists a map such that . It follows that belongs to the image of the map and hence i.e. .

For the last point, we let \(\{U_i \}_{i\in I}\) be an open covering of and \(\{g_{ij} \}_{i, j \in I}\) be the transition functions of on the intersections \(U_i \cap U_j\). For the sake of notation, compositions of maps are left understood in what follows. A Čech 1-cocycle representation of the class in terms of \(\{g_{ij} \}_{i,j \in I} \) can be obtained by letting \(\nabla _i \) be the (flat) connection on which is determined by a fixed trivialization relative to \(\{ U_i\}_{i\in I}.\) In particular, following [1] and [2] we let

where \(\phi _i \) is a trivialization on \(U_i\), and we define

by

$$\begin{aligned} {\mathfrak {a}}_{ij} = \nabla _j - \nabla _i. \end{aligned}$$

Observing that , one computes

This simplifies to

where we have used the Leibniz rule applied to . It follows that

Finally, upon using , one checks that , i.e. it defines a Čech 1-cocycle in the cohomology of . \(\square \)

Remark 4.4

With reference to the last part of Theorem 4.3, one can notice that a local holomorphic connection can be written in the form in a trivialization with a matrix-valued holomorphic 1-form on \(U_i\). These can be patched together to form a globally defined (holomorphic) affine connection if and only if

that can be rearranged as

(4.5)

Then, in view of Theorem 4.3, the left-hand side is (a Čech 1-cocycle representing) the Atiyah class of , and equation (4.5) can be written as

where the right-hand side is the Čech coboundary of . This shows via Čech cohomology that local connections can be patched together if and only if providing a different proof of the second point of Theorem 4.3 in a local-to-global fashion, as it is customary in Čech cohomology.

Remark 4.5

Further, notice that the same construction as above can be carried out for any locally-free sheaf on , not only the tangent sheaf . In this respect the non-vanishing of the corresponding Atiyah class, which we still denote as , is an obstruction to define a holomorphic connection on .

The previous Theorem 4.3 allows to identify the obstruction to splitting the -extension.

Theorem 4.6

(\(\Omega ^{1}_{ \tiny {{\textsf{M}}}}\) and the Atiyah class) Let be a complex supermanifold and let \({\mathsf {{M}}}\) be constructed as above. Then the -extension

(4.6)

is split if and only if  is trivial. In particular, the short exact sequence is split if and only if admits an affine connection.

Proof

By the previous Theorem 3.3 obstructions to splitting the short exact sequences lie indeed in . By Lemma 6.14 and the structure of the transition functions given in Theorem 3.1 one sees that the obstructions are represented as Čech 1-cocycles by elements of the form , (where the \(g_{ij}\)’s are the transition functions of the tangent sheaf ), which is identified with the Atiyah class by (4.4). \(\square \)

We now aim to relate the splitting of the short exact sequence (4.6) to the geometry of the complex supermanifold . To this end we first recall some basic constructions specific to the theory of complex supermanifolds, see [5] or [18]. To each complex supermanifold is attached the short exact sequence

(4.7)

where is the sheaf of nilpotent sections in and is the structure sheaf of the reduced space —and ordinary complex manifold—of the supermanifold . If (4.7) splits, then the supermanifold is said to be projected, because the splitting corresponds to the existence of a “projection” morphism such that , if is the canonical embedding of the reduced space into the supermanifold . Moreover, the quotient defines a locally-free sheaf of -modules of rank q —where q is the odd dimension of —and whose sections are seen to be odd. We call the quotient the fermionic sheaf of and we denote it by . We say that the supermanifold is split if its structure sheaf is globally isomorphic to the sheaf of exterior algebras over . Notice that a split supermanifold is in particular projected. The corresponding obstruction theory to splitting a supermanifold is currently a compelling active research topic, see for example [3, 22].

In this context, the fundamental obstruction class to splitting a supermanifold is given by a class

If is non-vanishing then is non-projected and in particular non-split. Whereas the fundamental obstruction class is always defined, higher obstruction classes

for \(i\geqslant 1\) are defined if and only if all of the previous ones are vanishing, see the discussion in [11] for example.

A different criterion, actually a sufficient condition, for the existence of a splitting of a supermanifold, has been given by Koszul in [17], relating the question about the splitting of to the existence of an affine connection on it.

Theorem 4.7

(Koszul) Let be a complex supermanifold. If admits an affine connection, then it is split. In particular, the affine connection defines a unique splitting of the supermanifold.

Proof

See [17], recently reviewed in [3].\(\square \)

This result can in turn be related with a recent result by Donagi and Witten [11]. Indeed, when restricted to the reduced space, the tangent and cotangent sheaf and split into a direct sum of an even and an odd part. The latter is isomorphic to the fermionic sheaf or its dual in the case of the cotangent and tangent sheaf respectively. More precisely, one finds

In this spirit, one of the key results in [11] concerns the decomposition of the Atiyah class of upon restriction of the tangent sheaf to the reduced manifold .

Theorem 4.8

(Donagi & Witten) Let be a complex supermanifold. Then, the restriction of the tangent sheaf to induces the following decomposition of the cohomology group :

In particular, with respect to the above decomposition the Atiyah class decomposes as follows:

(4.8)

where is the Atiyah class of the tangent sheaf of , is first obstruction class, and is the Atiyah class of the fermionic sheaf.

Proof

See [11]. \(\square \)

This result together with Koszul’s Theorem 4.7 leads to the following for the geometry of .

Theorem 4.9

(Splitting of ) Let be a complex supermanifold and let \({\mathsf {{M}}}\) be constructed as above. Then any of the following is an obstruction to split -extension (4.6):

  1. (1)

    , i.e. does not admit a holomorphic connection;

  2. (2)

    , i.e. does not admit a holomorphic connection;

  3. (3)

    , i.e. is non-projected or non-split.

In particular, a necessary condition for the -extension to split is that is a split supermanifold.

Proof

The -extension is split if and only if the Atiyah class of vanishes by Theorem 4.6 and the three obstructions to split the -extension follow from the decomposition at the Atiyah class given in (4.8) of Theorem 4.8. Finally, the vanishing of the Atiyah class implies the existence of an affine connection, which is equivalent to the existence of a splitting for by Theorem 4.7.\(\square \)

We conclude this section with some general remarks and speculation on the nature of the super Atiyah class for a complex supermanifold.

Remark 4.10

(Super Atiyah class & super characteristic classes) It should be clear by the above considerations that the vanishing of the super Atiyah class provides a very strong constraint on the geometry of a complex or algebraic supermanifold. Namely, the following is an immediate consequence of the Koszul’s result, Theorem 4.7, and the very definition of split supermanifold.

Corollary 4.11

Let be a complex supermanifold of dimension \(n\,{|}\,m\) such that , then is split. In particular all of the obstruction classes to splitting vanish, i.e. for any \(i = 1, \ldots , \lfloor m/2 \rfloor \),

This, together with Theorem 4.8, should make apparent the existence of a close relation between the super Atiyah class and the obstructions to splitting a complex supermanifold (see also the recent [3] in this regard).

It should be stressed indeed that, in a classical setting, for a compact complex Kähler manifold X, the Atiyah class of tangent bundle contains information about all the Chern classes \(c_k (X) = c_{k} ({\textsf{T}}_X)\) of the manifold [15]. In particular, if X admits a holomorphic connections, i.e. the Atiyah class of \({\textsf{T}}_X\) vanishes, then all the Chern classes vanish as well, i.e. \(c_k (X) = 0\) for any \(k>0\). It can be reasonably conjectured that, in a very similar fashion, also on a complex supermanifold the Atiyah class of the tangent bundle contains information about all of the characteristic classes related to . The difference relies in that among these are to be counted not only the Atiyah classes of the reduced manifold and of the fermionic sheaf —as seen in Theorem 4.8—and hence the related Chern classes, but also all of the obstruction classes to splitting the supermanifold, which indeed arises as well from the tangent bundle of the supermanifold. Whereas the fundamental obstruction arises from the super Atiyah class upon restriction to [11], it is quite reasonable to imagine that higher obstruction classes would arise upon allowing for higher fermionic terms in the filtration of by the ideal sheaf of nilpotent sections of the supermanifold

and thus considering the restriction of to the canonical superscheme contained in defined by the pair for \(\ell > 1\), i.e.

5 Examples and further results

In this section we discuss and comment some examples. First, it is obvious that the complex supermanifold \({\mathbb {C}}^{n|m}\) admits a splitting for the -extension for any values of n and m.

Example 5.1

(\({\mathbb {C}}^{n|m}\)) Let be the complex supermanifold with structure sheaf given by and let \({\textsf{M}}\) be the supermanifold constructed from \({\mathbb {C}}^{n|m}\) as in Definition 2.1. Then, for any n and m the -extension is split.

This follows from the fact that the tangent bundle \({\textsf{T}}_{{\mathbb {C}}^{n|m}}\) of \({\mathbb {C}}^{n|m}\) is trivial, and as such it admits a connection. Then, by Theorem 4.3 its super Atiyah class \({{\mathfrak {A}}}{{\mathfrak {t}}} ({\textsf{T}}_{{\mathbb {C}}^{n|m}})\) is trivial, and in turn, by Theorem 4.6 the -extension related to \({\mathbb {C}}^{n|m}\) splits.

A way more interesting example is provided by complex Lie supergroups—the reader is suggested to refer to [26] for the relevant definitions.

Example 5.2

(Complex Lie supergroups ) Let be a complex Lie supergroup. Then admits a holomorphic connection and hence it is split. In particular, if \({\textsf{M}}\) is the supermanifold constructed from as in Definition 2.1, the -extension splits.

The result follows from the fact that, as in the ordinary theory, a complex Lie supergroup is parallelizable, i.e. its tangent bundle is trivial. Just like in the ordinary theory this depends on the existence of a group structure on the supermanifold . In turn, since the tangent bundle of is trivial, then admits a connection. It follows from Theorem 4.7 that is split and from Theorems 4.3 and 4.6 that the related -extension splits.

It is to be stressed that the case of homogeneous supermanifolds, i.e. quotients of Lie supergroups by some closed Lie sub-(super)group, is more delicate: indeed, complex homogeneous supermanifolds can indeed be non-split, thus not admitting holomorphic connection, see [26].

We now move to (complex) projective superspaces \({{\mathbb {C}}}{{\mathbb {P}}}^{\,n|m}\), which are defined as the complex supermanifolds given by the pair where . Notice that according to the discussion after Theorem 4.6 above, complex projective superspaces \({{\mathbb {C}}}{{\mathbb {P}}}^{\,n|m}\) are by definition split supermanifolds, in particular their fermionic sheaf reads . See [4] for a dedicated paper.

Example 5.3

(\({{\mathbb {C}}}{{\mathbb {P}}}^{\,n|m}\)) Let \({{\mathbb {C}}}{{\mathbb {P}}}^{\,n|m}\) be any complex projective superspace and let \({\mathsf {{M}}}\) be the supermanifold constructed from \({{\mathbb {C}}}{{\mathbb {P}}}^{\,n|m}\) as above. Then, for any value of \(n\geqslant 1\) and \(m\geqslant 0\) the -extension is not split.

The result can be seen to follow from point (1) of Theorem 4.9. For this, one needs to prove that projective spaces \({{\mathbb {C}}}{{\mathbb {P}}}^{\,n}\) do not admit affine holomorphic connections. For \(n \geqslant 1\) the (dual of the) Euler exact sequence tensored by reads

$$\begin{aligned} \!\!\!\! \!\!\!\! 0 \rightarrow \!({\textsf{T}}^*_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}})^{\otimes 2} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n} }\! \rightarrow \!({\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}}^{*} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}} (-1)){\otimes }{{\mathbb {C}}}^{\oplus n+1}\! \rightarrow \! {\textsf{T}}^*_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}} \!\rightarrow 0. \end{aligned}$$
(5.1)

Here all tensor products are over . In the case \(n > 1\) one computes

$$\begin{aligned} H^0 ({{\mathbb {C}}}{{\mathbb {P}}}^{\,n}\! , ({\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}}^{*} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^n} (-1)))&= 0, \quad H^1 ({{\mathbb {C}}}{{\mathbb {P}}}^{\,n}\!, {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}}^{*} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}} (-1)) = {\mathbb {C}}^{n+1}\!, \\ H^0 ({{\mathbb {C}}}{{\mathbb {P}}}^{\,n}\! , ({\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}}^{*} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}} ))&= {\mathbb {C}}, \quad H^1 ({{\mathbb {C}}}{{\mathbb {P}}}^{\,n}\!, {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}}^{*} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}} (-1)) = 0. \end{aligned}$$

It follows that long cohomology sequence associated to (5.1) reads

$$\begin{aligned} 0 \rightarrow \!H^0 ({{\mathbb {C}}}{{\mathbb {P}}}^{\,n} \!, {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n}}^{*} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^n} ) \cong {\mathbb {C}} \xrightarrow {\,\delta \,} \!H^1 ({{\mathbb {C}}}{{\mathbb {P}}}, ({\textsf{T}}^*)_{{{\mathbb {C}}}{{\mathbb {P}}}^n}^{\otimes 2} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^{\,n} }) \cong {\mathbb {C}}^{(n+1)^2 + 1}\! \rightarrow \cdots \end{aligned}$$

and injectivity of the map implies that \({{\mathfrak {A}}}{{\mathfrak {t}}} ({\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^n}) \ne 0\). In the case of \({{\mathbb {C}}}{{\mathbb {P}}}^1\), one has that and . The long cohomology exact sequence reads

$$\begin{aligned} 0 \rightarrow H^0 \bigl ({{\mathbb {C}}}{{\mathbb {P}}}^1\!, {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^1}^{*} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^1} \bigr ) \cong {\mathbb {C}} \xrightarrow {\; \delta \;} H^1 \bigl ({{\mathbb {C}}}{{\mathbb {P}}}^1\!, ({\textsf{T}}^*_{{{\mathbb {C}}}{{\mathbb {P}}}^1})^{\otimes 2} {\otimes } {\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^1 }\bigr ) \rightarrow 0. \end{aligned}$$

Alternatively, one can just observe that for a line bundle the Atiyah class equals the Chern class, i.e. \({{\mathfrak {A}}}{{\mathfrak {t}}} ({\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^1}) = c ({\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^1})\) and \(c ({\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^1}) = c_1 ({\textsf{T}}_{{{\mathbb {C}}}{{\mathbb {P}}}^1}) = 2,\) see [15]. It then follows from Theorem 4.9 that the -extension related to \({{\mathbb {C}}}{{\mathbb {P}}}^{\,n|m}\) does not split.

Remark 5.4

Note that in the ordinary complex geometric setting, the vanishing of the Atiyah class \({{\mathfrak {A}}}{{\mathfrak {t}}}({E}) \) of a certain (Hermitian holomorphic) vector bundle E on a compact complex manifold X is equivalent for the vector bundle E to be flat. More precisely, the class of the curvature of the Chern connection of the vector bundle E corresponds to its Atiyah class \({{\mathfrak {A}}}{{\mathfrak {t}}} (E)\). This result is by no means true in a complex supergeometric setting. Indeed all of the Calabi–Yau projective superspaces \({{\mathbb {C}}}{{\mathbb {P}}}^{\,n|n+1}\) for \(n\geqslant 1\) admit a flat Chern connection via a generalization of the Fubini–Study metric [20], but none of them have vanishing super Atiyah class as shown above.

Another interesting example is provided by the possibly easiest—yet non-trivial—complete intersection in \({{\mathbb {C}}}{{\mathbb {P}}}^{\,2|2}\), corresponding to the super conic cut out by the equation

$$\begin{aligned} X_0^2 + X_1^2 + X_2^2 + \Theta _1 \Theta _2 = 0 \, \subset \, {{\mathbb {C}}}{{\mathbb {P}}}^{\,2|2}. \end{aligned}$$
(5.2)

The supergeometry related to the above equation in projective superspace is non-trivial and not that well-known outside an audience of experts in supergeometry. To help the reader make sense and appreciate this example, we have opted to discuss some of the more peculiar aspects of the geometry of the super conic in Appendix B. We thus refer to it for more information. Here we content ourselves to say that the resulting \(1\,{|}\,2\)-dimensional supermanifold is notably isomorphic to the non-projected supermanifold constructed out of the following three data

where the non-zero obstruction class is seen via the isomorphism . We have already seen that the Atiyah class of \({{\mathbb {C}}}{{\mathbb {P}}}^1\) is non-zero in Example 5.3, moreover the rank 2 vector bundle is obviously not flat, and hence it has a non-trivial Atiyah class. We can thus conclude that -extension related to the super conic is totally obstructed in the sense of Theorem 4.9, i.e. all of the obstructions in the points (1)–(3) in the statement of Theorem 4.9 are non-zero. We summarize this in the following.

Example 5.5

(Super conic ) Let be the complete intersection given by the equation \(X_0^2 + X_1^2 + X_2^2 + \Theta _1 \Theta _2 = 0\) in \({{\mathbb {C}}}{{\mathbb {P}}}^{\,2|2}\) and let \({\mathsf {{M}}}\) be the supermanifold constructed from as above. Then the related -extension is not split.

Finally, let us consider a slightly trickier example, namely that of a super elliptic curve modeled on an ordinary elliptic curve and whose rank 3 fermionic sheaf is given by the direct sum . We further assume that the fundamental obstruction class is vanishing, i.e. in the isomorphism . Under these hypotheses one has that all of the points (1)–(3) in Theorem 4.9 are indeed satisfied since also . Nonetheless, the -extension related to might still be non-split, since is projected but not necessarily split as a complex supermanifold. Indeed the higher obstruction to split takes values in the cohomology group which is computed to be isomorphic to \({\mathbb {C}}^{\oplus 3}\). A non-vanishing class obstruction class would correspond to odd transition functions of the kind \(\theta _i \mapsto \theta _i + c_i \theta _1 \theta _2 \theta _3\) for some complex number \(c_i\). In this case, if is a non-split supermanifold, the -extension related to is also non-split. The upshot of the example is that in the case of supermanifolds of odd dimension greater than 2, the presence of higher obstruction classes to split a supermanifold is quite a delicate issues. This is summarized in the following.

Example 5.6

(Super elliptic curve of dimension \(1\,{|}\,3\)) Let be a supermanifold of dimension \(1\,{|}\,3\) such that for E an elliptic curve (over \({\mathbb {C}}\)) and and let its fundamental obstruction class be zero. Then the higher obstruction to split

is defined and fully determines the geometry of . In particular, is also an obstruction to split the the -extension related to , i.e. if is non-split, then the -extension does not split as well.

6 Local theory: forms, natural operators and cohomology

In this section we study forms and natural, i.e. globally defined and invariant, operators acting on in the smooth and holomorphic category. We start by making contact between our framework and the setting developed by the author and collaborators in [5], where differential and integral forms on a real or complex supermanifold are recovered in a unified fashion starting from the triple tensor product of natural sheaves on given by , where is the sheaf of differential operators on . Notice that due to the pivotal presence of this is a non-commutative construction, better than just super-commutative.

This triple tensor product is acted upon by two globally defined mutually commuting operators \({\hat{d}}\) and \({\hat{\delta }}\), see [5]

and as such it yields a double complex of sheaves, . Differential and integral forms, together with their differentials and Poincaré lemmas are recovered via the two natural spectral sequences—we call them \(E^{{\hat{d}}}_i\) and \(E^{{\hat{\delta }}}_i\), depending on whether we are starting computing the cohomology with respect of \({\hat{d}} \) or \({\hat{\delta }}\)—related to this double complex . In particular, the following holds true.

Theorem 6.1

([5]) Let be a real or complex supermanifold. Then the spectral sequences \(E_i^{{\hat{\delta }}}\) and \(E_i^{{\hat{d}}}\) related to the double complex

  1. (1)

    yield the differential forms and integral forms on at the first page, i.e.

  2. (2)

    both converge to the locally constant sheaf   for \({\mathbb {K}}\) the real or complex numbers at the second page, i.e.

A consequence of the above is that the hypercohomologies of differential and integral forms are isomorphic, and both coincide with the Rham cohomology of the reduced manifold, see also the recent [22].

Corollary 6.2

([5]) Let be a real supermanifold. Then the hypercohomologies of (the sheaf of) differential forms and integral forms are isomorphic. In particular, one finds

One the main ingredients of the above construction is the non-commutative sheaf of differential operator . To is canonically associated a sheaf of super-commutative -algebras, by considering the filtration by the degree of the differential operators for any \(i \geqslant 0\). This is given by the quotient

It is not hard to see that . In this way, a “de-quantization” of the above triple tensor product reads

(6.1)

This sheaf of super-commutative -algebras can be put in relation with the sheaf , seen as a sheaf of -modules. Indeed one can observe that the -extension (3.4) is always locally split, so that over an open set \(\pi ^{-1} (U)\) of \({\textsf{M}}\) for U an open set in , one has

This holds true globally for a real supermanifold, as proved above. It follows that

(6.2)

to be compared to (6.1) above. Using the decomposition (6.2), the action of the de Rham differential can be given in an open set \(\pi ^{-1} (U)\) with local coordinate \( x_a\) and \(p_a\) as follows:

(6.3)

where , and . The sum over a is left understood.

Theorem 6.3

(Homology of \({\textsf{d}}\) / Poincaré lemma) Let \({\mathsf {{M}}}\) be defined as above and let the de Rham differential. Then

where is the sheaf of locally-constant functions on \({\mathsf {{M}}}\) for \({\mathbb {K}}\) the real or complex numbers.

Proof

Given the action of the de Rham differential \({\textsf{d}}\) in (6.3), the result follows from the ordinary Poincaré lemma for supermanifolds, see for example [22].\(\square \)

6.1 Odd symplectic form and its cohomology

Let us keep working in the smooth or holomorphic category and let us now consider the (non-degenerate) odd 2-form where the index a runs over both even and odd coordinates. We first observe the following.

Lemma 6.4

(Global definition on \(\omega \)) The odd 2-form is invariant, i.e. coordinate independent.

Proof

We use the coordinate transformations of Lemma 3.1. One finds that

and it is easy to verify that the contribution of the second term is zero.\(\square \)

Another way to see that \(\omega \) is actually invariant is to introduce its primitive form \( \eta \,{:}{=}\,(-1)^{|x_a| + 1} dx_a p_a. \)

Lemma 6.5

(Primitive form of \(\omega \) ) The primitive form of \(\omega \) is invariant, moreover one has \({\textsf{d }} \eta = \omega \). In particular, \(\omega \) is invariant.

Proof

Using again the the transformations of Lemma 3.1, it is enough to compute

Clearly \({\textsf{d}} \eta = \omega \). Since both \({\textsf{d}}\) and \(\eta \) are invariant so is \(\omega \).\(\square \)

The previous results allow to give the following definition, see for example [16, 25] or the dedicated chapter in the recent book [19].

Definition 6.6

(Odd symplectic form / odd symplectic supermanifold) We call \(\omega \,{:}{=}\,\sum _a dx_a dp_a\) the odd symplectic form associated to \({\mathsf {{M}}}\). In particular, we say that the pair \(({\textsf{M}}, \omega )\) defines an odd symplectic supermanifold.

Remark 6.7

Notice that with respect to the definition of odd symplectic supermanifolds available in the literature [25], the supermanifold \({\textsf{M}}\) is constructed by starting from a supermanifold and a vector bundle on it, better than from an ordinary manifold X and a vector bundle on it: in this sense it is a “generalized” odd symplectic supermanifold.

Left multiplication by the odd symplectic form \(\omega = \sum _a dx_a dp_a\) induces a well-defined invariant operator whose action with respect to the above decomposition is given by

Such as the de Rham differential \({\textsf{d}}\), also the multiplication by the odd symplectic form is nilpotent. We compute its homology in the next theorem.

Theorem 6.8

(Homology of \({\textsf{s}}\)) Let \({\mathsf {{M}}}\) be defined as above and let be the left multiplication by the odd symplectic form. Then

as a sheaf of -modules.

Proof

We need to construct a homotopy for the operator \({\textsf{s}}.\) Using the above local decomposition, can be represented by the sheaf of vector spaces generated by the elements where and for monomials \(\eta = dx^I \) and \(F = dp^J\) for multi-indices I and J.

On the other hand, one can observe that the decomposition (6.2) coincides with , and in view of this, the action of the operator \({\textsf{s}}\) reads

as \(dp^J\) corresponds to \(\partial ^J\) and \(dp_a\) corresponds to \(\partial _{x_a}\), having used the local splitting of the -extension. Notice that f is not touched by \({\textsf{s}}\). We thus introduce the following local operator:

We prove that this is a homotopy for the operator \({\textsf{s}}.\) In particular, one finds that

(6.4)

The summands in the previous expression read

where n is the even and m is the odd dimension of and \(\hbox {deg}_0\) and \(\hbox {deg}_1\) is the even and odd degree of \(\eta = dx^I\) and \(F = \partial ^J\). It follows that the above sum (6.4) gives

The homotopy \({\textsf{h}}\) fails if and only if one has \(\hbox {deg}_0 (\eta ) = \hbox {deg}_0 (\partial ^J) = 0\), \(\hbox {deg}_1 (\eta ) = n\) and \(\hbox {deg}_1 (\partial ^J) = m\), so that the non-zero element in homology takes the form , where f is any section of the structure sheaf .\(\square \)

The above theorem has the following corollary.

Corollary 6.9

Let \({\mathsf {{M}}}\) be defined as above and let be the left multiplication by the odd symplectic form. Then the homology of \(\text {\textsf{s}}\) is naturally isomorphic to the pull-back of the Berezinian sheaf on , i.e.

(6.5)

Proof

Allowing for the above identifications and the usual slight abuse of notation concerning the pull-backs, it is enough to observe that generates the Berezinian sheaf of the supermanifold , see [21] for details on this construction of the Berezinian sheaf. \(\square \)

Remark 6.10

The above result can be related to the notion of (super) semidensities, see [16, 19, 25]. Indeed the -extension exact sequence (3.4) allows to easily compute the Berezinian sheaf of the supermanifold \({\textsf{M}}\). Taking the Berezinians, the short exact sequence (3.4) yields

Observing that for any sheaf on one has , one sees that hence

(6.6)

Defining the sheaf of semidensities of the supermanifold \({\textsf{M}}\) to be the locally-free sheaf of -modules whose sections are “square roots” of the sections of the Berezinian sheaf, i.e. , it follows from (6.6) that . In turn, the above (6.5) can be re-written as

where the sheaf of semidensities is seen as a sheaf of -modules. Notice that reducing to the underlying ordinary manifold one would find , which is the ordinary notion for semidensities of odd symplectic supermanifolds constructed out of an ordinary manifold , see for example [19] and [25].

6.2 Deformed de Rham complex and BV Laplacian

Now, the crucial observation, originally due to Ševera in [25], is that the nilpotent operators \({\textsf{d}}\) and \({\textsf{s}}\) commutes with each other. This holds true also in the present setting, as the following shows.

Lemma 6.11

(\({{{\textsf{d}}}}\) commutes with \({{{\textsf{s}}}}\)) Let \({\mathsf {{d}}}\) and \({\mathsf {{s}}}\) be the de Rham differential and the multiplication by the odd symplectic form, then \([\, {\mathsf {{d}}}, {\mathsf {{s}}} \, ] = 0\). In particular the triple defines a double complex.

Proof

This is a local check. Using the above decomposition, one computes

which concludes the proof.\(\square \)

It follows from the previous Lemma 6.11 that, in particular, \({\textsf{d}}\) acts on the homology of \({\textsf{s}}\). This leads to the following definition.

Definition 6.12

(Deformed de Rham complex / spectral sequence ) We call the double complex the deformed de Rham (double) complex of \({\textsf{M}}\). We denote with the related spectral sequence \((E_i, \delta _i) \) that starts with the differential \(\delta _1 = {\textsf{s}}\) and we call it deformed de Rham spectral sequence.

Let us now study the deformed de Rham spectral sequence .

Theorem 6.13

(Semidensities & super BV operator) Let be defined as above. Then

  1. (1)

    the first page of the spectral sequence is isomorphic to semidensities on , i.e.

  2. (2)

    the second differential \(\delta _2 \) of the spectral sequence is zero. In particular, the second page of the spectral sequence is given again by

  3. (3)

    the third differential \(\delta _3\) of the spectral sequence is—up to exact terms—the super BV Laplacian

    where is a section of and is a section of . In particular, the spectral sequence converges at page three, which is isomorphic to the locally constant sheaf on , i.e.

    A representative of this homology class if given by

(6.7)

Proof

The first point of the theorem is just Corollary 6.9. As for the second point, notice that the corresponding differential is given by the induced action of the de Rham differential \({\textsf{d}}\) on . Referring to Theorem 6.8, one can observe that the induced action of \({\textsf{d}}\) maps to a zero-homotopic cohomology. More in particular, for immediate use, one can observe that for any , one has

where can be taken to be of the form .

The third differential can be easily inferred by noticing that, formally, , so that in particular, when acting on an element of one finds, upon the previous observation

Taking as above it is easy to compute that

We now look for a homotopy for this operator. To this end, without loss of generality, we let be of the form \(f (x,p)\,{:}{=}\,g_I(x) p^I \) for \(x_a = x_1 \ldots x_{n} \,|\, x_{n+1} \ldots x_{n+m}\) even and odd coordinates of and I a multi-index. We claim that the homotopy for is given by

(6.8)

where \(t \in [0,1]\), \({\textsf{P}}^*_t g (x) = g(tx) \) and \(\ell _f\) is a constant, which depends on f, that will be fixed later. An attentive computation yields the following:

This gives the following condition on \(\ell _f\) as to have a homotopy:

$$\begin{aligned} \ell _f = n+m + \deg _{0} (p^I) - \deg _{1} (p^I) - 2 \deg _{1} (g_I) -1, \end{aligned}$$

which yields

Observing that \(\deg _{0} (p^I) \geqslant 0\), \(0\leqslant \deg _{1 } (p^I) \leqslant n\) and \(0\leqslant \deg _{1} (g_I) \leqslant m\), one sees that the homotopy fails only for \(\deg _{0} (p^I) = 0\), \(\deg _{1} (p^1) = n \) and \(\deg _{1} (g_I) = m\). One thus finds that \(f(x, p) = x_{n+1} \ldots x_{n+m} p_{1} \ldots p_{n}\) so that

with \(k \in {\mathbb {R}}\) or \(k \in {\mathbb {C}}\) is a representative for . Finally, it is easy to see that the representative is \({\textsf{d}}\)-closed, so that it yields zero when acted by all the higher differentials, concluding the proof.\(\square \)

6.3 Remarks and outlooks

The above Theorem 6.13 extends to a “fully” supergeometric context the beautiful Ševera’s result [25] for odd symplectic supermanifolds, with possibly the bonus of showing explicitly the homotopy (6.8) of the super BV Laplacian—which is seen here as a morphism of sheaves—together with the related representative in sheaf cohomology (6.7). It is to be noted that the form of the homotopy shown above is somewhat general, as the structure of odd nilpotent operators in supergeometry often consists of a “multiplication” of an even and an odd part, such as the BV Laplacian above or the de Rham differential—notice indeed that the related complexes of integral forms and of differential forms are quasi-isomorphic, see Theorem 6.2. Similar structures for homotopies of differentials can be found also in [5, 14, 21, 22].

Finally, a remark—or better a warning—about the holomorphic category is in order. Let us consider a generic smooth supermanifold admitting a closed non-degenerate odd 2-form \(\omega \), i.e. an odd symplectic supermanifold . Then, by a well-known result due to Schwarz the supermanifold is globally symplectomorphic to the “standard” odd symplectic supermanifold constructed as \({\textsf{M}}\) above, starting from the reduced space of , and endowed with its standard odd symplectic form \(\sum _i dx_i dp_i\), see [24]. The proof of this fact heavily relies on that every smooth supermanifold is in fact split, hence it is itself the total space of a certain vector bundle whose fibers have odd parity. It is then natural to ask what happens in the holomorphic category, where complex supermanifolds can in fact be non-split. In particular, one can ask the following question: does Schwarz’s result hold true in the holomorphic category as well? In other words, is it possible to find an example of complex supermanifold admitting a closed non-degenerate odd 2-form which is not globally isomorphic to a supermanifold of the kind of \({\textsf{M}}\) for some manifold ? Clearly, a non-split complex supermanifold admitting a globally defined odd non-degenerate closed 2-form would provide such a counterexample to Schwarz result. This suggests, in turn, the following question: do the obstruction classes to splitting a complex supermanifold also obstruct the existence of a globally-defined odd non-degenerate closed 2-form? We leave these questions to future works.