Automorphism groups of compact complex supermanifolds

Abstract

Let ${\mathcal {M}}$ be a compact complex supermanifold. We prove that the set ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ of automorphisms of ${\mathcal {M}}$ can be endowed with the structure of a complex Lie group acting holomorphically on ${\mathcal {M}}$, so that its Lie algebra is isomorphic to the Lie algebra of even holomorphic super vector fields on ${\mathcal {M}}$. Moreover, we prove the existence of a complex Lie supergroup ${{\mathrm{Aut}}}({\mathcal {M}})$ acting holomorphically on ${\mathcal {M}}$ and satisfying a universal property. Its underlying Lie group is ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ and its Lie superalgebra is the Lie superalgebra of holomorphic super vector fields on ${\mathcal {M}}$. This generalizes the classical theorem by Bochner and Montgomery that the automorphism group of a compact complex manifold is a complex Lie group. Some examples of automorphism groups of complex supermanifolds over ${\mathbb {P}}_1({\mathbb {C}})$ are provided.

1 Introduction

The automorphism group of a compact complex manifold M carries the structure of a complex Lie group which acts holomorphically on M and whose Lie algebra consists of the holomorphic vector fields on M (see [6]). In this article, we investigate how this result can be extended to the category of compact complex supermanifolds.

Let ${\mathcal {M}}$ be a compact complex supermanifold, i.e. a complex supermanifold whose underlying manifold is compact. An automorphism of ${\mathcal {M}}$ is a biholomorphic morphism ${\mathcal {M}}\rightarrow {\mathcal {M}}$. A first candidate for the automorphism group of such a supermanifold is the set of automorphisms, which we denote by ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$. However, every automorphism $\varphi$ of a supermanifold ${\mathcal {M}}$ (with structure sheaf ${\mathcal {O}}_{\mathcal {M}}$) is “even” in the sense that its pullback $\varphi ^*:{\mathcal {O}}_{\mathcal {M}}\rightarrow {{\tilde{\varphi }}}_* ({\mathcal {O}}_{\mathcal {M}})$ is a parity-preserving morphism. Therefore, we can (at most) expect this set of automorphisms of ${\mathcal {M}}$ to carry the structure of a classical Lie group if we require its action on ${\mathcal {M}}$ to be smooth or holomorphic. We cannot obtain a Lie supergroup of positive odd dimension.

We prove that the group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, endowed with an analogue of the compact-open topology, carries the structure of a complex Lie group such that the action on ${\mathcal {M}}$ is holomorphic and its Lie algebra is the Lie algebra of even holomorphic super vector fields on ${\mathcal {M}}$. It should be noted that the group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is in general different from the group ${{\mathrm{Aut}}}(M)$ of automorphisms of the underlying manifold M. There is a group homomorphism ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}}) \rightarrow {{\mathrm{Aut}}}(M)$ given by assigning the underlying map to an automorphism of the supermanifold; this group homomorphism is in general neither injective nor surjective.

We define the automorphism group of a compact complex supermanifold  ${\mathcal {M}}$ to be a complex Lie supergroup which acts holomorphically on ${\mathcal {M}}$ and satisfies a universal property. In analogy to the classical case, its Lie superalgebra is the Lie superalgebra of holomorphic super vector fields on ${\mathcal {M}}$, and the underlying Lie group is ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, the group of automorphisms of ${\mathcal {M}}$. Using the equivalence of complex Harish-Chandra pairs and complex Lie supergroups (see [24]), we construct the appropriate automorphism Lie supergroup of ${\mathcal {M}}$.

More precisely, the outline of this article is the following: First, we introduce a topology on the set ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ of automorphisms on a compact complex supermanifold ${\mathcal {M}}$ (cf. Sect. 3). This topology is an analogue of the compact-open topology in the classical case, which coincides in the case of a compact complex manifold with the topology of uniform convergence. We prove that the topological space ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ with composition and inversion of automorphisms as group operations is a locally compact topological group which satisfies the second axiom of countability.

In Sect. 4, the non-existence of small subgroups of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is proven, which means that there exists a neighbourhood of the identity in ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ with the property that this neighbourhood does not contain any non-trivial subgroup. A result on the existence of Lie group structures on locally compact topological groups without small subgroups (see [25]) then implies that ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ carries the structure of a real Lie group.

In the case of a split compact complex supermanifold ${\mathcal {M}}$, the fact that ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ carries the structure of a Lie group follows more easily as described in Remark 8. In this case it can be proven that ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is the semi-direct product of a finite-dimensional vector space and the automorphism group of the vector bundle corresponding to ${\mathcal {M}}$, which is by [17] a complex Lie group.

Then, continuous one-parameter subgroups of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ and their action on the supermanifold ${\mathcal {M}}$ are studied (see Sect. 5). This is done in order to obtain results on the regularity of the ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$-action on ${\mathcal {M}}$ and characterize the Lie algebra of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$. We prove that the action of each continuous one-parameter subgroup of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on ${\mathcal {M}}$ is analytic. As a corollary we get that the Lie algebra of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is isomorphic to the Lie algebra $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ of even holomorphic super vector fields on ${\mathcal {M}}$, and ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ carries the structure of a complex Lie group so that its natural action on ${\mathcal {M}}$ is holomorphic.

Next, we show that the Lie superalgebra $\mathrm {Vec}({\mathcal {M}})$ of holomorphic super vector fields on a compact complex supermanifold ${\mathcal {M}}$ is finite-dimensional (see Sect. 6). Since ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ carries the structure of a complex Lie group, we already know that $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$, the even part of $\mathrm {Vec}({\mathcal {M}})$, is finite-dimensional. The key point in the proof in the case of a split supermanifold ${\mathcal {M}}$ is that the tangent sheaf of ${\mathcal {M}}$ is a coherent sheaf of ${\mathcal {O}}_M$-modules on the compact complex manifold M, where ${\mathcal {O}}_M$ is the sheaf of holomorphic functions on M.

Let $\alpha$ denote the action of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on the Lie superalgebra $\mathrm {Vec}({\mathcal {M}})$ by conjugation: $\alpha (\varphi )(X)=\varphi _*(X)=(\varphi ^{-1})^*\circ X\circ \varphi ^*$ for $\varphi \in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, $X\in \mathrm {Vec}({\mathcal {M}})$. The restriction of this representation $\alpha$ to $\mathrm {Vec}_{{{\bar{0}}}} ({\mathcal {M}})$, the even part of the Lie superalgebra $\mathrm {Vec}({\mathcal {M}})$, coincides with the adjoint action of the Lie group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on its Lie algebra, which is isomorphic to $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$. Hence $\alpha$ defines a Harish-Chandra pair $({{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}}), \mathrm {Vec}({\mathcal {M}}))$. The equivalence between Harish-Chandra pairs and complex Lie supergroups allows us to define the automorphism Lie supergroup of a compact complex supermanifold as follows (see Definition 2):

Definition

(Automorphism Lie supergroup) Define the automorphism group ${{\mathrm{Aut}}}({\mathcal {M}})$ of a compact complex supermanifold to be the unique complex Lie supergroup associated with the Harish-Chandra pair $({{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}}), \mathrm {Vec}({\mathcal {M}}))$ with representation $\alpha$.

The natural action of the automorphism Lie supergroup ${{\mathrm{Aut}}}({\mathcal {M}})$ on ${\mathcal {M}}$ is holomorphic, i.e. we have a morphism $\varPsi :{{\mathrm{Aut}}}({\mathcal {M}})\times {\mathcal {M}}\rightarrow {\mathcal {M}}$ of complex supermanifolds. The automorphism Lie supergroup ${{\mathrm{Aut}}}({\mathcal {M}})$ satisfies the following universal property (see Theorem 22):

Theorem

If ${\mathcal {G}}$ is a complex Lie supergroup with a holomorphic action $\varPsi _{{\mathcal {G}}}:{\mathcal {G}}\times {\mathcal {M}}\rightarrow {\mathcal {M}}$ on ${\mathcal {M}}$, then there is a unique morphism $\sigma :{\mathcal {G}}\rightarrow {{\mathrm{Aut}}}({\mathcal {M}})$ of Lie supergroups such that the diagram

is commutative.

The automorphism Lie supergroup of a compact complex supermanifold is the unique complex Lie supergroup satisfying the preceding universal property.

Using the “functor of points” approach to supermanifolds, an alternative definition of the automorphism group as a functor in analogy to [20, 22] is possible, which is studied in Sect. 8. If ${\mathcal {M}}$ is a compact complex supermanifold, this functor from the category of supermanifolds to the category of sets can be defined by the assignment

$$\begin{aligned}{\mathcal {N}}\mapsto \{\varphi :{\mathcal {N}}\times {\mathcal {M}}\rightarrow {\mathcal {N}}\times {\mathcal {M}}\,|\, \varphi \text { is invertible, and } \mathrm {pr}_{{\mathcal {N}}} \circ \varphi =\mathrm {pr}_{{\mathcal {N}}}\}, \end{aligned}$$

where $\mathrm {pr}_{{\mathcal {N}}}:{\mathcal {N}}\times {\mathcal {M}} \rightarrow {\mathcal {N}}$ denotes the projection onto the first component. The two approaches to the automorphism group are equivalent and the constructed automorphism group ${{\mathrm{Aut}}}({\mathcal {M}})$ represents the just defined functor.

In the classical case, another class of complex manifolds where the automorphism group carries the structure of a Lie group is given by the bounded domains in ${\mathbb {C}}^m$ (see [8]). An analogue statement is false in the case of supermanifolds. In Sect. 9, we give an example showing that in the case of a complex supermanifold ${\mathcal {M}}$ whose underlying manifold is a bounded domain in ${\mathbb {C}}^m$ there does in general not exist a Lie supergroup acting on ${\mathcal {M}}$ and satisfying the universal property of the preceding theorem.

In Sect. 10, the automorphism group ${{\mathrm{Aut}}}({\mathcal {M}})$ or its underlying Lie group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ are determined for some supermanifolds ${\mathcal {M}}$ with underlying manifold $M={\mathbb {P}}_1{\mathbb {C}}$.

2 Preliminaries and notation

Throughout, we work with the “Berezin-Leĭtes-Kostant-approach” to supermanifolds (cf. [1, 15, 16]). If a supermanifold is denoted by a calligraphic letter ${\mathcal {M}}$, then we denote the underlying manifold by the corresponding uppercase standard letter M, and the structure sheaf by ${\mathcal {O}}_{\mathcal {M}}$. We call a supermanifold ${\mathcal {M}}$ compact if its underlying manifold M is compact. By a complex supermanifold we mean a supermanifold ${\mathcal {M}}$ with structure sheaf ${\mathcal {O}}_{\mathcal {M}}$ which is locally, on small enough open subsets $U\subset M$, isomorphic to ${\mathcal {O}}_U\otimes \bigwedge {\mathbb {C}}^n$, where ${\mathcal {O}}_U$ denotes the sheaf of holomorphic functions on U. For a morphism $\varphi :{\mathcal {M}}\rightarrow {\mathcal {N}}$ between supermanifolds ${\mathcal {M}}$ and ${\mathcal {N}}$, the underlying map $M\rightarrow N$ is denoted by ${{\tilde{\varphi }}}$ and its pullback by $\varphi ^*:{\mathcal {O}}_{\mathcal {N}} \rightarrow {{\tilde{\varphi }}}_*{\mathcal {O}}_{\mathcal {M}}$. An automorphism of a complex supermanifold ${\mathcal {M}}$ is a biholomorphic morphism ${\mathcal {M}}\rightarrow {\mathcal {M}}$, i.e. an invertible morphism in the category of complex supermanifolds.

Let E be a vector bundle on a complex manifold M and ${\mathcal {E}}$ its sheaf of sections. Then we can associate a supermanifold ${\mathcal {M}}=(M,{\mathcal {O}}_{{\mathcal {M}}})$ by setting ${\mathcal {O}}_{\mathcal {M}}=\bigwedge {\mathcal {E}}$, which has a natural ${\mathbb {Z}}$-grading (and hence a ${\mathbb {Z}}/{2{\mathbb {Z}}}$-grading). Split supermanifolds are supermanifolds ${\mathcal {M}}$ such that there is a vector bundle on M with sheaf of sections ${\mathcal {E}}$ such that ${\mathcal {M}}\cong (M,\bigwedge {\mathcal {E}})$. If E is e.g. the trivial bundle of rank n on $M={\mathbb {C}}^m$, then we get the supermanifold ${\mathbb {C}}^{m|n}=({\mathbb {C}}^m,\bigwedge {\mathcal {E}})=({\mathbb {C}}^m,{\mathcal {O}}_{{\mathbb {C}}^m}\otimes \bigwedge {\mathbb {C}}^n)$.

For a complex supermanifold ${\mathcal {M}}$, let ${\mathcal {T}}_{\mathcal {M}}$ denote the tangent sheaf of ${\mathcal {M}}$. The Lie superalgebra of holomorphic vector fields on ${\mathcal {M}}$ is $\mathrm {Vec}({\mathcal {M}})={\mathcal {T}}_{\mathcal {M}}(M)$, it consists of the subspace $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ of even and the subspace $\mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}})$ of odd super vector fields on ${\mathcal {M}}$.

Let ${\mathcal {M}}$ be a complex supermanifold of dimension (m|n), and let ${\mathcal {I}}_{\mathcal {M}}$ be the subsheaf of ideals generated by the odd elements in the structure sheaf ${\mathcal {O}}_{\mathcal {M}}$ of a supermanifold ${\mathcal {M}}$. As described in [19], we have the filtration

$$\begin{aligned} {\mathcal {O}}_{\mathcal {M}}=({\mathcal {I}}_{\mathcal {M}})^0\supset ({\mathcal {I}}_{\mathcal {M}})^1\supset ({\mathcal {I}}_{\mathcal {M}})^2\supset \cdots \supset ({\mathcal {I}}_{\mathcal {M}})^{n+1}=0 \end{aligned}$$

of the structure sheaf ${\mathcal {O}}_{\mathcal {M}}$ by the powers of ${\mathcal {I}}_{\mathcal {M}}$. Define the quotient sheaves $\text {gr}_k({\mathcal {O}}_{\mathcal {M}})=({\mathcal {I}}_{\mathcal {M}})^k/ ({\mathcal {I}}_{\mathcal {M}})^{k+1}$. This gives rise to the ${\mathbb {Z}}$-graded sheaf $\text {gr}\,{\mathcal {O}}_{\mathcal {M}} ={\textstyle \bigoplus _k} \text {gr}_k({\mathcal {O}}_{\mathcal {M}})$. Furthermore, $\text {gr}\,{\mathcal {M}}= (M,\text {gr}\,{\mathcal {O}}_{\mathcal {M}})$ is a split complex supermanifold of the same dimension as ${\mathcal {M}}$.

Note that ${\mathcal {E}}:=\text {gr}_1({\mathcal {O}}_{\mathcal {M}})$ defines a vector bundle E on M. An automorphism $\varphi$ of ${\mathcal {M}}$ yields a pullback $\varphi ^*$ on ${\mathcal {O}}_{\mathcal {M}}$. Following [10], its reduction to the ${\mathcal {O}}_M$-module E yields a morphism of vector bundles $\varphi _0\in {{\mathrm{Aut}}}(E)$ over the reduction ${{\tilde{\varphi }}}\in {{\mathrm{Aut}}}(M)$. By [17] the automorphism group of a principal fibre bundle over a compact complex manifold carries the structure of a complex Lie group. Since every automorphism of a vector bundle canonically induces an automorphism of the associated principal fibre bundle and vice versa, the automorphism group of the associated principal fibre bundle and ${{\mathrm{Aut}}}(E)$ may be identified. Moreover, this identification also respects the topology of compact convergence on both groups. Hence, the group ${{\mathrm{Aut}}}(E)$ also carries the structure of a complex Lie group. On local coordinate domains U, V with ${{\tilde{\varphi }}}(U)\subset V$ we can identify ${\mathcal {O}}_{\mathcal {M}}|_V\cong \varGamma _{\varLambda E}|_V$ and ${\mathcal {O}}_{\mathcal {M}}|_U\cong \varGamma _{\varLambda E}|_U$ and following [21] decompose $\varphi ^*=\varphi _0^*\exp (Y)$ with ${\mathbb {Z}}$-degree preserving automorphism $\varphi _0^*:\varGamma _{\varLambda E}|_V \rightarrow \varGamma _{\varLambda E}|_U$ induced by $\varphi _0$ and where Y is an even super derivation on $\varGamma _{\varLambda E}|_V$ increasing the ${\mathbb {Z}}$-degree by 2 or more. Note that the exponential series $\exp (Y)$ is finite since Y is nilpotent.

More generally, there is a relation between nilpotent even super vector fields on a supermanifold and morphisms of this supermanifold satisfying a certain nilpotency condition. This is a direct consequence of a technical result on the relation of algebra homomorphisms and derivations (cf. [23], Proposition 2.1.3 and Lemma 2.1.4). If $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$ is a morphism of supermanifolds with underlying map ${{\tilde{\varphi }}}=\mathrm {id}_M$ and such that $\varphi ^*-\mathrm {id}_{\mathcal {M}}^*:{\mathcal {O}}_{\mathcal {M}}\rightarrow {\mathcal {O}}_{\mathcal {M}}$ is nilpotent, i.e. there is $N\in {\mathbb {N}}$ with $(\varphi ^*-\mathrm {id}_{\mathcal {M}}^*)^N=0$, then

$$\begin{aligned} X=\log (\varphi ^*)=\sum _{n=1}^N\frac{(-1)^{n+1}}{n} (\varphi ^*-\mathrm {id}_{\mathcal {M}}^*)^n \end{aligned}$$

is a nilpotent even super vector field on ${\mathcal {M}}$ and we have

$$\begin{aligned} \varphi ^*=\exp (X)=\sum _{n\ge 0}\frac{1}{n!} X^n. \end{aligned}$$

Furthermore, for any nilpotent even super vector fifeld X on ${\mathcal {M}}$, the (finite) sum $\exp (X)$ defines a map ${\mathcal {O}}_{\mathcal {M}}\rightarrow {\mathcal {O}}_{\mathcal {M}}$ which is the pullback of an invertible morphism ${\mathcal {M}}\rightarrow {\mathcal {M}}$ with the identity as underlying map, and the pullback of the inverse is $\exp (-X)$.

3 The topology on the group of automorphisms

Let ${\mathcal {M}}$ be a compact complex supermanifold. An automorphism of ${\mathcal {M}}$ is a biholomorphic morphism $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$. Denote by ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ the set of automorphisms of ${\mathcal {M}}$.

In this section, a topology on ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is introduced, which generalizes the compact-open topology and topology of compact convergence of the classical case. Then we show that ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is a locally compact topological group with respect to this topology.

Let $K\subseteq M$ be a compact subset such that there are local odd coordinates $\theta _1,\ldots , \theta _n$ for ${\mathcal {M}}$ on an open neighbourhood of K. Moreover, let $U\subseteq M$ be open and $f\in {\mathcal {O}}_{\mathcal {M}}(U)$, and let $U_{\nu }$ be open subsets of ${\mathbb {C}}$ for $\nu \in ({\mathbb {Z}}_2)^n$. Let $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$ be an automorphism with ${{\tilde{\varphi }}}(K)\subseteq U$. Then there are holomorphic functions $\varphi _{f,\nu }$ on a neighbourhood of K such that

$$\begin{aligned} \varphi ^*(f)=\sum _{\nu \in ({\mathbb {Z}}_2)^n}\varphi _{f,\nu } \theta ^\nu . \end{aligned}$$

Let

$$\begin{aligned} \varDelta (K, U, f,\theta _j, U_{\nu }) =\{\varphi \in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})|\,{\tilde{\varphi }}(K)\subseteq U, \, \varphi _{f,\nu }(K)\subseteq U_{\nu }\}, \end{aligned}$$

and endow ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ with the topology generated by sets of this form, i.e. the sets of the form $\varDelta (K, U,f,\theta _j, U_{\nu })$ form a subbase of the topology.

For any open subset $U\subseteq M$ such that there exist coordinates for ${\mathcal {M}}$ on U, fix a set of coordinates functions $f_1^U,\ldots , f_{m+n}^U \in {\mathcal {O}}_{{\mathcal {M}}}(U)$. Using Taylor expansion one can show that the sets of the form $\varDelta (K, U, f_l^U,\theta _j, U_{\nu })$ then also form a subbase of the topology.

Remark 1

In particular, the subsets of the form

$$\begin{aligned} \varDelta (K,U)=\{\varphi \in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}}) |\,{\tilde{\varphi }}(K)\subseteq U\} \end{aligned}$$

are open for $K\subseteq M$ compact and $U\subseteq M$ open. Hence the map ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\rightarrow {{\mathrm{Aut}}}(M)$, associating with an automorphism $\varphi$ of ${\mathcal {M}}$ the underlying automorphism ${\tilde{\varphi }}$ of M, is continuous.

Remark 2

The group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ endowed with the above topology is a second-countable Hausdorff space since M is second-countable.

Let $U\subseteq M$ be open. Then we can define a topology on ${\mathcal {O}}_{\mathcal {M}}(U)$ as follows: If $K\subseteq U$ is compact such that there exist odd coordinates $\theta _1,\ldots , \theta _n$ on a neighbourhood of K, write $f\in {\mathcal {O}}_{\mathcal {M}}(U)$ on K as $f=\sum _{\nu }f_\nu \theta ^\nu$. Let $U_\nu \subseteq {\mathbb {C}}$ be open subsets. Then define a topology on ${\mathcal {O}}_{\mathcal {M}}(U)$ by requiring that the sets of the form $\{f\in {\mathcal {O}}_{\mathcal {M}}(U)|\,f_\nu (K)\subseteq U_\nu \}$ are a subbase of the topology. A sequence of functions $f_k$ converges to f if and only if in all local coordinate domains with odd coordinates $\theta _1,\ldots ,\theta _n$ and $f_k=\sum _{\nu }f_{k,\nu }\theta ^\nu$, $f=\sum _{\nu }f_\nu \theta ^\nu$, the coefficient functions $f_{k,\nu }$ converge uniformly to $f_\nu$ on compact subsets. Note that for any open subsets $U_1, U_2\subseteq M$ with $U_1\subset U_2$ the restriction map ${\mathcal {O}}_{\mathcal {M}}(U_2)\rightarrow {\mathcal {O}}_{\mathcal {M}}(U_1)$, $f\mapsto f|_{U_1}$, is continuous.

Using Taylor expansion (in local coordinates) of automorphisms of ${\mathcal {M}}$ we can deduce the following lemma:

Lemma 3

A sequence of automorphisms $\varphi _k:{\mathcal {M}}\rightarrow {\mathcal {M}}$ converges to an automorphism $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$ with respect to the topology of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ if and only if the following condition is satisfied: For all $U,V\subseteq M$ open subsets of M such that V contains the closure of ${\tilde{\varphi }}(U)$, there is an $N\in {\mathbb {N}}$ such that $\tilde{\varphi _k}(U)\subseteq V$ for all $k\ge N$. Furthermore, for any $f\in {\mathcal {O}}_{\mathcal {M}}(V)$ the sequence $(\varphi _k)^*(f)$ converges to $\varphi ^*(f)$ on U in the topology of ${\mathcal {O}}_{\mathcal {M}}(U)$.

Lemma 4

If $U, V\subseteq M$ are open subsets, $K\subseteq M$ is compact with $V\subseteq K$, then the map

$$\begin{aligned} \varDelta (K,U)\times {\mathcal {O}}_{\mathcal {M}}(U) \rightarrow {\mathcal {O}}_{\mathcal {M}}(V),\, (\varphi ,f)\mapsto \varphi ^*(f) \end{aligned}$$

is continuous.

Proof

Let $\varphi _k\in \varDelta (K,U)$ be a sequence of automorphisms of ${\mathcal {M}}$ converging to $\varphi \in \varDelta (K,U)$, and $f_l\in {\mathcal {O}}_{\mathcal {M}}(U)$ a sequence converging to $f\in {\mathcal {O}}_{\mathcal {M}}(U)$. Choosing appropriate local coordinates and using Taylor expansion of the pullbacks $(\varphi _k)^*(f_l)$, it can be shown that $(\varphi _k)^*(f_l)$ converges to $\varphi ^*(f)$ as $k,l\rightarrow \infty$. This uses that the derivatives of a sequence of uniformly converging holomorphic functions also uniformly converge. $\square$

Lemma 5

The topological space ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is locally compact.

The following remark about invertible morphisms is useful for the proof of this lemma.

Remark 6

(See e.g. Proposition 2.15.1 in [15] or Corollary 2.3.3 in [16]) Let ${\mathcal {M}}$ be a complex supermanifold and $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$ any morphism. Let $\xi _1,\ldots , \xi _n$ and $\theta _1,\ldots ,\theta _n$ be local odd coordinates for ${\mathcal {M}}$, and superfunctions $\varphi _{j,k}$, $\varphi _{j,\nu }$ such that $\varphi ^*(\xi _j)=\sum _{k=1}^n \varphi _{j,k} \theta _k +\sum _{||\nu ||\ge 3} \varphi _{j,\nu }\theta ^\nu ,$ where $||\nu ||=||(\nu _1,\ldots ,\nu _n)||=\nu _1+\cdots +\nu _n\ge 3$. Then $\varphi$ is locally biholomorphic if and only if the underlying map ${{\tilde{\varphi }}}$ is locally biholomorphic and $\det \left( (\varphi _{j,k}(y))_{1\le j,k\le n}\right) \ne 0$. The morphism $\varphi$ is hence invertible if it is everywhere locally biholomorphic and ${{\tilde{\varphi }}}$ is biholomorphic.

Proof (of Lemma 5)

Let $\psi \in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$. For each fixed $x\in M$ there are open neighbourhoods $V_x$ and $U_x$ of x and ${\tilde{\psi }}(x)$ respectively such that ${\tilde{\psi }}(K_x)\subseteq U_x$ for $K_x:={\overline{V}}_x$. We may additionally assume that there are local odd coordinates $\xi _1,\ldots , \xi _n$ for ${\mathcal {M}}$ on $U_x$, and $\theta _1,\ldots , \theta _n$ local odd coordinates on an open neighbourhood of $K_x$. For any automorphism $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$ with ${\tilde{\varphi }}(K_x)\subseteq U_x$, let $\varphi _{j,k}$, $\varphi _{j,\nu }$ (for $||\nu ||=||(\nu _1,\ldots ,\nu _n)||=\nu _1+\cdots +\nu _n\ge 3$) be local holomorphic functions such that

$$\begin{aligned} \varphi ^*(\xi _j)=\sum _{k=1}^n \varphi _{j,k} \theta _k +\sum _{||\nu ||\ge 3} \varphi _{j,\nu }\theta ^\nu . \end{aligned}$$

Choose bounded open subsets $U_{j,k}, U_{j,\nu }\subset {\mathbb {C}}$, such that $\psi _{j,k}(x)\in U_{j,k}$ and $\psi _{j,\nu }(x)\in U_{j,\nu }$. Since $\psi$ is an automorphism, we have

$$\begin{aligned} \det \left( (\psi _{j,k}(y))_{1\le j,k\le n}\right) \ne 0 \end{aligned}$$

for all $y\in K_x$ by Remark 6. For later considerations shrink $U_{j,k}$ such that $\det (C)\ne 0$ for all $C=(c_{j,k})_{1\le j,k\le n}$ with $c_{j,k}\in U_{j,k}$. After shrinking $V_x$ we may assume $\psi _{j,k}(K_x)\subseteq U_{j,k}$ and $\psi _{j,\nu }(K_x)\subseteq U_{j,\nu }$. Hence $\psi$ is contained in the set $\varTheta (x)=\{\varphi \in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\,|\,{\tilde{\varphi }}(K_x) \subseteq {\overline{U}}_x,\,\varphi _{j,k}(K_x)\subseteq {\overline{U}}_{j,k}, \varphi _{j,\nu }(K_x) \subseteq {\overline{U}}_{j,\nu }\}$, which contains an open neighbourhood of $\psi$. Since M is compact, M is covered by finitely many of the sets $V_x$, say $V_{x_1},\ldots ,V_{x_l}$. Then $\psi$ is contained in $\varTheta =\varTheta (x_1)\cap \cdots \cap \varTheta (x_l)$. We will now prove that $\varTheta$ is sequentially compact:

Let $\varphi _k$ be any sequence of automorphisms contained in $\varTheta$. Then, using Montel’s theorem and passing to a subsequence, the sequence $\varphi _k$ converges to a morphism $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$. It remains to show that $\varphi$ is an automorphism of ${\mathcal {M}}$.

The underlying map ${\tilde{\varphi }}:M\rightarrow M$ is surjective since if $p\notin {\tilde{\varphi }}(M)$, then $\varphi \in \varDelta (M,M\setminus \{p\})$ and therefore $\varphi _k\in \varDelta (M,M\setminus \{p\})$ for k large enough which contradicts the assumption that $\varphi _k$ is an automorphism. This also implies that there is an $x\in M$ such that the differential $D{\tilde{\varphi }}(x)$ is invertible. Using Hurwitz’s theorem (see e.g. [18], p. 80) it follows $\det (D{\tilde{\varphi }}(x))\ne 0$ for all $x\in M$. Thus ${\tilde{\varphi }}$ is locally biholomorphic. Moreover, $\varphi$ is locally invertible due to the special form of the sets $\varTheta (x_i)$.

In order check that ${\tilde{\varphi }}$ is injective, let $p_1,p_2\in M$, $p_1\ne p_2$, such that $q={\tilde{\varphi }}(p_1)={\tilde{\varphi }}(p_2)$. Let $\varOmega _j$, $j=1,2$, be open neighbourhoods of $p_j$ with $\varOmega _1\cap \varOmega _2= \emptyset$. By [18], p. 79, Proposition 5, there exists $k_0$ with the property that $q\in {\tilde{\varphi }}_k(\varOmega _1)$ and $q\in {\tilde{\varphi }}_k(\varOmega _2)$ for all $k\ge k_0$. The bijectivity of the $\varphi _k$’s now yields a contradiction to $\varOmega _1\cap \varOmega _2=\emptyset$. $\square$

Proposition 7

The set ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is a topological group with respect to composition and inversion of automorphisms.

Proof

Let $\varphi _k$ and $\psi _l$ be two sequences of automorphisms of ${\mathcal {M}}$ converging to $\varphi$ and $\psi$ respectively. By the classical theory, $\tilde{\varphi _k}\circ \tilde{\psi _l}$ converges to ${\tilde{\varphi }}\circ {\tilde{\psi }}$, and $\tilde{\varphi _k}^{-1}$ to ${\tilde{\varphi }}^{-1}$.

Let $U, V, W\subseteq M$ be open subsets with ${\tilde{\varphi }}(V)\subseteq W$, $\tilde{\varphi _k}(V)\subseteq W$, ${\tilde{\psi }}(U)\subseteq V$, $\tilde{\psi _l}(U)\subseteq V$, for k and l sufficiently large and let $f\in {\mathcal {O}}_{\mathcal {M}}(W)$. Then the sequence $(\varphi _k)^*(f)\in {\mathcal {O}}_{\mathcal {M}}(V)$ converges to $\varphi ^*(f)$ on V, and by Lemma 4 $(\varphi _k\circ \psi _l)^*(f)=(\psi _l)^*\left( (\varphi _k)^*(f)\right)$ converges to $\psi ^*(\varphi ^*(f))=(\varphi \circ \psi )^*(f)$ on U as $k,l\rightarrow \infty$ , which shows that the multiplication is continuous.

Consider now the inversion map ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\rightarrow {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, $\varphi \mapsto \varphi ^{-1}$. Let $\varphi _k$ be a sequence in ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ converging to $\varphi \in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$. Note that since the automorphism group ${{\mathrm{Aut}}}(M)$ of the underlying manifold M is a topological group, the inversion map ${{\mathrm{Aut}}}(M)\rightarrow {{\mathrm{Aut}}}(M)$ is continuous. For any choice of local coordinate charts on $U, V\subseteq M$ such that the closure of ${\tilde{\varphi }}^{-1}(U)$ is contained in V we can conclude: Since ${\tilde{\varphi }}_k^{-1}$ converges to ${\tilde{\varphi }}^{-1}$, we have $\tilde{\varphi _k}^{-1}(U)\subseteq V$ for k sufficiently large. Identify ${\mathcal {O}}_{\mathcal {M}}(U) \cong \varGamma _{\varLambda E}(U)$, resp. ${\mathcal {O}}_{\mathcal {M}}(V) \cong \varGamma _{\varLambda E}(V)$ and decompose $\varphi ^*=\varphi ^*_0\exp (Y)$, $\varphi _k^*=\varphi _{k,0}^*\exp (Y_k)$ as in Section 2. Note that $\varphi _0^*$ is induced by an automorphism $\varphi _0$ of the vector bundle E. We can verify by an observation in local coordinates that the map ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\rightarrow {{\mathrm{Aut}}}(E)$, $\varphi \mapsto \varphi _0$, is continuous. Hence, the sequence $\varphi _{k,0}$ converges to $\varphi _0$ and $\varphi _{k,0}^*$ converges to $\varphi _{0}^*$. By [17] the inversion on ${{\mathrm{Aut}}}(E)$ is continuous. Therefore, $(\varphi _{k,0}^{-1})^*$ converges to  $(\varphi _{0}^{-1})^*$. Due to the finiteness of the logarithm and exponential series on nilpotent elements, $Y_k$ converges to Y. Hence, $(\varphi ^{-1}_k)^*=\exp ({-Y_k})(\varphi ^{*}_{k,0})^{-1}$ converges to $\exp ({-Y})(\varphi ^{*}_{0})^{-1}=(\varphi ^{*})^{-1}$. $\square$

Remark 8

Let ${\mathcal {M}}$ be a split supermanifold and let $E\rightarrow M$ be a vector bundle with associated sheaf of sections ${\mathcal {E}}$ such that the structure sheaf ${\mathcal {O}}_{\mathcal {M}}$ is isomorphic to $\bigwedge {\mathcal {E}}$. By [17] the group of automorphisms ${{\mathrm{Aut}}}(E)$ of the vector bundle E is a complex Lie group. Each automorphism $\varphi$ of the supermanifold ${\mathcal {M}}$ induces an automorphism $\varphi _0$ of the vector bundle E over the underlying map ${{\tilde{\varphi }}}$ of $\varphi$, and the map $\pi :{{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\rightarrow {{\mathrm{Aut}}}(E)$, $\varphi \mapsto \varphi _0$, is continuous. An automorphism of the bundle E lifts to an automorphism of the supermanifold ${\mathcal {M}}$ if we fix a splitting ${\mathcal {O}}_{\mathcal {M}}\cong \bigwedge {\mathcal {E}}$. If $\chi :E\rightarrow E$ is an automorphism with pullback $\chi ^*$ we define an automorphism of ${\mathcal {M}}$ by the pullback $f_1\wedge \ldots \wedge f_k\mapsto \chi ^*(f_1)\wedge \ldots \wedge \chi ^*(f_k)$ for $f_1\wedge \ldots \wedge f_k\in \bigwedge ^k{\mathcal {E}}$. This assignment defines a section of $\pi$. In particular, $\pi$ is surjective and we have an exact sequence

$$\begin{aligned} 0\rightarrow \ker \pi \rightarrow {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\rightarrow {{\mathrm{Aut}}}(E)\rightarrow 0, \end{aligned}$$

which splits. Consequently, the topological group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is a semidirect product

$$\begin{aligned} {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\cong \ker \pi \rtimes {{\mathrm{Aut}}}(E). \end{aligned}$$

The kernel of $\pi$ consists of those automorphisms $\varphi$ of ${\mathcal {M}}$ whose underlying map ${{\tilde{\varphi }}}$ is the identity on M and whose pullback $\varphi ^*$ satisfies

$$\begin{aligned} (\varphi ^*-\mathrm {id}^*)({\mathcal {E}})\subseteq \bigoplus _{k\ge 2} \left( \bigwedge \!{}^k\, {\mathcal {E}}\right) .\\ \end{aligned}$$

In this case $(\varphi ^*-\mathrm {id}^*)$ is nilpotent and there is an even super vector field X on ${\mathcal {M}}$ with $\exp (X)=\varphi ^*$ as mentioned in Sect. 2. The super vector field X is nilpotent and fulfills

$$\begin{aligned} X\left( \bigwedge \!{}^k\,{\mathcal {E}}\right) \subseteq \bigoplus _{l\ge k+2}\left( \bigwedge \!{}^l\,{\mathcal {E}}\right) \end{aligned}$$

for all k. More generally, the map

$$\begin{aligned} \left\{ X\in \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})\left| \, X\left( \bigwedge \!{}^k\,{\mathcal {E}}\right) \subseteq \bigoplus _{l\ge k+2}\left( \bigwedge \!{}^l\,{\mathcal {E}}\right) \text { for all } k \right\} \right.&\ \longrightarrow \ \ \ker \pi ,\ \\ X&\mapsto \exp (X), \end{aligned}$$

which assigns to a super vector field X the automorphism of ${\mathcal {M}}$ with pullback $\exp (X)$, is bijective. In Sect. 6, we will prove that the Lie superalgebra $\mathrm {Vec}({\mathcal {M}})$ of super vector fields on ${\mathcal {M}}$ and thus subspaces of $\mathrm {Vec}({\mathcal {M}})$ are finite-dimensional. Therefore, the topological group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\cong \ker \pi \rtimes {{\mathrm{Aut}}}(E)$ carries the structure of a complex Lie group.

In the general case of a not necessarily split supermanifold ${\mathcal {M}}$, the proof that ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ can be endowed with the structure of a complex Lie group is more difficult. In order to prove the corresponding result also for non-split supermanifolds, the structure of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is further studied in the next two sections.

4 Non-existence of small subgroups of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$

In this section, we prove that ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ does not contain small subgroups, i.e. that there exists an open neighbourhood of the identity in ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ such that each subgroup contained in this neighbourhood consists only of the identity. As a consequence, the topological group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ carries the structure of a real Lie group by a result of Yamabe (cf. [25]).

Before proving the non-existence of small subgroups, a few technical preparations are needed: Consider ${\mathbb {C}}^{m|n}$ and let $z_1,\ldots ,z_m,\xi _1,\ldots ,\xi _n$ denote coordinates on ${\mathbb {C}}^{m|n}$. Let $U\subseteq {\mathbb {C}}^m$ be an open subset. For $f=\sum _\nu f_{\nu }\xi ^{\nu } \in {\mathcal {O}}_{{\mathbb {C}}^{m|n}}(U)$ define

$$\begin{aligned} \left| \left| f\right| \right| _U =\left| \left| \sum _{\nu }f_\nu \xi ^\nu \right| \right| _U :=\sum _\nu \left| \left| f_\nu \right| \right| _U, \end{aligned}$$

where $||f_\nu ||_U$ denotes the supremum norm of the holomorphic function $f_\nu$ on U. For any morphism $\varphi :{\mathcal {U}}=(U,{\mathcal {O}}_{{\mathbb {C}}^{m|n}}|_U) \rightarrow {\mathbb {C}}^{m|n}$ define

$$\begin{aligned} ||\varphi ||_U:=\sum _{i=1}^m ||\varphi ^*(z_i)||_U +\sum _{j=1}^n ||\varphi ^*(\xi _j)||_U. \end{aligned}$$

Lemma 9

Let ${\mathcal {U}}=(U,{\mathcal {O}}_{{\mathbb {C}}^{m|n}}|_{U})$ be a superdomain in ${\mathbb {C}}^{m|n}$. For any relatively compact open subset $U'$ of U there exists $\varepsilon >0$ such that any morphism $\psi :{\mathcal {U}}\rightarrow {\mathbb {C}}^{m|n}$ with the property $||\psi -\mathrm {id}||_U<\varepsilon$ is biholomorphic as a morphism from ${\mathcal {U}}'=(U',{\mathcal {O}}_{{\mathbb {C}}^{m|n}}|_{U'})$ onto its image.

Proof

Let $r>0$ such that the closure of the polydisc

$$\begin{aligned} \varDelta ^n_r(z)=\{(w_1,\ldots , w_m)|\,|w_j-z_j|< r\} \end{aligned}$$

is contained in U for any $z=(z_1,\ldots , z_m)\in U'$. Let $v\in {\mathbb {C}}^m$ be any non-zero vector. Then we have $z+\zeta v\in U$ for any $z\in U'$ and $\zeta$ in the closure of $\varDelta _{\frac{r}{||v||}}(0)=\{t\in {\mathbb {C}}|\, |t|<\frac{r}{||v||}\}$. If for given $\varepsilon >0$ it is $||\psi -\mathrm {id}||_U<\varepsilon$ then we have in particular $||{{\tilde{\psi }}}-\mathrm {id}||_U<\varepsilon$ for the supremum norm of the underlying maps ${{\tilde{\psi }}},\mathrm {id}:U\rightarrow {\mathbb {C}}^m$. Then, for the differential $D{{\tilde{\psi }}}$ of ${{\tilde{\psi }}}$ and any non-zero vector $v\in {\mathbb {C}}^m$ and any $z \in U^\prime$ we have

$$\begin{aligned} \left| \left| D{{\tilde{\psi }}} (z)(v)-v\right| \right|&=\left| \left| \frac{d}{dt}\left( {{\tilde{\psi }}}(z+tv)-(z+tv)\right) \right| \right| \\&=\frac{1}{2\pi }\left| \left| \int _{\partial \varDelta _{\frac{r}{||v||}}(0)} \frac{{{\tilde{\psi }}}(z+\zeta v)-(z+\zeta v)}{\zeta ^2} d\zeta \right| \right| \\&\le \frac{1}{2\pi }\int _{\partial \varDelta _{\frac{r}{||v||}}(0)}\left| \left| \frac{{{\tilde{\psi }}}(z+\zeta v)-(z+\zeta v)}{\zeta ^2} \right| \right| d\zeta \\&<\frac{\varepsilon ||v||}{r}. \end{aligned}$$

This implies $|| D{{\tilde{\psi }}}(z)-\mathrm {id}||< \frac{\varepsilon }{r}$ with respect to the operator norm, for any $z\in U'$. Thus ${{\tilde{\psi }}}$ is locally biholomorphic on $U'$ if $\varepsilon$ is small enough. Moreover, $\varepsilon$ might now be chosen such that ${{\tilde{\psi }}}$ is injective (see e.g. [13], Chapter 2, Lemma 1.3).

Let $\psi _{j,k},\psi _{j,\nu }$ be holomorphic functions on U such that $\psi ^*(\xi _j)=\sum _{k=1}^n \psi _{j,k}\xi _k +\sum _{||\nu ||\ge 3} \psi _{j,\nu } \xi ^\nu .$ By Remark 6 it is now enough to show

$$\begin{aligned} \det ((\psi _{j,k})_{1\le j,k\le n}(z))\ne 0 \end{aligned}$$

for all $z\in U'$ and $\varepsilon$ small enough in order to prove that $\psi$ is a biholomorphism form ${\mathcal {U}}'$ onto its image. This follows from the fact that we assumed, via $||\psi -\mathrm {id}||_U<\varepsilon$, that $||\psi _{j,k}||_U< \varepsilon$ if $j\ne k$ and $||\psi _{j,j}-1||_U<\varepsilon$. $\square$

This lemma now allows us to prove that ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ contains no small subgroups; for a similar result in the classical case see [5], Theorem 1.

Proposition 10

The topological group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ has no small subgroups, i.e. there is a neighbourhood of the identity which contains no non-trivial subgroup.

Proof

Let $U\subset V\subset W$ be open subsets of M such that U is relatively compact in V and V is relatively compact in W. Suppose that ${\mathcal {W}}=(W, {\mathcal {O}}_{{\mathcal {M}}}|_W)$ is isomorphic to a superdomain in ${\mathbb {C}}^{m|n}$ and let $z_1,\ldots , z_m,\xi _1,\ldots ,\xi _n$ be local coordinates on ${\mathcal {W}}$. By definition $\varDelta ({\overline{V}},W)=\{\varphi \in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})|\,{\tilde{\varphi }} ({\overline{V}})\subseteq W\}$ and $\varDelta ({\overline{U}},V)$ are open neighbourhoods of the identity in ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$. Choose $\varepsilon >0$ as in the preceding lemma such that any morphism $\chi :{\mathcal {V}}\rightarrow {\mathbb {C}}^{m|n}$ with $||\chi -\mathrm {id}||_V<\varepsilon$ is biholomorphic as a morphism from ${\mathcal {U}}$ onto its image. Let $\varOmega \subseteq \varDelta ({\overline{V}},W)\cap \varDelta ({\overline{U}},V)$ be the subset whose elements $\varphi$ satisfy $||\varphi -\mathrm {id}||_V<\varepsilon$. The set $\varOmega$ is open and contains the identity. Since ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is locally compact by Lemma 5, it is enough to show that each compact subgroup $Q\subseteq \varOmega$ is trivial. Otherwise for non-compact Q, let $\varOmega '$ be an open neighbourhood of the identity with compact closure ${\overline{\varOmega }}'$ which is contained in $\varOmega$, and suppose $Q\subseteq \varOmega '$. Then ${\overline{Q}}\subseteq {\overline{\varOmega }}'\subset \varOmega$ is a compact subgroup, and Q is trivial if ${\overline{Q}}$ is trivial.

Define a morphism $\psi :{\mathcal {V}}\rightarrow {\mathbb {C}}^{m|n}$ by setting

$$\begin{aligned} \psi ^*(z_i)=\int _Q q^*(z_i)\,dq\ \ \text { and }\ \ \psi ^*(\xi _j)=\int _Q q^*(\xi _j)\,dq, \end{aligned}$$

where the integral is taken with respect to the normalized Haar measure on Q. This yields a holomorphic morphism $\psi :{\mathcal {V}}\rightarrow {\mathbb {C}}^{m|n}$ since each $q\in Q$ defines a holomorphic morphism ${\mathcal {V}}\rightarrow {\mathcal {W}}\subseteq {\mathbb {C}}^{m|n}$. Its underlying map is ${\tilde{\psi }}(z)=\int _Q {{\tilde{q}}} (z) \,dq$. The morphism $\psi$ satisfies

$$\begin{aligned} ||\psi ^*(z_i)-z_i||_V =\left| \left| \int _Q (q^*(z_i)-z_i)\,dq\right| \right| _V \le \int _Q ||q^*(z_i)-z_i||_V\, dq \end{aligned}$$

and similarly

$$\begin{aligned} ||\psi ^*(\xi _j)-\xi _j||_V \le \int _Q ||q^*(\xi _j)-\xi _j||_V \, dq. \end{aligned}$$

Consequently, we have

$$\begin{aligned} ||\psi -\mathrm {id}||_V&= \sum _{i=1}^m ||\psi ^*(z_i)-z_i||_V+\sum _{j=1}^n || \psi ^*(\xi _j)-\xi _j||_V\\&\le \int _Q\left( \sum _{i=1}^m||q^*(z_i)-z_i||_V\, + \sum _{j=1}^n ||q^*(\xi _j)-\xi _j||_V \right) \, dq \\&=\int _Q ||q-\mathrm {id}||_V\, dq < \varepsilon . \end{aligned}$$

Thus by the preceding lemma, $\psi |_U$ is a biholomorphic morphism onto its image. Furthermore, on U we have $\psi \circ q'=\psi$ for any $q'\in Q$ since

$$\begin{aligned} (\psi \circ q')^*(z_i)&=(q')^*(\psi ^*(z_i)) =(q')^*\left( \int _Q q^*(z_i)\,dq\right) =\int _Q (q')^*(q^*(z_i))\,dq\\&=\int _Q(q\circ q')^*(z_i)\,dq =\int _Q q^*(z_i)\,dq =\psi ^*(z_i) \end{aligned}$$

due to the invariance of the Haar measure, and also

$$\begin{aligned} (\psi \circ q')^*(\xi _j)=\psi ^*(\xi _j). \end{aligned}$$

The equality $\psi \circ q'=\psi$ on U implies $q'|_U=\mathrm {id}_{\mathcal {U}}$ because of the invertibility of $\psi$. By the identity principle it follows that $q'=\mathrm {id}_{{\mathcal {M}}}$ if M is connected, and hence $Q=\{\mathrm {id}_{{\mathcal {M}}}\}$.

In general, M has only finitely many connected components since M is compact. Therefore, a repetition of the preceding argument yields the existence of a neighbourhood of the identity of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ without any non-trivial subgroups. $\square$

By Theorem 3 in [25], the preceding proposition implies the following:

Corollary 11

The topological group $Aut_{{{\bar{0}}}}({\mathcal {M}})$ can be endowed with the structure of a real Lie group.

5 One-parameter subgroups of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$

In order to obtain results on the regularity of the action of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on the compact complex supermanifold ${\mathcal {M}}$ and to characterize the Lie algebra of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, we study continuous one-parameter subgroups of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$. Each continuous one-parameter subgroup ${\mathbb {R}}\rightarrow {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is an analytic map between the Lie groups ${\mathbb {R}}$ and ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$.

We prove that the action of each continuous one-parameter subgroup of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on ${\mathcal {M}}$ is analytic and induces an even holomorphic super vector field on ${\mathcal {M}}$. Consequently, the Lie algebra of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ may be identified with the Lie algebra $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ of even holomorphic super vector fields on ${\mathcal {M}}$, and ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ carries the structure of a complex Lie group whose action on the supermanifold ${\mathcal {M}}$ is holomorphic.

Definition 1

A continuous one-parameter subgroup $\varphi$ of automorphisms of ${\mathcal {M}}$ is a family of automorphisms $\varphi _t:{\mathcal {M}}\rightarrow {\mathcal {M}}$, $t\in {\mathbb {R}}$, such that the map $\varphi :{\mathbb {R}}\rightarrow {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, $t\mapsto \varphi _t$, is a continuous group homomorphism.

Remark 12

Let $\varphi _t:{\mathcal {M}}\rightarrow {\mathcal {M}}$, $t\in {\mathbb {R}}$, be a family of automorphisms satisfying $\varphi _{s+t}=\varphi _s\circ \varphi _t$ for all $s,t\in {\mathbb {R}}$, and such that ${{\tilde{\varphi }}}:{\mathbb {R}}\times M\rightarrow M$, ${{\tilde{\varphi }}}(t,p)= {\tilde{\varphi }}_t(p)$ is continuous. Then $\varphi _t$ is a continuous one-parameter subgroup if and only if the following condition is satisfied: Let $U, V\subset M$ be open subsets, and $[a,b]\subset {\mathbb {R}}$ such that ${{\tilde{\varphi }}}([a,b]\times U)\subseteq V$. Assume moreover that there are local coordinates $z_1,\ldots , z_m,\xi _1,\ldots ,\xi _n$ for ${\mathcal {M}}$ on U. Then for any $f\in {\mathcal {O}}_{{\mathcal {M}}}(V)$ there are continuous functions $f_\nu :[a,b]\times U\rightarrow {\mathbb {C}}$ with $(f_\nu )_t=f_\nu (t,\cdot )\in {\mathcal {O}}_{{\mathcal {M}}}(U)$ for fixed $t\in [a,b]$ such that

$$\begin{aligned} (\varphi _t)^*(f)=\sum _\nu f_\nu (t,z)\xi ^\nu . \end{aligned}$$

We say that the action of the one-parameter subgroup $\varphi$ on ${\mathcal {M}}$ is analytic if each $f_\nu (t,z)$ is analytic in both components.

This equivalent characterization of continuous one-parameter subgroups of automorphisms also allows us to define this notion for non-compact complex supermanifolds.

Proposition 13

Let $\varphi$ be a continuous one-parameter subgroup of automorphisms on ${\mathcal {M}}$. Then the action of $\varphi$ on ${\mathcal {M}}$ is analytic.

Remark 14

The statement of Proposition 13 also holds true for complex supermanifolds ${\mathcal {M}}$ with non-compact underlying manifold M as compactness of M is not needed for the proof.

For the proof of the proposition the following technical lemma is needed:

Lemma 15

Let $U\subseteq V\subseteq {\mathbb {C}}^m$ be open subsets, $p\in U$, $\varOmega \subseteq {\mathbb {R}}$ an open connected neighbourhood of 0, and let $\alpha :\varOmega \times U\rightarrow V$ be a continuous map satisfying

$$\begin{aligned} \alpha (t,z)=\alpha (t+s,z)-f(t,s,z) \end{aligned}$$

for (t, s, z) in a neighbourhood of (0, 0, p) and for some continuous function f which is analytic in (t, z). If $\alpha$ is holomorphic in the second component, then it is analytic on a neighbourhood of (0, p).

Proof

For small t, $h>0$, z near p, we have

$$\begin{aligned} h\cdot \alpha (t,z)&=\int _0^h\alpha (t+s,z)ds - \int _0^h f(t,s,z) ds\\&=\int _t^{h+t}\alpha (s,z)ds - \int _0^h \alpha (s,z)ds -\int _0^h (f(t,s,z)-\alpha (s,z))ds\\&=\int _h^{h+t}\alpha (s,z)ds-\int _0^t\alpha (s,z)ds -\int _0^h(f(t,s,z)-\alpha (s,z))ds\\&=\int _0^t(\alpha (s+h,z)-\alpha (s,z))ds -\int _0^h(f(t,s,z)-\alpha (s,z))ds\\&=\int _0^t f(s,h,z)ds -\int _0^h(f(t,s,z)-\alpha (s,z))ds. \end{aligned}$$

The assumption that f is a continuous function which is analytic in the first and third component therefore implies that $\alpha$ is analytic. $\square$

Proof (of Proposition 13)

Due to the action property $\varphi _{s+t}=\varphi _s\circ \varphi _t$ it is enough to show the statement for the restriction of $\varphi$ to $(-\varepsilon ,\varepsilon )\times {\mathcal {M}}$ for some $\varepsilon >0$. Let $U,V\subseteq M$ be open subsets such that U is relatively compact in V, and such that there are local coordinates $z_1,\ldots , z_m,\xi _1,\ldots , \xi _n$ on V for ${\mathcal {M}}$. Choose $\varepsilon >0$ such that ${{\tilde{\varphi }}}_t(U)\subseteq V$ for any $t\in (-\varepsilon ,\varepsilon )$. Let $\alpha _{i,\nu }$, $\beta _{j,\nu }$ be continuous functions on $(-\varepsilon ,\varepsilon )\times U$ with

$$\begin{aligned} (\varphi _t)^*(z_i)=\sum _{|\nu |=0} \alpha _{i,\nu }(t,z) \xi ^\nu \end{aligned}$$

and

$$\begin{aligned} (\varphi _t)^*(\xi _j)=\sum _{|\nu |=1} \beta _{j,\nu }(t,z) \xi ^\nu , \end{aligned}$$

where $|\nu |=|(\nu _1,\ldots ,\nu _n)|=(\nu _1+\ldots +\nu _n)\!\!\mod 2 \in {\mathbb {Z}}_2$. We have to show that $\alpha$ and $\beta$ are analytic in (t, z). The induced map $\psi ':(-\varepsilon ,\varepsilon )\times U\times {\mathbb {C}}^n\rightarrow V\times {\mathbb {C}}^n$ on the underlying vector bundle is given by

$$\begin{aligned} \left( t,\left( \begin{matrix} z_1\\ \vdots \\ z_m \\ v_1\\ \vdots \\ v_n \end{matrix}\right) \right) \mapsto \left( \begin{matrix} \alpha _{1,0}(t,z)\\ \vdots \\ \alpha _{m,0}(t, z)\\ \sum _{k=1}^n \beta _{1,k}(t,z)v_k\\ \vdots \\ \sum _{k=1}^n \beta _{n,k}(t,z)v_k \end{matrix}\right) , \end{aligned}$$

where $\beta _{j,k}=\beta _{j,e_k}$ if $e_k=(0,\ldots ,0,1,0,\ldots , 0)$ denotes the k-th unit vector. The map $\psi '$ is a local continuous one-parameter subgroup on $U\times {\mathbb {C}}^n$ because $\varphi$ is a continuous one-parameter subgroup. By a result of Bochner and Montgomery the map $\psi '$ is analytic in (t, z, v) (see [4], Theorem 4). Hence, the map $\psi :(-\varepsilon ,\varepsilon )\times {\mathcal {U}}\rightarrow {\mathcal {V}}$ given by $(\psi _t)^*(z_i)=\alpha _i(t,z)$, $(\psi _t)^*(\xi _j)=\sum _{k=1}^n \beta _{j,k}(t,z)\xi _k$ is analytic. Let X be the local vector field on ${\mathcal {U}}$ induced by $\psi$, i.e.

$$\begin{aligned} X(f)=\left. \frac{\partial }{\partial t}\right| _0 (\psi _t)^*(f). \end{aligned}$$

We may assume that X is non-degenerate, i.e. the evaluation of X in p, X(p), does not vanish for all $p\in U$. Otherwise, consider, instead of $\varphi$, the diagonal action on ${\mathbb {C}}\times {\mathcal {M}}$ acting by addition of t in the first component and $\varphi _t$ in the second, and note that this action is analytic precisely if $\varphi$ is analytic. For the differential $d\psi$ of $\psi$ in (0, p) we have

$$\begin{aligned} d\psi \left( \left. \frac{\partial }{\partial t}\right| _{(0,p)}\right) =\left. \frac{\partial }{\partial t}\right| _{(0,p)} \circ \psi ^* =X(p)\ne 0. \end{aligned}$$

Therefore, the restricted map $\psi |_{(-\varepsilon ,\varepsilon )\times \{p\}}$ is an immersion and its image $\psi ((-\varepsilon ,\varepsilon )\times \{p\})$ is a subsupermanifold of ${\mathcal {V}}$. Let ${\mathcal {S}}$ be a subsupermanifold of ${\mathcal {U}}$ transversal to $\psi ((-\varepsilon ,\varepsilon )\times \{p\})$ in p. The map $\psi |_{(-\varepsilon ,\varepsilon )\times {\mathcal {S}}}$ is a submersion in (0, p) since $d\psi (T_{(0,p)}(-\varepsilon ,\varepsilon )\times \{p\})) = T_p \psi ((-\varepsilon ,\varepsilon )\times \{p\})$ and $d\psi (T_{(0,p)}\{0\}\times {\mathcal {S}})=T_p{\mathcal {S}}$ because $\psi |_{\{0\}\times {\mathcal {U}}}=\mathrm {id}$. Hence $\chi :=\psi |_{(-\varepsilon ,\varepsilon )\times {\mathcal {S}}}$ is locally invertible around (0, p), and thus invertible as a map onto its image after possibly shrinking U and $\varepsilon$, and

$$\begin{aligned} \chi _*\left( \frac{\partial }{\partial t}\right) =(\chi ^{-1})^*\circ \frac{\partial }{\partial t}\circ \chi ^* =(\chi ^{-1})^*\circ \chi ^*\circ X=X. \end{aligned}$$

Therefore, after defining new coordinates $w_1,\ldots , w_m,\theta _1,\ldots , \theta _n$ for ${\mathcal {M}}$ on U via $\chi$, we have $X=\frac{\partial }{\partial w_1}$ and $(\varphi _t)^*$ is of the form

$$\begin{aligned} (\varphi _t)^*(w_1)&=w_1+t+\sum _{|\nu |=0, \nu \ne 0} \alpha _{1,\nu }(t,w)\theta ^\nu ,\\ (\varphi _t)^*(w_i)&=w_i+\sum _{|\nu |=0,\nu \ne 0} \alpha _{i,\nu }(t,w)\theta ^\nu \ \ \ \text { for } i\ne 1,\\ (\varphi _t)^*(\theta _j)&=\theta _j+\sum _{|\nu |=1,||\nu ||\ne 1} \beta _{j,\nu }(t,w)\theta ^\nu , \end{aligned}$$

for appropriate $\alpha _{i,\nu }$, $\beta _{j,\nu }$, where $||\nu ||=||(\nu _1,\ldots ,\nu _n)||=\nu _1+\cdots +\nu _n$.

For small s and t we have

$$\begin{aligned} \varphi _t^*\left( \varphi _s^*(w_i)\right)&=\varphi _t^*\left( w_i+\delta _{1,i}s+ \sum _{|\nu |=0,||\nu ||\ne 0} \alpha _{i,\nu }(s,w)\theta ^\nu \right) \nonumber \\&= w_i+\delta _{i,1}(t+s) +\sum _{|\nu |=0,||\nu ||\ne 0} \alpha _{i,\nu }(t,w)\theta ^\nu +\sum _{|\nu |=0,||\nu ||\ne 0} \varphi _t^*(\alpha _{i,\nu }(s,w)\theta ^\nu ). \end{aligned}$$

(1)

Let $f_{i,\nu }(t,s,w)$ be such that

$$\begin{aligned} \sum _{|\nu |=0,||\nu ||\ne 0} \varphi _t^*(\alpha _{i,\nu }(s,w)\theta ^\nu ) =\sum _{|\nu |=0,||\nu ||\ne 0} f_{i,\nu }(t,s,w) \theta ^\nu . \end{aligned}$$

(2)

For fixed $\nu _0$ the coefficient $f_{i,{\nu _0}}(t,s,w)$ of $\theta ^{\nu _0}$ depends only on $\alpha _{i,\nu _0}(s,w+t e_1)$, $\beta _{j,\mu }(t,w)$ for $\mu$ with $||\mu ||\le ||\nu _0||-1$, and $\alpha _{j,\nu }(t,w)$ and its partial derivatives in the second component for $\nu$ with $||\nu ||\le ||\nu _0||-2$. This can be shown by a calculation using the special form of $\varphi _t^*(w_j)$ and $\varphi _t^*(\theta _j)$ and general properties of the pullback of a morphism of supermanifolds. Assume now that the analyticity near (0, p) of $\alpha _{i,\nu }$, $\beta _{j,\mu }$ is shown for $||\nu ||, ||\mu ||< 2k$ and all i, j. Let $\nu _0$ be such that $||\nu _0||=2k$. Then $f_{i,\nu _0}(t,s,w)$ is a continuous function which is analytic in (t, w) near (0, p) for fixed s. Since $\varphi _t^*(\varphi _s^*(w_i)) =\varphi _{t+s}^*(w_i)$, using (1) and (2) we get

$$\begin{aligned} \alpha _{i,\nu _0}(t,w)+f_{i,\nu _0}(t,s,w) =\alpha _{i,\nu _0}(t+s,w), \end{aligned}$$

and thus $\alpha _{i,\nu _0}(t,w)$ is analytic near (0, p) by Lemma 15. Similarly, it can be shown that $\beta _{j,\mu _0}$ is analytic for $||\mu _0||=2k+1$ if $\alpha _{i,\nu }$, $\beta _{j,\mu }$ for $||\nu ||$, $||\mu ||< 2k+1$. $\square$

Corollary 16

The Lie algebra of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is isomorphic to the Lie algebra $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ of even super vector fields on ${\mathcal {M}}$, and ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is a complex Lie group.

Proof

If $\gamma :{\mathbb {R}}\rightarrow {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, $t\mapsto \gamma _t$ is a continuous one-parameter subgroup, then by Proposition 13 the action of $\varphi$ on ${\mathcal {M}}$ is analytic. Therefore, $\gamma$ induces an even holomorphic super vector field $X(\gamma )$ on ${\mathcal {M}}$ by setting

$$\begin{aligned} X(\gamma )=\left. \frac{\partial }{\partial t}\right| _0 (\gamma _t)^*, \end{aligned}$$

and $\gamma$ is the flow map of $X(\gamma )$. On the other hand, since M is compact, the underlying vector field of each $X\in \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ is globally integrable and the proof of Theorem 5.4 in [12] then shows that X is also globally integrable. Its flow defines a one-parameter subgroup $\gamma ^X$ of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, which is continuous. This yields an isomorphism of Lie algebras

$$\begin{aligned} \mathrm {Lie}({{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})) \rightarrow \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}}). \end{aligned}$$

Consequently, we have $\mathrm {Lie}({{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})) \cong \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ and since $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ is a complex Lie algebra, ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ carries the structure of a complex Lie group. $\square$

The Lie group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ naturally acts on ${\mathcal {M}}$; this action $\psi :{{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\times {\mathcal {M}}\rightarrow {\mathcal {M}}$ is given by $\mathrm {ev}_g\circ \psi ^*=g^*$ where $\mathrm {ev}_g$ denotes the evaluation in $g\in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ in the first component.

Corollary 17

The natural action of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on ${\mathcal {M}}$ defines a holomorphic morphism of supermanifolds $\psi :{{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\times {\mathcal {M}} \rightarrow {\mathcal {M}}$.

Proof

Since the action of each continuous one-parameter subgroup of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on ${\mathcal {M}}$ is holomorphic by the preceding considerations, and each $g\in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is a biholomorphic morphism $g:{\mathcal {M}}\rightarrow {\mathcal {M}}$, the action $\psi$ is a holomorphic. $\square$

If a Lie supergroup ${\mathcal {G}}$ (with Lie superalgebra ${\mathfrak {g}}$ of right-invariant super vector fields) acts on a supermanifold ${\mathcal {M}}$ via $\psi :{\mathcal {G}}\times {\mathcal {M}}\rightarrow {\mathcal {M}}$, this action $\psi$ induces an infinitesimal action $d\psi :{\mathfrak {g}}\rightarrow \mathrm {Vec}({\mathcal {M}})$ defined by $d\psi (X)=(X(e)\otimes \mathrm {id}_{\mathcal {M}}^*)\circ \psi ^*$ for any $X\in {\mathfrak {g}}$, where $X\otimes \mathrm {id}_{\mathcal {M}}^*$ denotes the canonical extension of the vector field X on ${\mathcal {G}}$ to a vector field on ${\mathcal {G}}\times {\mathcal {M}}$, and $(X(e)\otimes \mathrm {id}_{\mathcal {M}}^*)$ is its evaluation in the neutral element e of ${\mathcal {G}}$.

Corollary 18

Identifying the Lie algebra of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ with $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ as in Corollary 16, the induced infinitesimal action of the action $\psi :{{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\times {\mathcal {M}}\rightarrow {\mathcal {M}}$ in Corollary 17 is the inclusion $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})\hookrightarrow \mathrm {Vec}({\mathcal {M}})$.

6 The Lie superalgebra of vector fields

In this section, we prove that the Lie superalgebra $\mathrm {Vec}({\mathcal {M}})$ of holomorphic super vector fields on a compact complex supermanifold ${\mathcal {M}}$ is finite-dimensional.

First, we prove that $\mathrm {Vec}({\mathcal {M}})$ is finite-dimensional if ${\mathcal {M}}$ is a split supermanifold using that its tangent sheaf ${\mathcal {T}}_{{\mathcal {M}}}$ is a coherent sheaf of ${\mathcal {O}}_M$-modules, where ${\mathcal {O}}_M$ denotes again the sheaf of holomorphic functions on the underlying manifold M. Then the statement in the general case is deduced using a filtration of the tangent sheaf.

Remark that since ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is a complex Lie group with Lie algebra isomorphic to the Lie algebra $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ of even holomorphic super vector fields on ${\mathcal {M}}$ (see Corollary 16), we already know that the even part of $\mathrm {Vec}({\mathcal {M}}) =\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})\oplus \mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}})$ is finite-dimensional.

Lemma 19

Let ${\mathcal {M}}$ be a split complex supermanifold. Then its tangent sheaf ${\mathcal {T}}_{\mathcal {M}}$ is a coherent sheaf of ${\mathcal {O}}_M$-modules.

Proof

Since ${\mathcal {M}}$ is split, its structure sheaf ${\mathcal {O}}_{\mathcal {M}}$ is isomorphic to $\bigwedge {\mathcal {E}}$ as an ${\mathcal {O}}_M$-module, where ${\mathcal {E}}$ is the sheaf of sections of a holomorphic vector bundle on the underlying manifold M. Thus, the structure sheaf ${\mathcal {O}}_{\mathcal {M}}$, and hence also the tangent sheaf ${\mathcal {T}}_{\mathcal {M}}$, carry the structure of a sheaf of ${\mathcal {O}}_M$-modules. Let $U\subset M$ be an open subset such that there exist even coordinates $z_1,\ldots , z_m$ and odd coordinates $\xi _1,\ldots , \xi _n$. Any derivation $D\in {\mathcal {T}}_{\mathcal {M}}(U)$ on U can uniquely be written as

$$\begin{aligned} D=\sum _{\nu \in ({\mathbb {Z}}_2)^n}\left( \sum _{i=1}^m f_{i,\nu } (z) \xi ^\nu \frac{\partial }{\partial z_i} +\sum _{j=1}^n g_{j,\nu }(z)\xi ^\nu \frac{\partial }{\partial \xi _j}\right) \end{aligned}$$

where $f_{i,\nu }$, $g_{j,\nu }$ are holomorphic functions on U. Therefore, the restricted sheaf ${\mathcal {T}}_{\mathcal {M}}|_U$ is isomorphic to $({\mathcal {O}}_M|_U)^{2^n (m+n)}$ and ${\mathcal {T}}_{\mathcal {M}}$ is coherent over ${\mathcal {O}}_M$. $\square$

Proposition 20

The Lie superalgebra $\mathrm {Vec}({\mathcal {M}})$ of holomorphic super vector fields on a compact complex supermanifold ${\mathcal {M}}$ is finite-dimensional.

Proof

First, assume that ${\mathcal {M}}$ is split. Then the tangent sheaf ${\mathcal {T}}_{\mathcal {M}}$ is a coherent sheaf of ${\mathcal {O}}_M$-modules. Thus, the space of global sections of ${\mathcal {T}}_{\mathcal {M}}$, $\mathrm {Vec}({\mathcal {M}}) ={\mathcal {T}}_{\mathcal {M}}(M)$, is finite-dimensional since M is compact (cf. [9]).

Now, let ${\mathcal {M}}$ be an arbitrary compact complex supermanifold. We associate the split complex supermanifold $\text {gr}\,{\mathcal {M}}=(M, \text {gr}\, {\mathcal {O}}_{\mathcal {M}})$ as described in Section 2. Let ${\mathcal {I}}_{\mathcal {M}}$ denote as before the subsheaf of ideal in ${\mathcal {O}}_{\mathcal {M}}$ generated by the odd elements. Define the filtration of sheaves of Lie superalgebras

$$\begin{aligned} {\mathcal {T}}_{\mathcal {M}}=:({\mathcal {T}}_{\mathcal {M}})_{(-1)}\supset ({\mathcal {T}}_{\mathcal {M}})_{(0)}\supset ({\mathcal {T}}_{\mathcal {M}})_{(1)}\supset \cdots \supset ({\mathcal {T}}_{\mathcal {M}})_{(n+1)}=0 \end{aligned}$$

of the tangent sheaf ${\mathcal {T}}_{\mathcal {M}}$ by setting

$$\begin{aligned} ({\mathcal {T}}_{\mathcal {M}})_{(k)}=\{D\in {\mathcal {T}}_{\mathcal {M}}|\, D({\mathcal {O}}_{\mathcal {M}})\subset ({\mathcal {I}}_{\mathcal {M}})^k, \,D({\mathcal {I}}_{\mathcal {M}})\subset ({\mathcal {I}}_{\mathcal {M}})^{k+1}\} \end{aligned}$$

for $k\ge 0$. Moreover, define $\text {gr}_k({\mathcal {T}}_{\mathcal {M}})= ({\mathcal {T}}_{\mathcal {M}})_{(k)}/({\mathcal {T}}_{\mathcal {M}})_{(k+1)}$ and set

$$\begin{aligned} \text {gr}({\mathcal {T}}_{\mathcal {M}})=\bigoplus _{k\ge -1} \text {gr}_k({\mathcal {T}}_{\mathcal {M}}). \end{aligned}$$

By [19], Proposition 1, the sheaf $\text {gr}({\mathcal {T}}_{\mathcal {M}})$ is isomorphic to the tangent sheaf of the associated split supermanifold $\text {gr}\, {\mathcal {M}}$. By the preceding considerations, the space of holomorphic super vector fields on $\text {gr}\,{\mathcal {M}}$,

$$\begin{aligned} \text {Vec}(\text {gr}\,{\mathcal {M}})=\text {gr}({\mathcal {T}}_{\mathcal {M}})(M)= \bigoplus _{k\ge -1} \text {gr}_k({\mathcal {T}}_{\mathcal {M}})(M), \end{aligned}$$

is of finite dimension. The projection onto the quotient yields

$$\begin{aligned} \dim ({\mathcal {T}}_{\mathcal {M}})_{(k)}(M)-\dim ({\mathcal {T}}_{\mathcal {M}})_{(k+1)}(M) \le \dim (\text {gr}_k({\mathcal {T}}_{\mathcal {M}})(M)) \end{aligned}$$

and $\dim ({\mathcal {T}}_{\mathcal {M}})_{(n)}(M) =\dim (\text {gr}_n({\mathcal {T}}_{\mathcal {M}})(M))$ and hence by induction

$$\begin{aligned} \dim ({\mathcal {T}}_{\mathcal {M}})_{(k)}(M) \le \sum _{j\ge k}\dim (\text {gr}_j({\mathcal {T}}_{\mathcal {M}})(M)), \end{aligned}$$

which gives

$$\begin{aligned} \dim ({\mathcal {T}}_{\mathcal {M}}(M)) =\dim \left( ({\mathcal {T}}_{\mathcal {M}})_{(-1)}(M)\right) \le \dim \left( \text {gr}({\mathcal {T}}_{\mathcal {M}})(M)\right) . \end{aligned}$$

In particular, $\dim ({\mathcal {T}}_{\mathcal {M}}(M))$ is finite. $\square$

Remark 21

The proof of the preceding proposition also shows the following inequality:

$$\begin{aligned} \dim (\mathrm {Vec}({\mathcal {M}}))\le \dim (\mathrm {Vec}(\text {gr}\,{\mathcal {M}})) \end{aligned}$$

7 The automorphism group

In this section, the automorphism group of a compact complex supermanifold is defined. This is done via the formalism of Harish-Chandra pairs for complex Lie supergroups (cf. [24]). The underlying classical Lie group is ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ and the Lie superalgebra is $\mathrm {Vec}({\mathcal {M}})$, the Lie superalgebra of super vector fields on ${\mathcal {M}}$. Moreover, we prove that the automorphism group satisfies a universal property.

Consider the representation $\alpha$ of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on $\mathrm {Vec}({\mathcal {M}})$ given by

$$\begin{aligned} \alpha (g)(X)=g_*(X)=(g^{-1})^*\circ X\circ g^*\ \ \ \text { for } \ \ \ g\in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}}), \, X\in \mathrm {Vec}({\mathcal {M}}). \end{aligned}$$

This representation $\alpha$ preserves the parity on $\mathrm {Vec}({\mathcal {M}})$, and its restriction to $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ coincides with the adjoint action of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on its Lie algebra $\mathrm {Lie}({{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}}))\cong \mathrm {Vec}_{{{\bar{0}}}} ({\mathcal {M}})$. Moreover, the differential $(d\alpha )_{\mathrm {id}}$ at the identity $\mathrm {id}\in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is the adjoint representation of $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ on $\mathrm {Vec}({\mathcal {M}})$:

Let X and Y be super vector fields on ${\mathcal {M}}$. Assume that X is even and let $\varphi ^X$ denote the corresponding one-parameter subgroup. Then we have

$$\begin{aligned} (d\alpha )_{\mathrm {id}}(X)(Y)=\left. \frac{\partial }{\partial t}\right| _0 (\varphi ^X_t)_*(Y)=[X,Y]; \end{aligned}$$

see e.g. [2], Corollary 3.8. Therefore, the pair $({{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}}),\mathrm {Vec}({\mathcal {M}}))$ together with the representation $\alpha$ is a complex Harish-Chandra pair, and using the equivalence between the category of complex Harish-Chandra pairs and complex Lie supergroups (cf. [24], $\S$ 2), we can define the automorphism group of a compact complex supermanifold ${\mathcal {M}}$ as follows:

Definition 2

Define the automorphism group ${{\mathrm{Aut}}}({\mathcal {M}})$ of a compact complex supermanifold to be the unique complex Lie supergroup associated with the Harish-Chandra pair $({{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}}), \mathrm {Vec}({\mathcal {M}}))$ with adjoint representation $\alpha$.

Since the action $\psi :{{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\times {\mathcal {M}}\rightarrow {\mathcal {M}}$ induces the inclusion $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}}) \hookrightarrow \mathrm {Vec}({\mathcal {M}})$ as infinitesimal action (see Corollary 18), there exists a Lie supergroup action $\varPsi :{{\mathrm{Aut}}}({\mathcal {M}})\times {\mathcal {M}} \rightarrow {\mathcal {M}}$ with the identity $\mathrm {Vec}({\mathcal {M}}) \rightarrow \mathrm {Vec}({\mathcal {M}})$ as induced infinitesimal action and $\varPsi |_{{{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\times {\mathcal {M}}}=\psi$ (cf. Theorem 5.35 in [2]).

The automorphism group together with $\varPsi$ satisfies a universal property:

Theorem 22

Let ${\mathcal {G}}$ be a complex Lie supergroup with a holomorphic action $\varPsi _{{\mathcal {G}}}:{\mathcal {G}}\times {\mathcal {M}}\rightarrow {\mathcal {M}}$. Then there is a unique morphism $\sigma :{\mathcal {G}}\rightarrow {{\mathrm{Aut}}}({\mathcal {M}})$ of Lie supergroups such that the diagram

is commutative.

Proof

Let G be the underlying Lie group of ${\mathcal {G}}$. For each $g\in G$, we have a morphism $\varPsi _{\mathcal {G}}(g):{\mathcal {M}}\rightarrow {\mathcal {M}}$ by setting $(\varPsi _{\mathcal {G}}(g))^*=\mathrm {ev}_g\circ (\varPsi _{\mathcal {G}})^*$. This morphism $\varPsi _{\mathcal {G}}(g)$ is an automorphism of ${\mathcal {M}}$ with inverse $\varPsi _{\mathcal {G}}(g^{-1})$ and gives rise to a group homomorphism ${{\tilde{\sigma }}}:G\rightarrow {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, $g\mapsto \varPsi _{\mathcal {G}}(g)$.

Let ${\mathfrak {g}}$ denote the Lie superalgebra (of right-invariant super vector fields) of ${\mathcal {G}}$, and $d\varPsi _{\mathcal {G}}:{\mathfrak {g}}\rightarrow \mathrm {Vec}({\mathcal {M}})$ the infinitesimal action induced by $\varPsi _{\mathcal {G}}$. The restriction of $d\varPsi _{\mathcal {G}}$ to the even part ${\mathfrak {g}}_{{{\bar{0}}}}=\mathrm {Lie}(G)$ of ${\mathfrak {g}}$ coincides with the differential $(d{{\tilde{\sigma }}})_e$ of ${{\tilde{\sigma }}}$ at the identity $e\in G$.

Moreover, if $\alpha _{\mathcal {G}}$ denotes the adjoint action of G on ${\mathfrak {g}}$, and $\alpha$ denotes, as before, the adjoint action of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on $\mathrm {Vec}({\mathcal {M}})$, we have

$$\begin{aligned} d\varPsi _{\mathcal {G}}(\alpha _{\mathcal {G}}(g)(X))&=(\varPsi _{\mathcal {G}}(g^{-1}))^*\circ d\varPsi _{\mathcal {G}}(X) \circ (\varPsi _{\mathcal {G}}(g))^*\\&=({{\tilde{\sigma }}}(g^{-1}))^*\circ d\varPsi _{\mathcal {G}}(X)\circ ({{\tilde{\sigma }}}(g))^*\\&=\alpha ({{\tilde{\sigma }}}(g))( d\varPsi _{\mathcal {G}}(X)) \end{aligned}$$

for any $g\in G$, $X\in {\mathfrak {g}}$. Using the correspondence between Lie supergroups and Harish-Chandra pairs, it follows that there is a unique morphism $\sigma :{\mathcal {G}} \rightarrow {{\mathrm{Aut}}}({\mathcal {M}})$ of Lie supergroups with underlying map ${{\tilde{\sigma }}}$ and derivative $d\varPsi _{\mathcal {G}}: {\mathfrak {g}}\rightarrow \mathrm {Vec}({\mathcal {M}})$ (see e.g. [24], $\S$ 2), and $\sigma$ satisfies $\varPsi \circ (\sigma \times \mathrm {id}_{\mathcal {M}})=\varPsi _{\mathcal {G}}$.

The uniqueness of $\sigma$ follows from the fact that each morphism $\tau :{\mathcal {G}}\rightarrow {{\mathrm{Aut}}}({\mathcal {M}})$ of Lie supergroups fulfilling the same properties as $\sigma$ necessarily induces the map $d\varPsi _{\mathcal {G}}: {\mathfrak {g}}\rightarrow \mathrm {Vec}({\mathcal {M}})$ on the level of Lie superalgebras and its underlying map ${{\tilde{\tau }}}$ has to satisfy ${{\tilde{\tau }}}(g)=\varPsi _{\mathcal {G}}(g)={{\tilde{\sigma }}}(g)$. $\square$

Remark 23

Since the morphism $\sigma$ in Theorem 22 is unique, the automorphism group of a compact complex supermanifold ${\mathcal {M}}$ is the unique Lie supergroup satisfying the universal property formulated in Theorem 22.

Remark 24

We say that a real Lie supergroup ${\mathcal {G}}$ acts on ${\mathcal {M}}$ by holomorphic transformations if the underlying Lie group G acts on the complex manifold M by holomorphic transformations and if there is a homomorphism of Lie superalgebras ${\mathfrak {g}}\rightarrow \mathrm {Vec}({\mathcal {M}})$ which is compatible with the action of G on M. Using the theory of Harish-Chandra pairs, we also have the Lie supergroup ${\mathcal {G}}^{\mathbb {C}}$, the universal complexification of ${\mathcal {G}}$; see [14]. The underlying Lie group of ${\mathcal {G}}^{\mathbb {C}}$ is the universal complexification $G^{\mathbb {C}}$ of the Lie group G. Let ${\mathfrak {g}}={\mathfrak {g}}_{{{\bar{0}}}}\oplus {\mathfrak {g}}_{{{\bar{1}}}}$ denote the Lie superalgebra of ${\mathcal {G}}$, ${\mathfrak {g}}_{{{\bar{0}}}}$ the Lie algebra of G. Then the Lie algebra ${\mathfrak {g}}_{{{\bar{0}}}}^{\mathbb {C}}$ of $G^{\mathbb {C}}$ is a quotient of ${\mathfrak {g}}_{{{\bar{0}}}}\otimes {\mathbb {C}}$, and the Lie superalgebra of ${\mathcal {G}}^{\mathbb {C}}$ can be realized as ${\mathfrak {g}}_{{{\bar{0}}}}^{\mathbb {C}}\oplus ({\mathfrak {g}}_{{{\bar{1}}}}\otimes {\mathbb {C}})$. The action of G on ${\mathcal {M}}$ extends to a holomorphic $G^{\mathbb {C}}$-action on ${\mathcal {M}}$, and the homomorphism ${\mathfrak {g}}\rightarrow \mathrm {Vec}({\mathcal {M}})$ extends to a homomorphism ${\mathfrak {g}}_{{{\bar{0}}}}^{\mathbb {C}}\oplus ({\mathfrak {g}}_{{{\bar{1}}}}\otimes {\mathbb {C}}) \rightarrow \mathrm {Vec}({\mathcal {M}})$ of complex Lie superalgebras, which is compatible with the $G^{\mathbb {C}}$-action on ${\mathcal {M}}$. Thus, we have a holomorphic ${\mathcal {G}}^{\mathbb {C}}$-action on ${\mathcal {M}}$ extending the ${\mathcal {G}}$-action. Moreover, there is a morphism $\sigma :{\mathcal {G}}^{\mathbb {C}}\rightarrow {{\mathrm{Aut}}}({\mathcal {M}})$ of Lie supergroups as in Theorem 22.

Example 25

Let ${\mathcal {M}}={\mathbb {C}}^{0|1}$. Denoting the odd coordinate on ${\mathbb {C}}^{0|1}$ by $\xi$, each super vector field on ${\mathbb {C}}^{0|1}$ is of the form $X=a \xi \frac{\partial }{\partial \xi }+ b\frac{\partial }{\partial \xi }$ for $a,b\in {\mathbb {C}}$. The flow $\varphi :{\mathbb {C}}\times {\mathcal {M}}\rightarrow {\mathcal {M}}$ of $a \xi \frac{\partial }{\partial \xi }$ is given by $(\varphi _t)^*( \xi )=e^{at}\xi$, and the flow $\psi :{\mathbb {C}}^{0|1}\times {\mathcal {M}}\rightarrow {\mathcal {M}}$ of $b\frac{\partial }{\partial \xi }$ by $\psi ^*(\xi )=b\tau +\xi$. Let $X_0=\xi \frac{\partial }{\partial \xi }$ and $X_1=\frac{\partial }{\partial \xi }$. Then $\mathrm {Vec}({\mathbb {C}}^{0|1})={\mathbb {C}} X_0\oplus {\mathbb {C}} X_1= {\mathbb {C}}^{1|1}$, where the Lie algebra structure on ${\mathbb {C}}^{1|1}$ is given by $[X_0,X_1]= -X_1$ and $[X_1,X_1]=0$. Note that this Lie superalgebra is isomorphic to the Lie superalgebra of right-invariant vector fields on the Lie supergroup $({\mathbb {C}}^{1|1}, \mu _{0,1})$, where the multiplication $\mu =\mu _{0,1}$ is given by $\mu ^*(t)=t_1+t_2$ and $\mu ^*(\tau )=\tau _1+e^{t_1} \tau _2$; for the Lie supergroup structures on ${\mathbb {C}}^{1|1}$ see e.g. [12], Lemma 3.1. In particular, the Lie superalgebra $\mathrm {Vec}({\mathbb {C}}^{0|1})$ is not abelian.

Since each automorphism $\varphi$ of ${\mathbb {C}}^{0|1}$ is given by $\varphi ^*(\xi )=c\cdot \xi$ for some $c\in {\mathbb {C}}$, $c\ne 0$, we have ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathbb {C}}^{0|1})\cong {\mathbb {C}}^*$.

8 The functor of points of the automorphism group

In [22], the diffeomorphism supergroup of a real compact supermanifold is proven to carry the structure of a Fréchet Lie supergroup. This diffeomorphism supergroup is defined using the “functor of points” approach to supermanifolds, i.e. a supermanifold is a representable contravariant functor from the category of supermanifolds to the category of sets. Starting with a supermanifold ${\mathcal {M}}$ we define the corresponding functor $\mathrm {Hom}(-,{\mathcal {M}})$ by the assignment ${\mathcal {N}}\mapsto \mathrm {Hom}({\mathcal {N}},{\mathcal {M}})$, where $\mathrm {Hom}({\mathcal {N}},{\mathcal {M}})$ denotes the set of morphisms of supermanifolds ${\mathcal {N}}\rightarrow {\mathcal {M}}$, and for morphisms $\alpha :{\mathcal {N}}_1\rightarrow {\mathcal {N}}_2$ between supermanifolds ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ we define $\mathrm {Hom}(-,{\mathcal {M}})(\alpha ): \mathrm {Hom}({\mathcal {N}}_2,{\mathcal {M}}) \rightarrow \mathrm {Hom}({\mathcal {N}}_1,{\mathcal {M}})$ by $\varphi \mapsto \varphi \circ \alpha$.

In analogy to the definition in [22] for the diffeomorphism supergroup, a functor ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})$ associated with a complex supermanifold ${\mathcal {M}}$ can be defined. In the case of a compact complex supermanifold ${\mathcal {M}}$, the automorphism Lie supergroup as defined in Section 7 represents the functor ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})$, i.e. the functors ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})$ and $\mathrm {Hom}(-,{{\mathrm{Aut}}}({\mathcal {M}}))$ are isomorphic. This is proven in [3], Section 5.4. Here we give an outline of the main steps in the proof.

Definition 3

Let ${\mathcal {M}}$ be a complex supermanifold. We define the functor ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})$ from the category of supermanifolds to the category of groups as follows:

On objects, we define ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})$ by the assignment

$$\begin{aligned} {\mathcal {N}}\mapsto \{\varphi :{\mathcal {N}}\times {\mathcal {M}}\rightarrow {\mathcal {N}}\times {\mathcal {M}}\,|\, \varphi \text { is invertible, and } \mathrm {pr}_{{\mathcal {N}}}\circ \varphi =\mathrm {pr}_{{\mathcal {N}}}\}, \end{aligned}$$

where $\mathrm {pr}_{{\mathcal {N}}}:{\mathcal {N}}\times {\mathcal {M}}\rightarrow {\mathcal {N}}$ is the projection. For morphisms $\alpha :{\mathcal {N}}_1\rightarrow {\mathcal {N}}_2$, we set ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})(\alpha ): {\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})({\mathcal {N}}_2)\rightarrow {\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})({\mathcal {N}}_1)$,

$$\begin{aligned} \varphi \mapsto (\mathrm {id}_{{\mathcal {N}}_1}\times (\mathrm {pr}_{\mathcal {M}}\circ \varphi \circ (\alpha \times \mathrm {id}_{\mathcal {M}}))) \circ (\mathrm {diag}\times \mathrm {id}_{\mathcal {M}}), \end{aligned}$$

denoting by $\mathrm {diag}:{\mathcal {N}}_1\rightarrow {\mathcal {N}}_1\times {\mathcal {N}}_1$ the diagonal map and by $\mathrm {pr}_{\mathcal {M}}$ the projection onto ${\mathcal {M}}$. Thus ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})(\alpha )(\varphi )$ is the unique automorphism $\psi :{\mathcal {N}}_1\times {\mathcal {M}}\rightarrow {\mathcal {N}}_1\times {\mathcal {M}}$ with $\mathrm {pr}_{{\mathcal {N}}_1}\circ \psi =\mathrm {pr}_{{\mathcal {N}}_1}$ and $\mathrm {pr}_{{\mathcal {M}}}\circ \psi =\mathrm {pr}_{\mathcal {M}}\circ \varphi \circ (\alpha \times \mathrm {id}_{\mathcal {M}})$.

The group structure on ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})({\mathcal {N}})$ is defined by the composition and inversion of automorphisms ${\mathcal {N}}\times {\mathcal {M}}\rightarrow {\mathcal {N}}\times {\mathcal {M}}$, and the neutral element is the identity map ${\mathcal {N}}\times {\mathcal {M}} \rightarrow {\mathcal {N}}\times {\mathcal {M}}$.

Let $\chi :{\mathcal {N}}\rightarrow {{\mathrm{Aut}}}({\mathcal {M}})$ be an arbitrary morphism of complex supermanifolds and let $\varPsi :{{\mathrm{Aut}}}({\mathcal {M}})\times {\mathcal {M}}\rightarrow {\mathcal {M}}$ denote the natural action of ${{\mathrm{Aut}}}({\mathcal {M}})$ on ${\mathcal {M}}$. Then the composition

$$\begin{aligned} \varphi _\chi =(\mathrm {id}_{{\mathcal {N}}}\times (\varPsi \circ (\chi \times \mathrm {id}_{\mathcal {M}})))\circ (\mathrm {diag}\times \mathrm {id}_{\mathcal {M}}) \end{aligned}$$

is an invertible map ${\mathcal {N}}\times {\mathcal {M}}\rightarrow {\mathcal {N}} \times {\mathcal {M}}$ with $\mathrm {pr}_{{\mathcal {N}}}=\mathrm {pr}_{{\mathcal {N}}}\circ \varphi _\chi$. This defines a natural transformation:

Lemma 26

The assignments $\mathrm {Hom}({\mathcal {N}},{{\mathrm{Aut}}}({\mathcal {M}})) \rightarrow {\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})({\mathcal {N}})$, $\chi \mapsto \varphi _\chi$, define a natural transformation $\mathrm {Hom}(-,{{\mathrm{Aut}}}({\mathcal {M}}))\rightarrow {\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})$.

This statement of the lemma can be verified by direct calculations; see also Lemma 5.4.2 in [3].

The natural transformation between $\mathrm {Hom}(-,{{\mathrm{Aut}}}({\mathcal {M}}))$ and ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})$ is actually an isomorphism of functors. The injectivity of the assignment $\chi \mapsto \varphi _\chi$ follows from the fact that the ${{\mathrm{Aut}}}({\mathcal {M}})$-action on ${\mathcal {M}}$ is effective. As a generalization of the classical definition of effectiveness, we call an action $\varPsi$ of a Lie supergroup ${\mathcal {G}}$ on a supermanifold ${\mathcal {M}}$ effective if for arbitrary morphisms $\chi _1,\chi _2:{\mathcal {N}}\rightarrow {\mathcal {G}}$ of supermanifolds the equality

$$\begin{aligned} \varPsi \circ (\chi _1\times \mathrm {id}_{\mathcal {M}})=\varPsi \circ (\chi _2\times \mathrm {id}_{\mathcal {M}}) \end{aligned}$$

implies $\chi _1=\chi _2$; cf. Section 2.5 in [3].

In the proof of the surjectivity a “normal form” of the pullback of automorphisms $\varphi :{\mathbb {C}}^{0|k}\times {\mathcal {M}}\rightarrow {\mathbb {C}}^{0|k}\times {\mathcal {M}}$ with $\mathrm {pr}_{{\mathbb {C}}^{0|k}}\circ \varphi =\mathrm {pr}_{{\mathbb {C}}^{0|k}}$ is used. Let ${\mathcal {M}}$ be a complex supermanifold and $\varphi :{\mathbb {C}}^{0|k}\times {\mathcal {M}}\rightarrow {\mathbb {C}}^{0|k}\times {\mathcal {M}}$ be an invertible morphism with $\mathrm {pr}_{{\mathbb {C}}^{0|k}}\circ \varphi =\mathrm {pr}_{{\mathbb {C}}^{0|k}}$. Let $\iota :{\mathcal {M}}\hookrightarrow \{0\}\times {\mathcal {M}} \subset {\mathbb {C}}^{0|k}\times {\mathcal {M}}$ denote the canonical inclusion. The composition ${{\bar{\varphi }}}=\mathrm {pr}_{\mathcal {M}}\circ \varphi \circ \iota$ is an automorphism of ${\mathcal {M}}$. Then $\varphi$ is uniquely determined by ${{\bar{\varphi }}}$ and a set of super vector fields on ${\mathcal {M}}$:

Lemma 27

Let $\varphi :{\mathbb {C}}^{0|k}\times {\mathcal {M}}\rightarrow {\mathbb {C}}^{0|k}\times {\mathcal {M}}$ be an invertible morphism with $\mathrm {pr}_{ {\mathbb {C}}^{0|k}}\circ \varphi =\mathrm {pr}_{{\mathbb {C}}^{0|k}}$. Let $\tau _1,\ldots , \tau _k$ denote coordinates on ${\mathbb {C}}^{0|k}\subset {\mathbb {C}}^{0|k}\times {\mathcal {M}}$. Then there are super vector fields $X_{\nu }$ on ${\mathcal {M}}$, of parity $|\nu |$ for $\nu \in ({\mathbb {Z}}_2)^k$, $\nu \ne 0$, such that

$$\begin{aligned} \varphi ^*=(\mathrm {id}_{{\mathbb {C}}^{0|k}}\times {{\bar{\varphi }}})^* \exp \left( \sum _{\nu \ne 0} \tau ^\nu X_{\nu }\right) , \end{aligned}$$

By $\tau ^\nu X_{\nu }$ we mean the super vector field on ${\mathbb {C}}^{0|k}\times {\mathcal {M}}$ which is induced by the extension of the super vector field $X_{\nu }$ on ${\mathcal {M}}$ to a super vector field on the product ${\mathbb {C}}^{0|k}\times {\mathcal {M}}$ followed by the multiplication with $\tau ^\nu =\tau _1^{\nu _1}\ldots \tau _k^{\nu _k}$. In other words for $U\subseteq M$ open we have $\tau ^\nu X_\nu (f)=0$ for $f\in {\mathcal {O}}_{{\mathbb {C}}^{0|k}}(\{0\}) \subset {\mathcal {O}}_{{\mathbb {C}}^{0|k}\times {\mathcal {M}}}(\{0\}\times U)$ and $(\tau ^\nu X_\nu )(g)=\tau ^\nu X_{\nu }(g)$ for $g\in {\mathcal {O}}_{{\mathcal {M}}}(U)\subset {\mathcal {O}}_{{\mathbb {C}}^{0|k}\times {\mathcal {M}}}(\{0\}\times U)$ considering $X_\nu (g)$ as a function on the product.

Moreover,

$$\begin{aligned} \exp \left( \sum _{\nu \ne 0} \tau ^\nu X_\nu \right) =\sum _{n \ge 0}\frac{1}{n!} \left( \sum _{\nu \ne 0} \tau ^\nu X_\nu \right) ^n \end{aligned}$$

is a finite sum since $\left( \sum _{\nu \ne 0} \tau ^\nu X_\nu \right) ^{k+1}=0$.

A version of this lemma is also proven in [22], Theorem 5.1. A different proof using the relation between nilpotent even super vector fields on a supermanifold and morphisms of this supermanifold satisfying a certain nilpotency condition as formulated in Sect. 2 is also possible; for details see also [3], Lemma 5.4.3.

Using the normal form of the lemma, we can prove that the assignment $\chi \mapsto \varphi _\chi$ defines a surjective map by directly constructing a morphism $\chi$ with $\varphi _\chi =\varphi$ for any $\varphi :{\mathcal {N}}\times {\mathcal {M}}\rightarrow {\mathcal {N}}\times {\mathcal {M}}$ with $\mathrm {pr}_{{\mathcal {N}}}\circ \varphi =\mathrm {pr}_{{\mathcal {N}}}$. It is here enough to prove this statement locally (in ${\mathcal {N}}$) and thus to consider the case where ${\mathcal {N}}=N\times {\mathbb {C}}^{0|k}$ for a classical complex manifold N. In the following we indicate how such a morphism $\chi$ can be defined; for the proof that $\chi$ fulfills the desired property $\varphi _\chi =\varphi$ see Proposition 5.4.4 in [3].

Let $\varphi :N\times {\mathbb {C}}^{0|k}\times {\mathcal {M}}\rightarrow N\times {\mathbb {C}}^{0|k}\times {\mathcal {M}}$ be an invertible morphism with $\mathrm {pr}_{N\times {\mathbb {C}}^{0|k}}\circ \varphi =\mathrm {pr}_{N\times {\mathbb {C}}^{0|k}}$. Each $z\in N$ induces an invertible morphism $\varphi _z:{\mathbb {C}}^{0|k}\times {\mathcal {M}}\rightarrow {\mathbb {C}}^{0|k}\times {\mathcal {M}}$ with $\mathrm {pr}_{{\mathbb {C}}^{0|k}}\circ \varphi _z=\mathrm {pr}_{{\mathbb {C}}^{0|k}}$, and the family $\varphi _z$, $z\in N$, uniquely determines $\varphi$.

Let $X_{\nu , z}$ be super vector fields on ${\mathcal {M}}$ of parity $|\nu |$, $\nu \in ({\mathbb {Z}}_2)^k$, $\nu \ne 0$, and ${{\bar{\varphi }}}_z:{\mathcal {M}}\rightarrow {\mathcal {M}}$ automorphisms such that $\varphi _z^*=(\mathrm {id}_{{\mathbb {C}}^{0|k}}\times {{\bar{\varphi }}}_z)^* \exp \left( \sum _{\nu \ne 0} \tau ^\nu X_{\nu ,z}\right)$ as in Lemma 27. Since $\varphi$ is holomorphic, the coefficients of the super vector fields $X_{\nu ,z}$ and the pullbacks ${{\bar{\varphi }}}_z^*$ in local coordiantes depend holomorphically on $z\in N$. Each ${{\bar{\varphi }}}_z$ is the automorphism of ${\mathcal {M}}$ induced by the evalutation in $(z,0)\in N\times {\mathbb {C}}^{0|k}$ and an element of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ by definition. Let $\mathrm {ev}_{{{\bar{\varphi }}}_z}$ denote the evaluation in ${{\bar{\varphi }}}_z$, i.e. $\mathrm {ev}_{{{\bar{\varphi }}}_z}$ is the pullback of the canonical inclusion $\{{{\bar{\varphi }}}_z\}\hookrightarrow {{\mathrm{Aut}}}({\mathcal {M}})$, and let $\mathrm {pr}_{{{\mathrm{Aut}}}({\mathcal {M}})}:N\times {\mathbb {C}}^{0|k}\times {{\mathrm{Aut}}}({\mathcal {M}}) \rightarrow {{\mathrm{Aut}}}({\mathcal {M}})$ be the projection. We define $\chi :N\times {\mathbb {C}}^{0|k}\rightarrow {{\mathrm{Aut}}}({\mathcal {M}})$ as the morphism whose underlying map is $\{z\}\hookrightarrow \{{{\bar{\varphi }}}_z\}\subset {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ and whose pullback evaluated in $z\in N$ is

$$\begin{aligned} \chi _z^*=(\mathrm {id}_{{\mathbb {C}}^{0|k}}^*\otimes \mathrm {ev}_{{{\bar{\varphi }}}_z})\circ \exp \left( \sum _{\nu \ne 0}\tau ^ \nu (X_{\nu ,z})_R\right) \circ \mathrm {pr}_{{{\mathrm{Aut}}}({\mathcal {M}})}^*, \end{aligned}$$

where $(X_{\nu ,z})_R$ denotes the right-invariant super vector field on ${{\mathrm{Aut}}}({\mathcal {M}})$ corresponding to the super vector field $X_{\nu ,z}$ on ${\mathcal {M}}$ which is an element of the Lie superalgebra $\mathrm {Vec}({\mathcal {M}})$ of ${{\mathrm{Aut}}}({\mathcal {M}})$.

The next proposition is then a consequence of Lemma 26 and the surjectivity of the assignment $\chi \mapsto \varphi _\chi$.

Proposition 28

(See [3], Corollary 5.4.5) The functors ${\overline{{{\mathrm{Aut}}}}}({\mathcal {M}})$ and $\mathrm {Hom}(-,{{\mathrm{Aut}}}({\mathcal {M}}))$ are isomorphic. This isomorphism is realized by the natural transformation introduced in Lemma 26.

9 The case of a superdomain with bounded underlying domain

In the classical case, the automorphism group of a bounded domain $U\subset {\mathbb {C}}^m$ is a (real) Lie group (see Theorem 13 in “Sur les groupes de transformations analytiques” in [8]). If ${\mathcal {U}}\subset {\mathbb {C}}^{m|n}$ is a superdomain whose underlying set U is a bounded domain in ${\mathbb {C}}^m$, it is in general not possible to endow its set of automorphisms with the structure of a Lie group such that the action on ${\mathcal {U}}$ is smooth, as will be illustrated in an example. In particular, there is no Lie supergroup satisfying the universal property as the automorphism group of a compact complex supermanifold ${\mathcal {M}}$ does as formulated in Theorem 22.

Example 29

Consider a superdomain ${\mathcal {U}}$ of dimension (1|2) whose underlying set is a bounded domain $U\subset {\mathbb {C}}$. Let $z,\theta _1,\theta _2$ denote coordinates for ${\mathcal {M}}$. For any holomorphic function f on U, define the even super vector field $X_f=f(z)\theta _1\theta _2\frac{\partial }{\partial z}$. The reduced vector field ${\tilde{X}}_f=0$ is completely integrable and thus the flow of $X_f$ can be defined on ${\mathbb {C}}\times {\mathcal {U}}$ (cf. [12] Lemma 5.2). The flow is given by $(\varphi _t)^*(z)=z+t\cdot f(z)\theta _1\theta _2$ and $(\varphi _t)^*(\theta _j)=\theta _j$. For all holomorphic functions f and g we have $[X_f,X_g]=0$, and thus their flows locally commute (cf. [2], Corollary 3.8). Therefore, $\{X_f|\,f\in {\mathcal {O}}(U)\}\cong {\mathcal {O}}(U)$ is an uncountably infinite-dimensional abelian Lie algebra. If the set of automorphisms of ${\mathcal {U}}$ carried the structure of a Lie group such that its action on ${\mathcal {U}}$ was smooth, its Lie algebra would necessarily contain $\{X_f|\,f\in {\mathcal {O}}(U)\}\cong {\mathcal {O}}(U)$ as a Lie subalgebra, which is not possible.

10 Examples

In this section, we determine the automorphism group ${{\mathrm{Aut}}}({\mathcal {M}})$ for some complex supermanifolds ${\mathcal {M}}$ with underlying manifold $M={\mathbb {P}}_1{\mathbb {C}}$.

Let $L_1$ denote the hyperplane bundle on $M={\mathbb {P}}_1{\mathbb {C}}$ with sheaf of sections ${\mathcal {O}}(1)$, and $L_k=(L_1)^{\otimes k}$ the line bundle of degree k, $k\in {\mathbb {Z}}$, on ${\mathbb {P}}_1{\mathbb {C}}$, and sheaf of sections ${\mathcal {O}}(k)$. Each holomorphic vector bundle on ${\mathbb {P}}_1{\mathbb {C}}$ is isomorphic to a direct sum of line bundles $L_{k_1}\oplus \ldots \oplus L_{k_n}$ (see [11]). Therefore, if ${\mathcal {M}}$ is a split supermanifold with $M={\mathbb {P}}_1{\mathbb {C}}$ and $\dim {\mathcal {M}}=(1|n)$, there exist $k_1,\ldots , k_n\in {\mathbb {Z}}$ such that the structure sheaf ${\mathcal {O}}_{{\mathcal {M}}}$ of ${\mathcal {M}}$ is isomorphic to

$$\begin{aligned} \bigwedge ({\mathcal {O}}(k_1)\oplus \ldots \oplus {\mathcal {O}}(k_n)). \end{aligned}$$

Let $U_j=\{[z_0:z_1]\in {\mathbb {P}}_1{\mathbb {C}}\,|\, z_j\ne 0\}$, $j=1,2$, and ${\mathcal {U}}_j=(U_j,{\mathcal {O}}_{{\mathcal {M}}}|_{U_j})$. Moreover, define ${U_0}^*=U_0\setminus \{[1:0]\}$ and ${U_1}^*=U_1\setminus \{[0:1]\}$, and let ${{\mathcal {U}}_j}^*=({U_j}^*,{\mathcal {O}}_{{\mathcal {M}}}|_{{U_j}^*})$. We can now choose local coordinates $z,\theta _1,\ldots , \theta _n$ for ${\mathcal {M}}$ on $U_0$, and local coordinates $w,\eta _1,\ldots ,\eta _n$ on $U_1$ so that the transition map $\chi :{{\mathcal {U}}_0}^*\rightarrow {{\mathcal {U}}_1}^*$, which determines the supermanifold structure of ${\mathcal {M}}$, is given by

$$\begin{aligned} \chi ^*(w)=\frac{1}{z} \ \ \text { and }\ \ \chi ^*(\eta _j)={z^{k_j}}\theta _j. \end{aligned}$$

Example 30

Let ${\mathcal {M}}=({\mathbb {P}}_1{\mathbb {C}}, {\mathcal {O}}_{\mathcal {M}})$ be a complex supermanifold of dimension (1|1). Since the odd dimension is 1, the supermanifold ${\mathcal {M}}$ has to be split. Let $-k\in {\mathbb {Z}}$ be the degree of the associated line bundle. Choose local coordinates $z,\theta$ for ${\mathcal {M}}$ on $U_0$ and $w,\eta$ on $U_1$ as above so that the transition map $\chi :{{\mathcal {U}}_0}^*\rightarrow {{\mathcal {U}}_1}^*$ is given by $\chi ^*(w)=\frac{1}{z}$ and $\chi ^*(\eta )=\frac{1}{z^k}\theta$.

We first want to determine the Lie superalgebra $\mathrm {Vec}({\mathcal {M}})$ of super vector fields on ${\mathcal {M}}$. A calculation in local coordinates verifying the compatibility condition with the transition map $\chi$ yields that the restriction to $U_0$ of any super vector field on ${\mathcal {M}}$ is of the form

$$\begin{aligned} \left( (\alpha _0+\alpha _1z+\alpha _2z^2)\frac{\partial }{\partial z} +(\beta +k\alpha _2 z)\theta \frac{\partial }{\partial \theta }\right) +\left( p(z)\frac{\partial }{\partial \theta } +q(z)\theta \frac{\partial }{\partial z}\right) , \end{aligned}$$

where $\alpha _0,\alpha _1,\alpha _2,\beta \in {\mathbb {C}}$, p is a polynomial of degree at most k, and q is a polynomial of degree at most $2-k$. If $k<0$ (respectively $2-k<0$), the polynomial p (respectively q) is 0. The Lie algebra $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ of even super vector fields is isomorphic to ${{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}})\oplus {\mathbb {C}}$, where an isomorphism ${{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}})\oplus {\mathbb {C}} \rightarrow \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ is given by

$$\begin{aligned} \left( \left( \begin{matrix} a&{}b\\ c&{}-a \end{matrix}\right) , d\right) \mapsto (-b-2az+cz^2)\frac{\partial }{\partial z} +((d-ka)+kcz)\theta \frac{\partial }{\partial \theta }. \end{aligned}$$

Note that since the odd dimension of ${\mathcal {M}}$ is 1 each automorphism $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$ gives rise to an automorphism of the line bundle $L_{-k}$ and vice versa. Hence, the automorphism group ${{\mathrm{Aut}}}(L_{-k})$ of the line bundle $L_{-k}$ and ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ coincide.

A calculation yields that the group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ of automorphisms ${\mathcal {M}}\rightarrow {\mathcal {M}}$ can be identified with $\mathrm {PSL}_2({\mathbb {C}})\times {\mathbb {C}}^*$ if k is even and with $\mathrm {SL}_2({\mathbb {C}})\times {\mathbb {C}}^*$ if k is odd. Consider the element $\left( \left( {\begin{matrix} a &{} b\\ c&{} d\end{matrix}}\right) ,s\right)$, where $s\in {\mathbb {C}}^*$ and $\left( {\begin{matrix} a &{} b\\ c&{} d\end{matrix}}\right)$ is either an element of $\mathrm {SL}_2({\mathbb {C}})$ or the representative of the corresponding class in $\mathrm {PSL}_2({\mathbb {C}})$. The action of the corresponding element $\varphi \in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on ${\mathcal {M}}$ is then given by

$$\begin{aligned} \varphi ^*(z)=\frac{c+dz}{a+bz}\ \ \text { and } \ \ \varphi ^*(\theta )=\left( \frac{1}{(a+bz)^k}+s\right) \theta \end{aligned}$$

as a morphism over appropriate subsets of $U_0$ and by

$$\begin{aligned} \varphi ^*(w)=\frac{aw+b}{cw+d}\ \ \text { and } \ \ \varphi ^*(\eta )=\left( \frac{1}{(cw+d)^k}+s\right) \eta \end{aligned}$$

over appropriate subsets of $U_1$.

The Lie supergroup structure on ${{\mathrm{Aut}}}({\mathcal {M}})$ is now uniquely determined by ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$, $\mathrm {Vec}({\mathcal {M}})$, and the adjoint action of ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ on $\mathrm {Vec}({\mathcal {M}})$. Since ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ is a connected Lie group, it is enough to calculate the adjoint action of $\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}}) \cong {{\mathfrak {s}}}{{\mathfrak {l}}}_2{{\mathbb {C}}} \oplus {\mathbb {C}}$ on $\mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}})$.

Let $P_l$ denote the space of polynomials of degree at most l, and set $P_l=\{0\}$ for $l<0$. The space of odd super vector fields $\mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}})$ is isomorphic to $P_{k}\oplus P_{2-k}$ via $\left( p(z)\frac{\partial }{\partial \theta } +q(z)\theta \frac{\partial }{\partial z}\right) \mapsto (p(z),q(z))$.

The element $H=\left( {\begin{matrix} 1&{}0\\ 0&{}-1 \end{matrix}}\right) \in {{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}})\subset {{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}})\oplus {\mathbb {C}} \cong \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ corresponds to $-2z\frac{\partial }{\partial z}-k\theta \frac{\partial }{\partial \theta }$. The adjoint action of this super vector field on the first factor $P_k$ of $\mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}})$ is given by by $-2z\frac{\partial }{\partial z}+k\cdot \mathrm {Id}$, and on the second factor $P_{2-k}$ by $-2z\frac{\partial }{\partial z}+ (2-k)\cdot \mathrm {Id}$. Calculating the weights of the ${{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}})$-representation on $P_k$ and $P_{2-k}$, we get that $P_k$ is the unique irreducible $(k+1)$-dimensional representation and $P_{2-k}$ the unique irreducible $(3-k)$-dimensional representation. Moreover, a calculation yields that $d\in {\mathbb {C}}$ corresponding to $d\cdot \theta \frac{\partial }{\partial \theta }\in \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$ acts on $P_k$ by multiplication with $-d$ and on $P_{2-k}$ by multiplication with d.

If $k<0$ or $k>2$, we have

$$\begin{aligned} \left[ \mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}}), \mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}})\right] =0. \end{aligned}$$

In the case $k=0$, we have $P_k\cong {\mathbb {C}}$. Since $[\frac{\partial }{\partial \theta },q(z)\theta \frac{\partial }{\partial z}] =q(z)\frac{\partial }{\partial z}$ for any $q\in P_2$, we get

$$\begin{aligned} \left[ \mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}}), \mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}})\right] =\left. \left\{ a(z)\frac{\partial }{\partial z}\,\right| \,a\in P_2\right\} \cong {{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}}), \end{aligned}$$

and the map $P_0\times P_2\rightarrow \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$, $(X,Y)\mapsto [X,Y]$, corresponds to ${\mathbb {C}}\times P_2\rightarrow \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$, $(p,q(z))\mapsto p\cdot q(z)\frac{\partial }{\partial z}$.

Similarly, if $k=2$, we have $P_{2-k}\cong {\mathbb {C}}$, and

$$\begin{aligned}{}[\mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}}),&\mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}})] =\left\{ \left. (\alpha _0+\alpha _1z+\alpha _2 z^2)\frac{\partial }{\partial z} +(\alpha _1+2\alpha _2 z)\theta \frac{\partial }{\partial \theta }\,\right| \, \alpha _j\in {\mathbb {C}}\right\} \\&\cong {{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}}) \end{aligned}$$

since $[p(z)\frac{\partial }{\partial \theta }, \theta \frac{\partial }{\partial z}] =p(z)\frac{\partial }{\partial z}+ p'(z)\theta \frac{\partial }{\partial \theta }$, and the map $P_2\times P_0\rightarrow \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$, $(X,Y)\mapsto [X,Y]$, corresponds to $P_2\times {\mathbb {C}}\rightarrow \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$, $(p(z),q)\mapsto q\cdot p(z)\frac{\partial }{\partial z}+q\cdot p'(z)\theta \frac{\partial }{\partial \theta }$.

If $k=1$, then $P_k\oplus P_{2-k}\cong {\mathbb {C}}^2\oplus {\mathbb {C}}^2$. We have

$$\begin{aligned} \left[ \frac{\partial }{\partial \theta },\theta \frac{\partial }{\partial z}\right]= & {} \frac{\partial }{\partial z}, \left[ z\frac{\partial }{\partial \theta },\theta \frac{\partial }{\partial z}\right] =z\frac{\partial }{\partial z}+\theta \frac{\partial }{\partial \theta },\\ \left[ \frac{\partial }{\partial \theta },z\theta \frac{\partial }{\partial z}\right]= & {} z\frac{\partial }{\partial z}, \left[ z\frac{\partial }{\partial \theta },z\theta \frac{\partial }{\partial z}\right] =z^2\frac{\partial }{\partial z}+z\theta \frac{\partial }{\partial \theta }, \end{aligned}$$

and consequently $[\mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}}),\mathrm {Vec}_{{{\bar{1}}}}({\mathcal {M}})] =\mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$.

Remark that ${{\mathrm{Aut}}}({\mathcal {M}})$ carries the structure of a split Lie supergroup if and only if $k<0$ or $k>2$ (cf. Proposition 4 in [24]).

Example 31

Let ${\mathcal {M}}=({\mathbb {P}}_1{\mathbb {C}},{\mathcal {O}}_{{\mathcal {M}}})$ be a split complex supermanifold of dimension $\dim {\mathcal {M}}=(1|2)$ associated with ${\mathcal {O}}(-k_1)\oplus {\mathcal {O}}(-k_2)$, $k_1,k_2\in {\mathbb {Z}}$. We will determine the group ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ of automorphisms ${\mathcal {M}}\rightarrow {\mathcal {M}}$.

We choose coordinates $z,\theta _1,\theta _2$ for ${\mathcal {U}}_0$ and $w,\eta _1,\eta _2$ for ${\mathcal {U}}_1$ as described above such that the transition map $\chi$ is given by $\chi ^*(w)=z^{-1}$ and $\chi ^*(\eta _j)={z^{-k_j}}\theta _j$.

The action of $\mathrm {PSL}_2({\mathbb {C}})$ on ${\mathbb {P}}_1{\mathbb {C}}$ by Möbius transformations lifts to an action of $\mathrm {SL}_2({\mathbb {C}})$ on ${\mathcal {M}}$ by letting $A=\left( {\begin{matrix} a&{}b\\ c&{}d \end{matrix}}\right) \in \mathrm {SL}_2({\mathbb {C}})$ act by the automorphism $\varphi _A:{\mathcal {M}}\rightarrow {\mathcal {M}}$ with pullback

$$\begin{aligned} \varphi _A^*(z)=\frac{c+dz}{a+bz}\ \ \text { and }\ \ \varphi _A^*(\theta _j)=(a+bz)^{-k_j}\theta _j \end{aligned}$$

as a morphism over appropriate subsets of $U_0$, and

$$\begin{aligned} \varphi _A^*(w)=\frac{aw+b}{cw+d}\ \ \text { and }\ \ \varphi _A^*(\eta _j)=(cw+d)^{-k_j}\eta _j \end{aligned}$$

over appropriate subsets of $U_1$. Using the transition map $\chi$ one might also calculate the representation of $\varphi$ in coordinates as a morphism over subsets $U_0\rightarrow U_1$ and $U_1\rightarrow U_0$.

If $k_1$ and $k_2$ are both even, we have $\varphi _A=\mathrm {Id}_{\mathcal {M}}$ for $A=\left( {\begin{matrix} -1&{}0\\ 0&{}-1 \end{matrix}}\right)$ and thus we get an action of $\mathrm {PSL}_2({\mathbb {C}})$ on ${\mathcal {M}}$.

Consider the homomorphism of Lie groups $\varPsi :{{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}}) \rightarrow {{\mathrm{Aut}}}({\mathbb {P}}_1{\mathbb {C}})$ assigning to each automorphism $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$ the underlying biholomorphic map ${{\tilde{\varphi }}}:{\mathbb {P}}_1{\mathbb {C}}\rightarrow {\mathbb {P}}_1{\mathbb {C}}$. This homomorphism $\varPsi$ is surjective since ${{\mathrm{Aut}}}({\mathbb {P}}_1{\mathbb {C}}) \cong \mathrm {PSL}_2({\mathbb {C}})$ and since the $\mathrm {PSL}_2({\mathbb {C}})$-action on ${\mathbb {P}}_1{\mathbb {C}}$ lifts to an action (of $\mathrm {SL}_2({\mathbb {C}})$) on the supermanifold ${\mathcal {M}}$. The kernel $\ker \varPsi$ of the homomorphism $\varPsi$ consists of those automorphisms $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$ whose underlying map ${{\tilde{\varphi }}}$ is the identity ${\mathbb {P}}_1{\mathbb {C}}\rightarrow {\mathbb {P}}_1{\mathbb {C}}$. This kernel $\ker \varPsi$ is a normal subgroup, $\mathrm {SL}_2({\mathbb {C}})$ acts on $\ker \varPsi$, and we have

$$\begin{aligned} {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\cong \ker \varPsi \rtimes \mathrm {SL}_2({\mathbb {C}}) \end{aligned}$$

if $k_1$ and $k_2$ are not both even, and ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\cong \ker \varPsi \rtimes \mathrm {PSL}_2({\mathbb {C}})$ if $k_1$ and $k_2$ are even. Thus, it remains to determine $\ker \varPsi$.

Let $\varphi :{\mathcal {M}}\rightarrow {\mathcal {M}}$ be an automorphism with ${{\tilde{\varphi }}}=\mathrm {Id}$. Let f and $b_{jk}$, $j,k=1,2$, be holomorphic functions on $U_0\cong {\mathbb {C}}$ such that the pullback of $\varphi$ over $U_0$ is given by

$$\begin{aligned} \varphi ^*(z)=z+f(z)\theta _1\theta _2\ \ \text { and }\ \ \varphi ^*(\theta )=B(z)\theta , \end{aligned}$$

where $B(z)=\left( {\begin{matrix} b_{11}(z)&{}b_{12}(z)\\ b_{21}(z)&{}b_{22}(z) \end{matrix}}\right)$ and $\varphi ^*(\theta )=B(z)\theta$ is an abbreviation for

$$\begin{aligned} \varphi ^*(\theta _j)=b_{j1}(z)\theta _1+b_{j2}(z)\theta _2\ \text {for}\ j=1,2. \end{aligned}$$

Similarly, let g and $c_{jk}$ be holomorphic functions on $U_1\cong {\mathbb {C}}$ such that the pullback of $\varphi$ over $U_1$ is given by

$$\begin{aligned} \varphi ^*(w)=w+g(w)\eta _1\eta _2\ \ \text { and }\ \ \varphi ^*(\eta )=C(z)\eta , \end{aligned}$$

where $C(z)=\left( {\begin{matrix} c_{11}(z)&{}c_{12}(z)\\ c_{21}(z)&{}c_{22}(z) \end{matrix}}\right)$. The compatibility condition with the transition map $\chi$ gives now the relation

$$\begin{aligned} f(z)=-z^{2-(k_1+k_2)}g\left( \frac{1}{z}\right) \ \text { for } z\in {\mathbb {C}}^*. \end{aligned}$$

Therefore, f and g are both polynomials of degree at most $2-(k_1+k_2)$, and they are 0 in the case $k_1+k_2>2$. For the matrices B and C we get the relation

$$\begin{aligned} B(z)=\begin{pmatrix} z^{k_1}&{}0\\ 0&{}z^{k_2} \end{pmatrix}C\left( \frac{1}{z}\right) \begin{pmatrix} z^{-k_1}&{}0\\ 0&{}z^{-k_2} \end{pmatrix}\ \text { for } z\in {\mathbb {C}}^*. \end{aligned}$$

If $k_1=k_2$, this implies $B(z)=C\left( \frac{1}{z}\right)$ for all $z\in {\mathbb {C}}^*$. Thus, $B(z)=B$ and $C(w)=C$ are constant matrices, and $B=C\in \mathrm {GL}_2({\mathbb {C}})$ since $\varphi$ was assumed to be invertible. Consequently, we have

$$\begin{aligned} \ker \varPsi \cong P_{2-(k_1+k_2)}\rtimes \mathrm {GL}_2({\mathbb {C}}) \end{aligned}$$

in the case $k_1=k_2$, where $P_{2-(k_1+k_2)}$ denotes the space of polynomials of degree at most $2-(k_1+k_2)$ if $k_1+k_2< 2$ and $P_{2-(k_1+k_2)}=\{0\}$ otherwise. The group structure on the semidirect product is given by $(f_1(z),B_1)\cdot (f_2(z),B_2)=(\det B_1 f_1(z)+f_2(z), B_1B_2)$.

Let now $k_1\ne k_2$. After possibly changing coordinates we may assume $k_1>k_2$. Then we have

$$\begin{aligned} B(z) =\begin{pmatrix} z^{k_1}&{}0\\ 0&{}z^{k_2} \end{pmatrix}C\left( \frac{1}{z}\right) \begin{pmatrix} z^{-k_1}&{}0\\ 0&{}z^{-k_2} \end{pmatrix} =\begin{pmatrix} c_{11}\left( \frac{1}{z}\right) &{} z^{k_1-k_2}c_{12}\left( \frac{1}{z}\right) \\ z^{k_2-k_1}c_{21}\left( \frac{1}{z}\right) &{} c_{22}\left( \frac{1}{z}\right) \end{pmatrix} \end{aligned}$$

for all $z\in {\mathbb {C}}^*$. This implies that $b_{11}=c_{11}$ and $b_{22}=c_{22}$ are constants. Since we assume $k_1>k_2$, we also get $b_{21}=c_{21}=0$ and $b_{12}$ and $c_{12}$ are polynomials of degree at most $k_1-k_2$. Therefore,

$$\begin{aligned} \ker \varPsi \cong P_{2-(k_1+k_2)}\rtimes \left\{ \left. \begin{pmatrix} \lambda &{} p(z)\\ 0&{}\mu \end{pmatrix} \,\right| \, \lambda ,\mu \in {\mathbb {C}}^*,\, p\in P_{k_1-k_2}\right\} , \end{aligned}$$

and the group structure is again given by

$$\begin{aligned} (f_1(z),B_1)\cdot (f_2(z),B_2)=(\det B_1 f_1(z)+f_2(z), B_1B_2) \end{aligned}$$

for $f_1,f_2\in P_{2-(k_1+k_2)}$, $B_1,B_2\in \left\{ \left. \left( {\begin{matrix} \lambda &{} p(z)\\ 0&{}\mu \end{matrix}}\right) \,\right| \, \lambda ,\mu \in {\mathbb {C}}^*,\, p\in P_{k_1-k_2}\right\}$.

The semidirect product $\ker \varPsi \rtimes \mathrm {SL}_2({\mathbb {C}})$ (or $\ker \varPsi \rtimes \mathrm {PSL}_2({\mathbb {C}})$) is a direct product if and only if $k_1=k_2$ and $k_1+k_2\ge 2$.

Example 32

Let ${\mathcal {M}}=({\mathbb {P}}_1{\mathbb {C}}, {\mathcal {O}}_{\mathcal {M}})$ be the complex supermanifold of dimension $\dim {\mathcal {M}}=(1|2)$ given by the transition map $\chi :{{\mathcal {U}}_0}^*\rightarrow {{\mathcal {U}}_1}^*$ with pullback

$$\begin{aligned} \chi ^*(w)=\frac{1}{z} +\frac{1}{z^3}\theta _1\theta _2\ \ \text { and }\ \ \chi ^*(\eta _j)=\frac{1}{z^2} \theta _j. \end{aligned}$$

The supermanifold ${\mathcal {M}}$ is not split and the associated split supermanifold corresponds to ${\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2)$; see e.g. [7].

As in the previous example, the action of $\mathrm {PSL}_2({\mathbb {C}})$ on ${\mathbb {P}}_1{\mathbb {C}}$ by Möbius transformations lifts to an action of $\mathrm {PSL}_2({\mathbb {C}})$ on ${\mathcal {M}}$. Let A denote the class of $\left( {\begin{matrix} a&{}b\\ c&{}d \end{matrix}}\right) \in \mathrm {SL}_2({\mathbb {C}})$ in $\mathrm {PSL}_2({\mathbb {C}})$. Then A acts by the morphism $\varphi _A:{\mathcal {M}}\rightarrow {\mathcal {M}}$ whose pullback as a morphism over appropriate subsets of $U_0$ is given by

$$\begin{aligned} \varphi _A^*(z)=\frac{c+dz}{a+bz}-\frac{b}{(a+bz)^3}\theta _1\theta _2 \ \ \text { and }\ \ \varphi _A^*(\theta _j)=\frac{1}{(a+bz)^2}\theta _j. \end{aligned}$$

Let $\varPsi :{{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\rightarrow {{\mathrm{Aut}}}({\mathbb {P}}_1{\mathbb {C}}) \cong \mathrm {PSL}_2({\mathbb {C}})$ denote again the Lie group homomorphism which assigns to an automorphism of ${\mathcal {M}}$ the underlying automorphism of ${\mathbb {P}}_1{\mathbb {C}}$. The assignment $A\mapsto \varphi _A\in {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ defines a section $\mathrm {PSL}_2({\mathbb {C}})\rightarrow {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ of $\varPsi$, and we have

$$\begin{aligned} {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\cong \ker \varPsi \rtimes \mathrm {PSL}_2({\mathbb {C}}). \end{aligned}$$

The section $\mathrm {PSL}_2({\mathbb {C}})\rightarrow {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})$ induces on the level of Lie algebras the morphism $\sigma :{{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}})\hookrightarrow \mathrm {Vec}_{{{\bar{0}}}}({\mathcal {M}})$, which maps an element $\left( {\begin{matrix} a&{}b\\ c&{}-a \end{matrix}}\right) \in {{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}})$ to the super vector field on ${\mathcal {M}}$ whose restriction to ${\mathcal {U}}_0$ is

$$\begin{aligned} \left( c-2az-bz^2-b\theta _1\theta _2\right) \frac{\partial }{\partial z} -2(a+bz)\left( \theta _1\frac{\partial }{\partial \theta _1} +\theta _2\frac{\partial }{\partial \theta _2}\right) . \end{aligned}$$

We now calculate the kernel $\ker \varPsi$. Let $\varphi \in \ker \varPsi$. Its underlying map ${{\tilde{\varphi }}}$ is the identity and we thus have

$$\begin{aligned} \varphi ^*(z)=z+a_0(z)\theta _1\theta _2\ \ \text { and }\ \ \varphi ^*(\theta )=A_0(z)\theta \end{aligned}$$

on $U_0$ and

$$\begin{aligned} \varphi ^*(w)=w+a_1(w)\eta _1\eta _2\ \ \text { and }\ \ \varphi ^*(\eta )=A_1(w)\eta \end{aligned}$$

on $U_1$ for holomorphic functions $a_0$ and $a_1$ and invertible matrices $A_0$ and $A_1$ whose entries are holomorphic functions. The notation $\varphi ^*(\theta )=A_0(z)\theta$ (and similarly $\varphi ^*(\eta )=A_1(w)\eta$) is again an abbreviation for $\varphi ^*(\theta _j)=(A_0(z))_{j1}\theta _1+(A_0(z))_{j2}\theta _2$, where $A_0(z)=\left( (A_0(z))_{jk}\right) _{1\le j,k\le 2}$. A calculation with the transition map $\chi$ then yields the relations

$$\begin{aligned} A_1(w)=A_0\left( \frac{1}{w}\right) \ \ \text { and }\ \ a_1(w)=\frac{1}{w} \left( \left( \det A_0\left( \frac{1}{w} \right) -1\right) -\frac{1}{w} a_0\left( \frac{1}{w} \right) \right) \end{aligned}$$

for any $w\in {\mathbb {C}}^*$. Since $a_0$, $a_1$, $A_0$, and $A_1$ are holomorphic on ${\mathbb {C}}$, we get that $A_0=A_1$ are constant matrices, $\det A_0=1$, and $a_0=a_1=0$. Therefore, $\ker \varPsi \cong \mathrm {SL}_2({\mathbb {C}})$, and its Lie algebra is

$$\begin{aligned} \left\{ \left. \left( a_{11}\theta _1+a_{12}\theta _2\right) \frac{\partial }{\partial \theta _1} +\left( a_{21}\theta _1+a_{22}\theta _2\right) \frac{\partial }{\partial \theta _2}\,\right| \, \begin{pmatrix} a_{11}&{}a_{12}\\ a_{21}&{}a_{22} \end{pmatrix}\in {{\mathfrak {s}}}{{\mathfrak {l}}}_2({\mathbb {C}})\right\} . \end{aligned}$$

Since $\mathrm {Lie}(\ker \varPsi )$ and $\sigma (\mathrm {Lie}(\mathrm {PSL}_2({\mathbb {C}}))$ commute, the semidirect product $\ker \varPsi \rtimes \mathrm {PSL}_2({\mathbb {C}})$ is direct and we have

$$\begin{aligned} {{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\cong \mathrm {SL}_2({\mathbb {C}})\times \mathrm {PSL}_2({\mathbb {C}}). \end{aligned}$$

Remark in particular that this group is different from the automorphism group of the corresponding split supermanifold ${\mathcal {N}}$, which is associated with ${\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2)$, with ${{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {N}}) \cong \mathrm {GL}_2({\mathbb {C}})\times \mathrm {PSL}_2({\mathbb {C}})$.