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.