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.