Automorphism group functors of algebraic superschemes

Abstract

The famous theorem of Matsumura-Oort states that if $X$ is a proper scheme, then the automorphism group functor $\mathfrak{Aut}(X)$ of $X$ is a locally algebraic group scheme. In this paper we generalize this theorem to the category of superschemes, that is if $\mathbb{X}$ is a proper superscheme, then the automorphism group functor $\mathfrak{Aut}(\mathbb{X})$ of $\mathbb{X}$ is a locally algebraic group superscheme. Moreover, we also show that if $H^1(X, \mathcal{T}_X)=0$, where $X$ is the geometric counterpart of $\mathbb{X}$ and $\mathcal{T}_X$ is the tangent sheaf of $X$, then $\mathfrak{Aut}(\mathbb{X})$ is a smooth group superscheme.