Automorphism group functors of algebraic superschemes

TL;DR

This paper extends Matsumura-Oort's theorem to superschemes, proving that automorphism group functors of proper superschemes are locally algebraic group superschemes and are smooth under certain cohomological conditions.

Contribution

It generalizes the classical automorphism group functor result from schemes to superschemes, including smoothness criteria based on cohomology.

Findings

Automorphism group functor of proper superschemes is a locally algebraic group superscheme.

If $H^1(X, \mathcal{T}_X)=0$, then the automorphism group functor is smooth.

Extension of classical automorphism group results to the superscheme setting.

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.