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 \(H^1(X, {\mathchoice{\text{ T }}{\text{ T }}{\text{ T }}{\text{ T }}}_X)=0\), where X is the geometric counterpart of \({\mathbb {X}}\) and \({\mathchoice{\text{ T }}{\text{ T }}{\text{ T }}{\text{ T }}}_X\) is the tangent sheaf of X, then \(\mathfrak {Aut}({\mathbb {X}})\) is a smooth group superscheme.