An algebraic variant of the Fischer-Grauert Theorem

TL;DR

This paper extends the Fischer-Grauert theorem to algebraic geometry, showing that certain smooth projective families over algebraically closed fields are locally trivial in the étale topology, with applications to families with automorphism group schemes.

Contribution

It introduces an algebraic analogue of the Fischer-Grauert theorem for families over algebraically closed fields of infinite transcendence degree, using Kodaira-Spencer map techniques.

Findings

Families with isomorphic fibers and reduced automorphism groups are locally trivial in the étale topology.

Formal triviality is established at every geometric point via vanishing Kodaira-Spencer map.

Examples of fiberwise trivial families with non-vanishing Kodaira-Spencer map are provided in positive characteristic.

Abstract

A well-known theorem of W. Fischer and H. Grauert states that analytic fiber spaces with all fibers isomorphic to a fixed compact connected complex manifold are locally trivial. Motivated by this result, we show that if $k$ is an algebraically closed field of infinite transcendence degree over its prime field, then every smooth projective family over a reduced $k$-scheme of finite type with isomorphic fibers having reduced automorphism group schemes is locally trivial in the \'etale topology. We do so by reducing the problem to the case when the base is a smooth integral curve, and then, using the vanishing of the Kodaira-Spencer map, we prove formal triviality of such families at every geometric point of the base. We also provide examples of smooth projective fibrewise trivial families in positive characteristic whose Kodaira-Spencer map are nowhere vanishing.