On the fundamental group scheme of rationally chain connected varieties

TL;DR

This paper proves that the fundamental group scheme of a normal rationally chain connected variety over an algebraically closed field is finite and étale, extending known results about the étale fundamental group to the fundamental group scheme.

Contribution

It establishes the finiteness and étaleness of the fundamental group scheme for such varieties, including Fano varieties, which was previously known only for the étale fundamental group.

Findings

Fundamental group scheme of these varieties is finite.

Fundamental group scheme is étale.

Includes Fano varieties as special cases.

Abstract

Let $k$ be an algebraically closed field. Chambert-Loir proved that the \'etale fundamental group of a normal rationally chain connected variety over $k$ is finite. We prove that the fundamental group scheme of a normal rationally chain connected variety over $k$ is finite and \'etale. In particular, the fundamental group scheme of a Fano variety is finite and \'etale.