On the S-fundamental group scheme II

TL;DR

This paper proves that the S-fundamental group scheme of a product of complete varieties is the product of their individual S-fundamental groups, and computes the abelian part for specific varieties, advancing understanding of these group schemes.

Contribution

It confirms a conjecture about the product structure of S-fundamental groups and provides explicit computations for abelian varieties and varieties with trivial étale fundamental groups.

Findings

The S-fundamental group of a product equals the product of individual groups.

Explicit description of the abelian part of the S-fundamental group scheme.

Determination of the S-fundamental group scheme for abelian varieties.

Abstract

The S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a product of their S-fundamental groups as conjectured by V. Mehta and the author. We also compute the abelian part of the S-fundamental group scheme and the S-fundamental group scheme of an abelian variety or a variety with trivial etale fundamental group.