Gieseker conjecture for homogeneous spaces

TL;DR

This paper proves Gieseker's conjecture for homogeneous spaces, establishing that the absence of non-trivial tame coverings implies no non-trivial regular singular D-modules, using K"unneth formulas and base change techniques.

Contribution

It extends Gieseker's conjecture to homogeneous spaces by proving new K"unneth formulas and base change results for stratified fundamental groups.

Findings

Proved Gieseker conjecture for homogeneous spaces.

Established a K"unneth formula for regular singular stratified fundamental groups.

Developed a base change theorem for Gauss-Manin stratifications in non-proper cases.

Abstract

We prove Gieseker conjecture for an homogeneous space $X$, saying that if $X$ has no non-trivial tame coverings then it has no non-trivial regular singular $\mathscr{O}_X$-coherent $\mathscr{D}_{X/k}$-modules. In order to do so we prove a K\"unneth formula for the regular singular stratified fundamental group and a base change for Gauss-Manin stratifications in the non-proper case.