Generic vanishing on homogeneous spaces in arbitrary characteristic

TL;DR

This paper extends a characteristic-zero vanishing theorem to arbitrary characteristic for proper homogeneous spaces, providing new inequalities, trace-function identities, and Lang-Weil estimates over finite fields.

Contribution

It generalizes the vanishing theorem to arbitrary characteristic and introduces trace-function identities and Lang-Weil estimates over finite fields.

Findings

Established a non-negativity inequality for intersection Euler characteristics in arbitrary characteristic.

Derived trace-function identities on dense open subsets of algebraic groups over finite fields.

Provided Lang-Weil estimates for the non-generic locus in the context of homogeneous spaces.

Abstract

Let $X$ be a proper homogeneous space for a connected algebraic group $G$ over an algebraically closed field. For locally closed smooth affine subvarieties $W,Z\subset X$, we show that \[ (-1)^{\dim X-\dim W+\dim Z}\chi(gW\cap Z)\geq 0 \] for generic $g\in G$. This extends the characteristic-zero theorem of Sch\"urmann--Simpson--Wang. Over finite fields, our methods give a trace-function identity on a dense open subset of $G$ and a Lang--Weil estimate for the non-generic locus.