Some questions arising from the study of cohomology on flag varieties

TL;DR

This paper discusses open questions related to the cohomology of line bundles on flag varieties in positive characteristic, highlighting differences from the well-understood characteristic zero case and emphasizing combinatorial and algebraic aspects.

Contribution

It identifies and explores specific open problems in the study of cohomology on flag varieties over fields of positive characteristic, connecting algebraic geometry and representation theory.

Findings

Highlights open questions in positive characteristic cohomology

Connects combinatorial and algebraic aspects of the problem

Contrasts with the classical Borel-Weil-Bott theorem in characteristic zero

Abstract

A fundamental problem at the confluence of algebraic geometry and representation theory is to describe the cohomology of line bundles on flag varieties over a field of characteristic p. When p=0, the solution is given by the celebrated Borel-Weil-Bott Theorem, while for p>0 the problem is widely open. In this note we describe a collection of open questions that arise from the study of particular cases of the general theory, focusing on their combinatorial and commutative algebra aspects.