Maximal compatible splitting and diagonals of Kempf varieties

TL;DR

This paper constructs a Frobenius splitting with maximal multiplicity on the diagonal of flag varieties, leading to new results on Schubert varieties and confirming conjectures in positive characteristic.

Contribution

It introduces an algebraic analogue of maximal multiplicity vanishing and constructs a Frobenius splitting with maximal multiplicity on flag variety diagonals, impacting several conjectures.

Findings

Frobenius splitting of tangent bundles

Frobenius splitting of blow-ups along diagonals

Verification of LMP and Wahl conjectures in positive characteristic

Abstract

Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles, of blow-ups along the diagonal in flag varieties along with the LMP and Wahl conjectures in positive characteristic for the special linear group.