A Frobenius splitting and cohomology vanishing for the cotangent bundles of the flag varieties of GL_n

TL;DR

This paper demonstrates a Frobenius splitting of the cotangent bundle of flag varieties in GL_n over fields of positive characteristic, leading to cohomology vanishing results and a generalization of known splittings.

Contribution

It provides an explicit Frobenius splitting formula for cotangent bundles of flag varieties, extending previous results and establishing cohomology vanishing for associated line bundles.

Findings

Frobenius splitting of cotangent bundles with top degree (p-1)dim(G/P)

Explicit formula for the splitting function f

Vanishing of higher cohomology groups for line bundles associated to dominant weights

Abstract

Let k be an algebraically closed field of characteristic p>0, let G=GL_n be the general linear group over k, let P be a parabolic subgroup of G, and let u_P be the Lie algebra of its unipotent radical. We show that the Kumar-Lauritzen-Thomsen splitting of the cotangent bundle Gx^Pu_P of G/P has top degree (p-1)\dim(G/P). The component of that degree is therefore given by the (p-1)-th power of a function f. We give a formula for f and deduce that it vanishes on the exceptional locus of the resolution Gx^Pu_P-->\ov{\mc O} where \ov{\mc O} is the closure of the Richardson orbit of P. As a consequence we obtain that the higher cohomology groups of a line bundle on Gx^Pu_P associated to a dominant weight are zero. The splitting of Gx^Pu_P given by f^{p-1} can be seen as a generalisation of the Mehta-Van der Kallen splitting of Gx^Bu.