Chow-Witt rings and topology of flag varieties

TL;DR

This paper computes the Witt-sheaf cohomology rings of partial flag varieties, revealing that all torsion in their integral cohomology is 2-torsion and deriving implications for topology and enumerative geometry.

Contribution

It provides explicit computations of Witt-sheaf cohomology rings for flag varieties and establishes new results on torsion in their integral cohomology.

Findings

All torsion in integral cohomology is 2-torsion.

Computed Poincaré polynomials for flag varieties with twisted coefficients.

Described Chow-Witt rings and applications to enumerative geometry.

Abstract

The paper computes the Witt-sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray-Hirsch-type theorem for Witt-sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2-torsion, which was not known in full generality previously. This allows for example to compute the Poincar\'e polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow-Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces.