Trivial Witt groups of flag varieties

TL;DR

This paper establishes a straightforward criterion based on Dynkin diagrams for when the Witt groups of line bundles over flag varieties are trivial, showing they vanish except in trivial cases.

Contribution

It provides a simple condition on line bundle classes in Picard groups mod 2 that determines the vanishing of Witt groups on flag varieties.

Findings

Witt groups W^i(G/P,L) are zero under the given condition

Witt groups W^i(G/B,L) vanish unless L is trivial in Pic(G/B)/2

The criterion simplifies understanding of Witt groups in algebraic geometry

Abstract

Let G be a split semi-simple linear algebraic group over a field, let P be a parabolic subgroup and let L be a line bundle on the projective homogeneous variety G/P. We give a simple condition on the class of L in Pic(G/P)/2 in terms of Dynkin diagrams implying that the Witt groups W^i(G/P,L) are zero for all integers i. In particular, if B is a Borel subgroup, then W^i(G/B,L) is zero unless L is trivial in Pic(G/B)/2.