Failure of Bott vanishing for (co)adjoint partial flag varieties

TL;DR

This paper proves that (co)adjoint partial flag varieties of all classical and exceptional types do not satisfy Bott vanishing, confirming a longstanding conjecture and highlighting limitations of Bott vanishing in complex geometric structures.

Contribution

It establishes the failure of Bott vanishing for (co)adjoint partial flag varieties across all classical and exceptional Dynkin types, extending previous partial results.

Findings

(co)adjoint partial flag varieties do not satisfy Bott vanishing

Confirmation of the Buch-Thomsen-Lauritzen-Mehta conjecture

Failure applies to all classical and exceptional types

Abstract

Bott vanishing is a strong vanishing result for the cohomology of exterior powers of the cotangent bundle twisted by ample line bundles. Buch-Thomsen-Lauritzen-Mehta conjectured that partial flag varieties (which are not products of projective spaces) do not satisfy Bott vanishing, despite all their other nice properties. The cominuscule case is an easy application of the Borel-Weil-Bott theorem, following results of Snow. We show that the (co)adjoint partial flag varieties of all classical and exceptional Dynkin types also do not satisfy Bott vanishing, thus confirming the conjecture for this class of varieties.