A proof of Wahl's conjecture in the symplectic case

TL;DR

This paper proves Wahl's conjecture for symplectic (type C) flag varieties by constructing a canonical Frobenius splitting, confirming a conjecture and providing new proofs and cohomological insights.

Contribution

It constructs a canonical Frobenius splitting of $X imes X$ for type C flag varieties, verifying Wahl's conjecture and offering new proofs for type A.

Findings

Verification of Wahl's conjecture in type C

New proof of Wahl's conjecture in type A

Cohomological consequences derived from the splitting

Abstract

Let $X$ denote a flag variety of type $A$ or type $C$. We construct a canonical Frobenius splitting of $X \times X$ which vanishes with maximal multiplicty along the diagonal. This way we verify a conjecture by Lakshmibai, Mehta and Parameswaran in type $C$, and obtain a new proof in type $A$. In particular, we obtain a proof of Wahl's conjecture in type $C$, and a new proof in type $A$. We also present certain cohomological consequences.