Splitting criteria for vector bundles on the symplectic isotropic Grassmannian

TL;DR

This paper generalizes Ottaviani's cohomological splitting criterion for vector bundles from classical Grassmannians to symplectic isotropic Grassmannians, providing necessary and sufficient conditions in specific cases.

Contribution

It extends splitting criteria to symplectic isotropic Grassmannians, including necessary and sufficient conditions for certain cases and computational verification.

Findings

Ottaviani's conditions are necessary for Lagrangian Grassmannian of isotropic k-planes for k ≤ 6

Ottaviani's conditions are not necessary for k=7

New splitting criteria are established for the Lagrangian Grassmannian

Abstract

We extend a theorem of Ottaviani on cohomological splitting criterion for vector bundles over the Grassmannian to the case of the symplectic isotropic Grassmanian. We find necessary and sufficient conditions for the case of the Grassmanian of symplectic isotropic lines. For the general case the generalization of Ottaviani's conditions are sufficient for vector bundles over the symplectic isotropic Grassmannian. By a calculation in the program LiE, we find that Ottaviani's conditions are necessary for Lagrangian Grassmannian of isotropic $k$-planes for $k\leq 6$, but they fail to be necessary for the case of the Lagrangian Grassmannian of isotropic 7-planes. Finally, we find a related set of necessary and sufficient splitting criteria for the Lagrangian Grassmannian.