Secondary staircase complexes on isotropic Grassmannians

TL;DR

This paper introduces secondary staircase complexes on isotropic Grassmannians, providing new tools to analyze derived categories and exceptional collections in symplectic geometry.

Contribution

It constructs equivariant vector bundles as truncations of staircase complexes and demonstrates their relation to symplectic wedge powers, advancing the understanding of derived categories.

Findings

Constructed secondary staircase complexes on isotropic Grassmannians.

Showed these complexes are quasi-isomorphic to symplectic wedge powers.

Lays groundwork for studying fullness of exceptional collections.

Abstract

We introduce a class of equivariant vector bundles on isotropic symplectic Grassmannians $\mathrm{IGr}(k,2n)$ defined as appropriate truncations of staircase complexes and show that these bundles can be assembled into a number of complexes quasi-isomorphic to the symplectic wedge powers of the symplectic bundle on $\mathrm{IGr}(k,2n)$. We are planning to use these secondary staircase complexes to study fullness of exceptional collections in the derived categories of isotropic Grassmannians and Lefschetz exceptional collections on $\mathrm{IGr}(3,2n)$.