Full exceptional collections on the isotropic Grassmannians

TL;DR

This paper proves the fullness of Kuznetsov-Polishchuk exceptional collections on symplectic homogeneous spaces, introducing new classes of complexes that could be of independent interest.

Contribution

It establishes the fullness of certain exceptional collections on symplectic Grassmannians and introduces new complex constructions.

Findings

Exceptional collections are full on symplectic homogeneous spaces.

New classes of complexes are constructed: generalized staircase, symplectic staircase, and secondary staircase complexes.

Complexes may be of independent mathematical interest.

Abstract

We prove that the Kuznetsov--Polishchuk exceptional collections on rational homogeneous spaces of the symplectic groups $\mathrm{Sp}(2n,\mathbb{C})$ are full and consist of vector bundles. To achieve this, we construct several classes of complexes, which we call generalized staircase complexes, symplectic staircase complexes and secondary staircase complexes -- each of which may be of independent interest.