Geometry of horospherical varieties of Picard rank one

TL;DR

This paper investigates the geometry and quantum cohomology of non-homogeneous horospherical varieties of Picard rank one, including their cohomology, Gromov-Witten invariants, and derived categories, with applications to Dubrovin's conjecture.

Contribution

It provides a detailed analysis of the quantum cohomology, Gromov-Witten invariants, and derived categories of horospherical varieties, including explicit presentations and exceptional collections.

Findings

Quantum cohomology often semisimple in these varieties.

Many Gromov-Witten invariants are enumerative.

Constructed full exceptional collections for specific cases.

Abstract

We study the geometry of non-homogeneous horospherical varieties. These have been classified by Pasquier and include the well-known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin's conjecture. In particular, we describe the cohomology groups of these varieties as well as a Chevalley formula, and prove that many Gromov-Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. The final section is devoted to the derived categories of coherent sheaves on horospherical varieties. We first discuss a general construction of exceptional bundles on these varieties. We then study in detail the case of the horospherical variety associated to the exceptional group $G_2$, and construct a full rectangular Lefschetz exceptional collection in the derived category.