Cohomology of hyperplane sections of (co)adjoint varieties

TL;DR

This paper investigates the geometry and cohomology of hyperplane sections of (co)adjoint varieties, revealing their unique stabilizing properties and computing their quantum cohomology, thus extending understanding of their algebraic and geometric structures.

Contribution

It characterizes hyperplane sections stabilized by maximal tori and derives explicit formulas for their classical and quantum cohomology rings.

Findings

Hyperplane sections are uniquely stabilized by maximal tori.

Formulas for classical cohomology rings in terms of Schubert classes.

Quantum cohomology of these sections is semi-simple.

Abstract

In this paper we study general hyperplane sections of adjoint and coadjoint varieties. We show that these are the only sections of homogeneous varieties such that a maximal torus of the automorphism group of the ambient variety stabilizes them. We then study their geometry, provide formulas for their classical cohomology rings in terms of Schubert classes and compute the quantum Chevalley formula. This allows us to obtain results about the semi-simplicity of the (small) quantum cohomology, analogous to those holding for (co)adjoint varieties.