Horospherical two-orbit varieties as zero loci

TL;DR

This paper provides geometric descriptions of horospherical two-orbit varieties as zero loci of sections of vector bundles, and computes their cohomology rings, including quantum versions, with specific focus on the G2 and Spin7 varieties.

Contribution

It introduces a novel geometric realization of these varieties as zero loci and computes their cohomology rings, advancing understanding of their structure.

Findings

Cohomology ring of the G2-variety computed.

Quantum cohomology of the G2-variety analyzed.

Geometric realization as zero loci established.

Abstract

We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed-invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary variety. As an application, we compute the cohomology ring of the $G_2$-variety, including its quantum version. We also consider the Spin$_7$-variety, which deserves a different treatment.