A two-step approach to Chow quotients

TL;DR

This paper introduces a novel two-step method to analyze Chow quotients of projective varieties by complex torus actions, linking their complex geometry to toric varieties and automorphisms.

Contribution

It proposes a new approach encoding Chow quotient geometry via a toric variety and automorphism subgroup, demonstrated on rational homogeneous varieties.

Findings

Chow quotients have complex geometry even for simple varieties.

The approach encodes Chow quotient geometry in a toric variety and automorphisms.

Application to specific rational homogeneous varieties illustrates the method.

Abstract

The Chow quotient of a projective variety by the action of a complex torus is known to have a very complicated geometry, even in the case of simple varieties, such as rational homogeneous varieties. In this paper we propose an approach in which the geometry of the Chow quotient is encoded in a projective toric variety and a finite subgroup of its birational automorphisms. We then illustrate how to apply our strategy in the case of some particular rational homogeneous varieties.