The variety of reductions for a reductive symmetric pair

TL;DR

This paper introduces and analyzes the variety of reductions for reductive symmetric pairs, connecting geometric properties with representation theory, and producing new Fano varieties through this framework.

Contribution

It defines the variety of reductions for symmetric pairs, develops a theoretical foundation, and links it to representation theory and Fano varieties, expanding the understanding of these geometric objects.

Findings

Varieties of reductions generalize Severi varieties

Rigidity of semi-simple elements in algebraic subalgebras

Discovery of two new Fano varieties

Abstract

We define and study the variety of reductions for a reductive symmetric pair (G,theta), which is the natural compactification of the set of the Cartan subspaces of the symmetric pair. These varieties generalize the varieties of reductions for the Severi varieties studied by Iliev and Manivel, which are Fano varieties. We develop a theoretical basis to the study these varieties of reductions, and relate the geometry of these variety to some problems in representation theory. A very useful result is the rigidity of semi-simple elements in deformations of algebraic subalgebras of Lie algebras. We apply this theory to the study of other varieties of reductions in a companion paper, which yields two new Fano varieties.