The variety of reductions for a reductive symmetric pair

Abstract

We define and study the variety of reductions for a complex reductive symmetric pair (G, θ), which is the natural compactification of the set of its Cartan subspaces. 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 of these varieties of reductions, and relate their geometry to some problems in representation theory. A very useful result is the rigidity of semisimple elements in deformations of algebraic subalgebras of Lie algebras. We use it to show that the closure of a decomposition class is a union of decomposition classes.We apply this theory to the study of other varieties of reductions in a companion paper, which yields two new Fano varieties.