On the Local structure theorem and equivariant geometry of cotangent bundles

TL;DR

This paper improves local structure theorems for reductive group actions on algebraic varieties, extends results on horospheres, and describes the moment map image of cotangent bundles using geometric methods.

Contribution

It provides an improved local structure theorem, constructs a family of nongeneric horospheres, and describes the moment map image without differential operators.

Findings

Enhanced local structure theorem for group actions

Construction of a family of nongeneric horospheres

Geometric description of the moment map image

Abstract

Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We give a slightly improved version of local structure theorems obtained by F.Knop and D.A.Timashev that describe an action of some parabolic subgroup of $G$ on an open subset of $X$. We also extend various results of E.B.Vinberg and D.A.Timashev on the set of horospheres in $X$. We construct a family of nongeneric horospheres in $X$ and a variety $\Hor$ parameterizing this family, such that there is a rational $G$-equivariant symplectic covering of cotangent vector bundles $T^*\Hor \dashrightarrow T^*X$. As an application we get a description of the image of the moment map of $T^*X$ obtained by F.Knop by means of geometric methods that do not involve differential operators.