On Neeman's gradient flows

TL;DR

This paper explores Neeman's gradient flow in algebraic geometry, demonstrating its applications such as deformation retraction of cones in Euclidean space and extending related results to real reductive groups using Lojasiewicz inequalities.

Contribution

It provides new insights into Neeman's gradient flow, including applications to deformation retractions and an extension of existing theorems to real reductive group actions.

Findings

The cone on a Zariski closed subset is a deformation retract of Euclidean space.

Extension of Schwarz's explanation to real reductive algebraic group actions.

Use of Lojasiewicz gradient inequality in the analysis.

Abstract

In his brilliant but sketchy paper on the strucure of quotient varieties of affine actions of reductive algebraic groups over C, Amnon Neeman introduced a gradiant flow with remarkable properties. The purpose of this paper is to study several applications of this flow. In particular we prove that the cone on a Zariski closed subset of n-1 dimensional real projective space is a deformation retract of n dimensional Euclidean space. We also give an exposition of an extension to real reductive algebraic group actions of Schwarz's excellent explanation of Neeman's sketch of a proof of his deformation theorem. This exposition precisely explains the use of Lojasiewicz gradient inequality. The result described above for cones makes use of these ideas.