On Neeman’s gradient flows

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.