Refinements of the Morse stratification of the normsquare of the moment map

Abstract

Let X be any nonsingular complex projective variety on which a complex reductive group G acts linearly, and let X^{ss} and X^s be the sets of semistable and stable points of X in the sense of Mumford's geometric invariant theory. Then X has a G-equivariantly perfect stratification by locally closed nonsingular G-invariant subvarieties with X^{ss} as an open stratum, which can be obtained as the Morse stratification of the normsquare of a moment map for the action of a maximal compact subgroup K of G. In this note this stratification is refined to obtain stratifications of X by locally closed nonsingular G-invariant subvarieties with X^s as an open stratum. The strata can be defined inductively in terms of the sets of stable points of closed nonsingular subvarieties of X acted on by reductive subgroups of G, and their projectivised normal bundles. When G is abelian another way to obtain a refined stratification is by perturbing the moment map; the refinement is then itself equivariantly perfect, and its strata can be described in terms of the sets of stable points of linear actions of reductive subgroups of G for which semistability and stability coincide. This is useful even when G is nonabelian, since important questions about the cohomology of the GIT quotient (or Marsden-Weinstein reduction) X//G can be reduced to questions about the quotient of X by a maximal torus of G.