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

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

2003-05-24

arxiv.org/abs/math/0305347v1

Mathematics

Algebraic Geometry

33 pages

Scientific paper

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.

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