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

Let X be any nonsingular complex projective variety with a linear action of a complex reductive group G, and let X and X be the sets of semistable and stable points of X in the sense of Mumford’s geometric invariant theory [17]. We can choose a maximal compact subgroup K of G and an inner product on the Lie algebra k of K which is invariant under the adjoint action. Then X has a G-equivariantly perfect stratification {Sβ : β ∈ B} by locally closed nonsingular G-invariant subvarieties with X as an open stratum, which can be obtained as the Morse stratification for the normsquare of a moment map μ : X → k for the action of K on X [9]. In this note the Morse stratification {Sβ : β ∈ B} is refined to obtain stratifications of X by locally closed nonsingular G-invariant subvarieties with X 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. These refinements of the Morse stratification are not in general equivariantly perfect; that is, the associated equivariant Morse inequalities are not necessarily equalities. However when G is abelian we can modify the moment map (or equivalently modify the linearisation of the action) by the addition of any constant, since the adjoint action is trivial. Perturbation of the moment map by adding a small constant then provides an equivariantly perfect refinement of the stratification {Sβ : β ∈ B}, and a generic perturbation gives us a refined stratification whose strata can be described inductively in terms of the sets of stable points of linear actions of reductive subgroups of G for which stability and semistability coincide. This is useful even when G is not abelian, since important questions about the cohomology of the Marsden-Weinstein reduction μ(0)/K (or equivalently the geometric invariant theoretic quotient X//G) can be reduced to questions about the quotient of X by a maximal torus of G. The same constructions work when X is a compact Kähler manifold with a Hamiltonian action ofK. Even whenX is symplectic but not Kähler, refinements of the Morse stratification for ||μ|| can be obtained by choosing a suitable almost complex structure and Riemannian metric.