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Suppose X is a compact symplectic manifold acted on by a compact Lie group K (which may be nonabelian) in a Hamiltonian fashion, with moment map µ : X → Lie(K) * and Marsden-Weinstein reduction M X = µ −1 (0)/K. There is then a natural surjective map κ 0 from the equivariant cohomology H * K (X) of X to the cohomology H * (M X ). In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of M X of any η 0 ∈ H * (M X ) whose degree is the dimension of M X , provided that 0 is a regular value of the moment map µ on X. This formula is given in terms of any class η ∈ H * K (X) for which κ 0 (η) = η 0 , and involves the restriction of η to K-orbits KF of components F ⊂ X of the fixed point set of a chosen maximal torus T ⊂ K. Since κ 0 is surjective, in principle the residue formula enables one to determine generators and relations for the cohomology ring H * (M X ), in terms of generators and relations for H * K (X). There are two main ingredients in the proof of our formula: one is the localization theorem [3,7] for equivariant cohomology of manifolds acted on by compact abelian groups, while the other is the equivariant normal form for the symplectic form near the zero locus of the moment map.We also make use of the techniques appearing in our proof of the residue formula to give a new proof of the nonabelian localization formula of Witten ([35], Section 2) for Hamiltonian actions of compact groups K on symplectic manifolds X; this theorem expresses η 0 [M X ] in terms of certain integrals over X.

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Abstract:We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X s and semistable points X ss , often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric quotient of X s , and (3) the existence of a canonical "enveloping quotient" variety of X ss , denoted X//H, which (4) has a projective completion given by a reductive GIT quotient and (5) is itself projective and isomorphic to Proj(k[X] H ) when k [X] H is finitely generated.