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Intersection cohomology of symplectic quotients by circle actions. (English) Zbl 1070.55002

Let \(T= U(1)\) and \(M\) be a Hamiltonian \(T\)-space with proper moment map \(\mu: M\to\mathbb{R}\). When \(0\) is not a regular value of \(\mu\), the symplectic quotient \(X= Z/T\), \(Z= \mu^{-1}(0)\), is a singular stratified space. The authors prove that the middle perversity intersection cohomology \(IH^*(X)\) is isomorphic to a subspace of the equivariant cohomology \(H^*_T(Z)\).
More precisely, let \(F+1,\dots, F_r\) be the \(T\)-fixed components in \(Z\). For \(1\leq i\leq r\), the normal space of \(F_i\) decomposes into positive and negative weight spaces \(w^+_i\), \(w^-_i\). Let \(d_i={1\over 2}\min(\dim w^+_i, \dim w^-_i)\). Then \(H^*_T(F_i)\cong H^*(F_i)\otimes H^*_T\) and \[ IH^*(X)\cong\{x\in H^*_T(Z)\mid x|_{F_i}\in H^*(F_i)\otimes H_T^{2d_i-2}\}. \]

MSC:

55N33 Intersection homology and cohomology in algebraic topology
55N91 Equivariant homology and cohomology in algebraic topology
53D20 Momentum maps; symplectic reduction
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