Intersection cohomology of \(S^1\) symplectic quotients and small resolutions (Q1568661)

Let the circle \(S^1\) act on a compact connected symplectic manifold \(M\) with moment map \(\Phi:M \rightarrow R\) so that \(0\) is in the interior of \(\Phi(M)\). Let \(M_{\text{red}}:=\Phi^{-1}(0)/S^1\) denote the reduced space, which is a stratified space, when \(0\) is a singular value of \(\Phi\). The authors find two explicit formulas for the intersection cohomology \(\mathbb{H}^{\ast} (M_{\text{red}},R)\) (as a graded vector space with pairing) of \(M_{\text{red}}\) in terms of \(H^{\ast}_{S^1}(M,R)\), the \(S^1\) equivariant cohomology of \(M\) and the fixed-point data. They prove that there exists a surjective map \(\kappa:H^{\ast}_{S^1}(M,R) \rightarrow \mathbb{H}^{\ast} (M_{\text{red}},R)\), such that the pairing of \(\kappa(\alpha)\) and \(\kappa(\beta)\) in \(\mathbb{H}^{\ast} (M_{\text{red}},R)\) is given by the explicit formula involving \(\alpha\), \(\beta\) and fixed point-data for any \(\alpha, \beta \in H^{\ast}_{S^1}(M,R)\). They also formulate the alternate version of the result on the existence of the ring structure on \(\mathbb{H}^{\ast}(M_{\text{red}},R)\) compatible with the pairing such that \(\mathbb{H}^{\ast}(M_{\text{red}},R)\) is isomorphic to the quotient of \(H^{\ast}_{S^1}(M,R)\) as a graded ring. The authors prove these results by constructing an orbifold which is a small resolution of \(M_{\text{red}}\). The orbifold is obtained as the quotient of the zero fiber of the perturbation of the moment map \(\Phi\). By construction, the perturbed map is Morse-Bott and its critical points are exactly fixed points of the \(S^1\) action on \(M\). Hence, by standard techniques used to compute the cohomology of symplectic quotients, the authors compute the cohomology of the orbifold which is isomorphic to \(\mathbb{H}^{\ast}(M_{\text{red}},R)\) as a graded vector space with pairing.