Lagrangian Intersections, Symplectic Reduction and Kirwan Surjectivity

Abstract

Given a smooth holomorphic symplectic variety $X$ with a Hamiltonian $G$-action, $G$-invariant Lagrangians $C^{\prime }s$ induce Lagrangians in the symplectic quotient $X//G$. Given clean intersections $B=C_1\cap C_2$ whose conormal sequence splits, we show that $$C_1/G{\times }_{X//G}{C}_2/G\cong T^{\vee }[-1](B/G).$$ When $det(N_{B/C_2})$ is torsion, we have $Ex{t}_{X//G}^{\bullet }({\mathcal{O}}_{C_1/G},{\mathcal{O}}_{C_2/G})\cong H_G^{\bullet }(B,det(N_{B/C_2}{)}_{\hat{I}}´)$ provided that the Hodge-to-de Rham degeneracy holds. Furthermore, we have a generalized version of Kirwan surjectivity $Ex{t}_{X//G}^{\bullet }({\mathcal{O}}_{C_1/G},{\mathcal{O}}_{C_2/G})\twoheadrightarrow Ex{t}_{X^{ss}//G}^{\bullet }({\mathcal{O}}_{C_1^{ss}/G},{\mathcal{O}}_{C_2^{ss}/G})$ if $B$ is proper. When $C_1=C_2$, this is the Kirwan surjectivity, which is now interpreted as the symmetry commutes with reduction problem in 3d B-model. We also obtain similar results for $K_{C_1/G}^{1/2}$ and $K_{C_2/G}^{1/2}$.

27 pages