Lagrangian Intersections, Symplectic Reduction and Kirwan Surjectivity
Naichung Conan Leung, Ying Xie, Yu Tung Yau
TL;DR
The paper develops a derived-intersection framework for complex Lagrangian intersections under a Hamiltonian $G$-action, showing that $C_1/G\times_{X//G} C_2/G\cong \operatorname{Tot}_{B/G}(\mathbb{L}_{B/G}[-1])$ when the conormal sequence splits. It then computes Ext groups on the equivariant quotient $X//G$ in terms of equivariant cohomology on the intersection locus, including twists by 2-torsion Local systems arising from spin structures and canonical-square roots. A key contribution is the twisted Kirwan surjectivity, which ensures surjectivity of restriction maps from $X//G$ to semi-stable quotients and extends to line bundle twists and spin cases. The results provide concrete tools for calculating Fukaya-category-like invariants in the 3d B-model context and connect symplectic reduction with equivariant cohomology via derived geometry. Overall, the work bridges derived intersection theory, shifted symplectic geometry, and GIT to produce computable invariants for equivariant Lagrangian intersections.
Abstract
Given a smooth holomorphic symplectic variety $X$ with a Hamiltonian $G$-action, $G$-invariant Lagrangians $C's$ 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 $Ext^{\bullet}_{X// G}(\mathcal{O}_{C_1/G}, \mathcal{O}_{C_2/G})\cong H^{\bullet}_G(B, det(N_{B/C_2})_δ)$ provided that the Hodge-to-de Rham degeneracy holds. Furthermore, we have a generalized version of Kirwan surjectivity $Ext^{\bullet}_{X// G}(\mathcal{O}_{C_1/G}, \mathcal{O}_{C_2/G})\twoheadrightarrow Ext^{\bullet}_{X^{ss}// G}(\mathcal{O}_{C_1^{ss}/G}, \mathcal{O}_{C_2^{ss}/G})$ 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 $K_{C_2/G}^{1/2}$.