Lagrangian Intersections, Symplectic Reduction and Kirwan Surjectivity

TL;DR

This paper explores the relationships between Lagrangian subvarieties, symplectic reduction, and Kirwan surjectivity in the context of holomorphic symplectic varieties with group actions, providing new geometric and algebraic insights.

Contribution

It establishes new isomorphisms and surjectivity results for Ext groups of Lagrangians under symplectic reduction, extending Kirwan surjectivity to derived categories and half-canonical bundles.

Findings

Derived isomorphism between reduced Lagrangians and conormal bundles.

Extension of Kirwan surjectivity to Ext groups in derived categories.

Results applicable to half-canonical bundles and symmetry reduction.

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 $Ext^{\bullet}_{X// G}(\mathcal{O}_{C_1/G}, \mathcal{O}_{C_2/G})\cong H^{\bullet}_G(B, det(N_{B/C_2})_{\delta})$ 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 $K_{C_2/G}^{1/2}$.