Cohomology of Quotients in Real Symplectic Geometry

TL;DR

This paper extends Kirwan's theorems to real symplectic quotients, providing formulas for their $bZ_2$-Betti numbers using Morse theory and equivariant formality in the context of involutions.

Contribution

It proves analogues of Kirwan's theorems for real Hamiltonian systems, enabling calculation of $bZ_2$-Betti numbers of real symplectic quotients.

Findings

$| abla ||^2|$ is a perfect Morse function on the fixed point set.

Explicit description of critical sets using real Hamiltonian subsystems.

Establishment of equivariant formality for the group action on fixed point sets.

Abstract

Given a Hamiltonian system $ (M,\omega, G,\mu) $ where $(M,\omega)$ is a symplectic manifold, $G$ is a compact connected Lie group acting on $(M,\omega)$ with moment map $ \mu:M \rightarrow\mathfrak{g}^{*}$, then one may construct the symplectic quotient $(M//G, \omega_{red})$ where $M//G := \mu^{-1}(0)/G$. Kirwan used the norm-square of the moment map, $|\mu|^2$, as a G-equivariant Morse function on $M$ to derive formulas for the rational Betti numbers of $M//G$. A real Hamiltonian system $(M,\omega, G,\mu, \sigma, \phi) $ is a Hamiltonian system along with a pair of involutions $(\sigma:M \rightarrow M, \phi:G \rightarrow G) $ satisfying certain compatibility conditions. These imply that the fixed point set $M^{\sigma}$ is a Lagrangian submanifold of $(M,\omega)$ and that $M^{\sigma}//G^{\phi} := (\mu^{-1}(0) \cap M^{\sigma})/G^{\phi}$ is a Lagrangian submanifold of $(M//G, \omega_{red})$. In this paper we prove analogues of Kirwan's Theorems that can be used to calculate the $\mathbb{Z}_2$-Betti numbers of $M^{\sigma}//G^{\phi} $. In particular, we prove (under appropriate hypotheses) that $|\mu|^2$ restricts to a $G^{\phi}$-equivariantly perfect Morse-Kirwan function on $M^{\sigma}$ over $\mathbb{Z}_2$ coefficients, describe its critical set using explicit real Hamiltonian subsystems, prove equivariant formality for $G^{\phi}$ acting on $M^{\sigma}$, and combine these results to produce formulas for the $\mathbb{Z}_2$-Betti numbers of $M^{\sigma}//G^{\phi}$.