Transversality theory, cobordisms, and invariants of symplectic quotients

TL;DR

This paper develops methods to compute invariants of symplectic quotients by compact tori, using cobordism and characteristic classes, and provides explicit formulas for integrals over these quotients based on fixed point data.

Contribution

It introduces an explicit cobordism approach for symplectic quotients by tori and derives formulas for cohomology integrals using fixed point localization.

Findings

Explicit cobordism between symplectic quotients and projective bundles

Formulas for cohomology integrals localized at fixed points

Characteristic classes of bundles are explicitly determined

Abstract

This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients considered here are compact symplectic manifolds (or more generally orbifolds), which arise as the symplectic quotients of a symplectic manifold by a compact torus. (A companion paper examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus.) Let X be a symplectic manifold, with a Hamiltonian action of a compact torus T. The main topological result of this paper describes an explicit cobordism that exists between a symplectic quotient of X by T, and a collection of iterated projective bundles over components of the set of T-fixed-points. The characteristic classes of these bundles can be determined explicitly, and another result uses this to give formulae for integrals of cohomology classes over the symplectic quotient, in terms of data localized at the T-fixed points of X.