Symplectic structures on stratified pseudomanifolds

TL;DR

This paper develops a sheaf-theoretic framework for defining symplectic structures on stratified pseudomanifolds, extending classical concepts to singular spaces and analyzing their properties via quotient constructions.

Contribution

It introduces a sheaf-theoretic definition of symplectic forms on stratified pseudomanifolds and applies it to singular symplectic quotients, generalizing symplectic geometry to singular spaces.

Findings

Defined symplectic structures on stratified pseudomanifolds using sheaf theory

Extended symplectic form concepts to quotient spaces of singular symplectic quotients

Proved the existence and uniqueness of cohomologically symplectic structures on singular reduced spaces

Abstract

The purpose of this paper is to investigate the definition of symplectic structure on a smooth stratified pseudomanifold in the framework of local $\C^{\infty}$-ringed space theory. We introduce a sheaf-theoretic definition of symplectic form and cohomologically symplectic structure on smooth stratified pseudomanifolds. In particular, we give an indirect definition of symplectic form on the quotient space of a smooth $G$-stratified pseudomanifold. Based on the structure theorem of singular symplectic quotients by Sjamaar--Lerman, we show that the singular reduced space $M_{0}=\mu^{-1}(0)/G$ of a symplectic Hamiltonian $G$-manifold $(M,\omega,G,\mu)$ admits a natural (indirect) symplectic form and a unique cohomologically symplectic structure.