Symplectic structures on stratified pseudomanifolds

The purpose of this paper is to investigate the definition of symplectic structure on a smooth stratified pseudomanifold in the framework of local ${\text{C}}^{\mathrm{\infty }}$-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 $M_0={\mu }^{-1}(0)\mathrm{/}G$ of a symplectic Hamiltonian $G$-manifold $(M,\omega ,G,\mu )$ admits a natural (indirect) symplectic form and a unique cohomologically symplectic structure.