Log symplectic manifolds and $[Q,R]=0$

TL;DR

This paper establishes that log symplectic manifolds with certain singularities admit a Spin_c structure and demonstrates a $[Q,R]=0$ theorem for their Dirac operators in the Hamiltonian setting.

Contribution

It proves the existence of Spin_c structures on log symplectic manifolds with normal crossing singularities and extends the $[Q,R]=0$ theorem to this context.

Findings

Log symplectic manifolds with simple normal crossing singularities are Spin_c.

The index of the Spin_c Dirac operator satisfies a $[Q,R]=0$ theorem in the Hamiltonian case.

Abstract

We show, under an orientation hypothesis, that a log symplectic manifold with simple normal crossing singularities has a stable almost complex structure, and hence is Spin$_c$. In the compact Hamiltonian case we prove that the index of the Spin$_c$ Dirac operator twisted by a prequantum line bundle satisfies a $[Q,R]=0$ theorem.