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

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.