Quantization of b-symplectic manifolds

ABSTRACT: b-symplectic (or log-symplectic) manifolds are Poisson manifolds with the
property that the Poisson bivector field arises from a symplectic structure on the complement
of a real hypersurface, and has a prescribed vanishing on that hypersurface. This is a "least
degenerate" case of Poisson geometry which we can try to use as a laboratory to apply t
echniques developed in symplectic geometry to obtain intuition about quantization in the
Poisson setting.

Recent work with V. Guillemin and E. Miranda on desingularization of b-symplectic forms
leads to a proposed scheme for quantization of these manifolds in the case where the b-symplectic
form has certain integrality properties and where an abelian group acts in a Hamiltonian fashion
and satisfies a technical nondegeneracy condition.  We show that in this case we obtain a finite
quantization, which we conjecture should arise from the index of a Fredholm operator acting on'
sections of an appropriate line bundle.

This is joint work with V. Guillemin and E. Miranda.