Quantization of presymplectic manifolds and circle actions

We prove several versions of "quantization commutes with reduction" for cir cle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin(c) structure. Our theorems work whenever the quantization data and the reduction data are compatible; this condition always holds if we start from a presymplectic (in particular, sym plectic) manifold.