Geometric quantization on presymplectic manifolds

In this paper, we discuss the possibilities of adapting geometric quantization to presymplectic manifolds, i.e., differentiable manifoldsM ^2n+k (k>0) endowed with a closed 2-form ω of rank2n. We show that such an adaptation is possible in various manners, and that, as a general idea, it reduces the quantization onM to quantization on the symplectic quotientM/V, whereV is the foliation defined by the annihilator of ω.