Geometric quantization on presymplectic manifolds

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