Abstract

We extend known prequantization procedures for Poisson and presym- plectic manifolds by defining the prequantization of a Dirac manifold P as a principal U(1)-bundle Q with a compatible Dirac-Jacobi structure. We study the action of Poisson algebras of admissible functions on P on various spaces of locally (with respect to P) defined functions on Q, via hamiltonian vector fields. Finally, guided by examples arising in complex analysis and contact geometry, we propose an extension of the notion of prequantization in which the action of U(1) on Q is permitted to have some fixed points. Dedicated to the memory of Professor Shiing-Shen Chern.