  We study two quantization schemes for compact symplectic manifolds with
almost complex structures. The first of these is the Spin-c quantization. We
prove the analog of Kodaira vanishing for the Spin-c Dirac operator, which
shows that the index space of this operator provides an honest (not virtual)
vector space semiclassically. We also introduce a new quantization scheme,
based on a rescaled Laplacian, for which we are able to prove strong
semiclassical properties. The two quantizations are shown to be close
semiclassically.
