Mathematical Research Letters

Volume 3 (1996)

Number 6

Almost Complex Structures and Geometric Quantization

Pages: 845 – 861



David Borthwick

Alejandro Uribe


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.

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