Mathematical Research Letters

Volume 3, Issue 6, November 1996 pp. 845-861.

Almost Complex Structures and Geometric Quantization

Authors: David Borthwick and Alejandro Uribe
Author institution: University of Michigan, Ann Arbor

Summary: 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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