Quantization of Abelian Varieties: distributional sections and the transition from Kähler to real polarizations

Details Quantization of Abelian Varieties: distributional sections and the transition from Kähler to real polarizations Quantization of Abelian Varieties: distributional sections and the transition from Kähler to real polarizations

2009-07-30

arxiv.org/abs/0907.5324v3

Mathematics

Symplectic Geometry

24 pages; revised version, to appear in JFA

Scientific paper

We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil-Brezin expansion to describe distributional sections, we give a unified analytical description of the quantization spaces for all nonnegative polarizations. The Blattner-Kostant-Sternberg (BKS) pairing maps between half-form corrected quantization spaces for different polarizations are shown to be transitive and related to an action of $Sp(2g,\R)$. Moreover, these maps are shown to be unitary.

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