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

2009/07/30 by Thomas Baier, Baier, Thomas, José M. Mourão +3
Mathematics · #53D50 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:53D50

paper · pdf · doi:10.48550/arxiv.0907.5324

24 pages; revised version, to appear in JFA

openalex publication_date 2009/07/30 · arxiv created 2010/01/26 · arxiv updated 2010/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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