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Degenerating Kähler structures and geometric quantization

João P. Nunes

Center for Mathematical Analysis, Geometry and Dynamical Systems and Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

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https://doi.org/10.1142/S0129055X1430009XCited by:2

Abstract

We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in real and Kähler polarizations in some families of symplectic manifolds, that can be explicitly quantized and that constitute an important class of examples: cotangent bundles of Lie groups, abelian varieties and toric varieties. Applications to theta functions and moduli spaces of vector bundles on curves are also reviewed.

Keywords:

AMSC: 53D50, 32Q15, 81S10, 81R30, 22E30

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Vol. 26, No. 09

Metrics

History

Received 12 April 2013

Revised 18 July 2014

Accepted 21 September 2014

Published: 14 October 2014

Keywords

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Degenerating Kähler structures and geometric quantization

Abstract

References

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