Surveys in Differential Geometry

Volume 21 (2016)

Theta functions and mirror symmetry

Pages: 95 – 138

DOI: http://dx.doi.org/10.4310/SDG.2016.v21.n1.a3

Authors

Mark Gross (DPMMS, Centre for Mathematical Sciences, Cambridge, United Kingdom)

Bernd Siebert (Department Mathematik, Universität Hamburg, Germany)

Abstract

This is a survey covering aspects of varied work of the authors with Mohammed Abouzaid, Paul Hacking, and Sean Keel. While theta functions are traditionally canonical sections of ample line bundles on abelian varieties, we motivate, using mirror symmetry, the idea that theta functions exist in much greater generality. This suggestion originates with the work of the late Andrei Tyurin. We outline how to construct theta functions on the degenerations of varieties constructed in previous work of the authors, and then explain applications of this construction to homological mirror symmetry and constructions of broad classes of mirror varieties.

Keywords

Calabi–Yau manifolds

2010 Mathematics Subject Classification

14J33

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Published 7 June 2016