Current Developments in Mathematics

Volume 2012

Mirror symmetry and the Strominger-Yau-Zaslow conjecture

Pages: 133 – 191

DOI: http://dx.doi.org/10.4310/CDM.2012.v2012.n1.a3

Author

Mark Gross (Department of Mathematics, University of California at San Diego, La Jolla, Calif., U.S.A.)

Abstract

We trace progress and thinking about the Strominger-Yau-Zaslow conjecture since its introduction in 1996. We begin with the original differential geometric conjecture and its refinements, and explain how insights gained in this context led to the algebrogeometric program developed by the author and Siebert. The objective of this program is to explain mirror symmetry by studying degenerations of Calabi-Yau manifolds. This introduces logarithmic and tropical geometry into the mirror symmetry story, and gives a clear path towards a conceptual understanding of mirror symmetry within an algebrogeometric context. After explaining the overall philosophy, we explain how recent results fit into this program.

2010 Mathematics Subject Classification

14J32

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