Surveys in Differential Geometry

Volume 22 (2017)

Speculations on homological mirror symmetry for hypersurfaces in $(\mathbb{C}^{\ast})^n$

Pages: 1 – 47

DOI: https://dx.doi.org/10.4310/SDG.2017.v22.n1.a1

Author

Denis Auroux (Department of Mathematics, University of California at Berkeley; and School of Mathematics, Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Abstract

Given an algebraic hypersurface $H = f^{-1} (0)$ in $(\mathbb{C}^{\ast})^n$, homological mirror symmetry relates the wrapped Fukaya category of $H$ to the derived category of singularities of the mirror Landau–Ginzburg model. We propose an enriched version of this picture which also features the wrapped Fukaya category of the complement $(\mathbb{C}^{\ast})^n \setminus H$ and the Fukaya–Seidel category of the Landau–Ginzburg model $(\mathbb{C}^{\ast})^n , f)$. We illustrate our speculations on simple examples, and sketch a proof of homological mirror symmetry for higher-dimensional pairs of pants.

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This work was partially supported by NSF grants DMS-1264662 and DMS-1406274; by a Simons Foundation grant (#385573, Simons Collaboration on Homological Mirror Symmetry); by the Eilenberg Chair at Columbia University; and by the Schmidt Fellowship and the IAS Fund for Mathematics.

Published 13 September 2018