Mirror symmetry for very affine hypersurfaces

Vivek Shende (Centre for Quantum Mathematics, Universitet Syddansk, Odense, Denmark; and Department of Mathematics, University of California, Berkeley, Cal., U.S.A.)

Abstract

We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasi-projective toric DM stack is equivalent to the wrapped Fukaya category of a hypersurface in $(C^\times)^n$. Hypersurfaces with every Newton polytope can be obtained. Our proof has the following ingredients. Using recent results on localization, we may trade wrapped Fukaya categories for microlocal sheaf theory along the skeleton of the hypersurface. Using Mikhalkin–Viro patchworking, we identify the skeleton of the hypersurface with the boundary of the Fang–Liu–Treumann–Zaslow skeleton. By proving a new functoriality result for Bondal’s coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki’s recent theorem on mirror symmetry for toric varieties.

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Received 26 October 2018

Received revised 20 January 2021

Accepted 2 February 2022

Published 21 February 2023