Homological mirror symmetry for hypersurfaces in (ℂ∗)n

Abouzaid, Mohammed¹ ; Auroux, Denis²

¹ Department of Mathematics, Columbia University, New York, NY, United States, Department of Mathematics, Stanford University, Stanford, CA, United States
² Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 28 (2024) no. 6, pp. 2825-2914.

Résumé

We prove a homological mirror symmetry result for maximally degenerating families of hypersurfaces in ${({ℂ}^{\ast })}^n$ (B–model) and their mirror toric Landau–Ginzburg A–models. The main technical ingredient of our construction is a “fiberwise wrapped” version of the Fukaya category of a toric Landau–Ginzburg model. With the definition in hand, we construct a fibered admissible Lagrangian submanifold whose fiberwise wrapped Floer cohomology is isomorphic to the ring of regular functions of the hypersurface. It follows that the derived category of coherent sheaves of the hypersurface quasiembeds into the fiberwise wrapped Fukaya category of the mirror. We also discuss an extension to complete intersections.

Détail

DOI : 10.2140/gt.2024.28.2825

Keywords: homological mirror symmetry, Fukaya categories, Landau–Ginzburg models

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