Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

By M. Abouzaid, D. Auroux, L. Katzarkov. Publications mathématiques (Bures-sur-Yvette), 2012

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We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface H$H$ in a toric variety V$V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of V×C$V\times \mathbf {C}$ along H×0$H\times0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to H$H$. The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

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