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

Abstract

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 $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times \mathbb{C}$ along $H\times 0$, 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 $H$. 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.

83 pages; v2: added appendix discussing the analytic structure on moduli of objects in the Fukaya category; v3: further clarifications in response to referee report; v4: further clarifications throughout, especially sections 4 and 7 and appendix A; added appendix B on the geometry of reduced spaces

References in corpus (5)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Homological Mirror Symmetry for manifolds of general type
• Mirror Symmetry And Quantum Geometry
• On the reconstruction problem in mirror symmetry

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• Lagrangian sections on mirrors of toric Calabi-Yau 3-folds
• Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians