A smooth affine hypersurface Z of complex dimension n is homotopy equivalent to an n-dimensional cell complex. Given a defining polynomial f for Z as well as a regular triangulation of its Newton polytope, we provide a purely combinatorial construction of a compact topological space S as a union of components of real dimension n, and prove that S embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S.
Hypersurfaces in algebraic tori have been studied by Danilov-Khovanski @cite_14 and Batyrev @cite_19 . Danilov-Khovankski computed mixed Hodge numbers, while Batyev studied the variation of mixed Hodge structures. Log geometry has been extensively employed by Gross and Siebert @cite_21 in their seminal work studying the degenerations appearing in mirror symmetry. Their strategy is crucial to our work, even though we take a somewhat different track by working in a non-compact setting for hypersurfaces that are not necessarily Calabi-Yau. The non-compactness allows us to deal with log-smooth log structures. Mirror symmetry for general hypersurfaces was recently studied in @cite_16 (projective case) and @cite_4 (affine case) using polyhedral decompositions of the Newton polytope. This relates to the Gross-Siebert program by embedding the hypersurface in codimension two in the special fiber of a degenerating Calabi-Yau family. In this family, the hypersurface coincides with the log singular locus --- see @cite_0 for the simplicial case.
@cite_14 @cite_4 @cite_21 @cite_0 @cite_19 @cite_16
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An algorithm is given for computing the mixed Hodge structure (more precisely, the Hodge–Deligne numbers) for cohomology of complete intersections in toric varieties in terms of Newton polyhedra specifying the complete intersection. In some particular cases the algorithm leads to explicit formulas. Bibliography: 8 titles. 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 @math in a toric variety @math we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of @math along @math , 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 @math . 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. We construct from a real ane manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect that our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. We give a spectral sequence to compute the logarithmic Hodge groups on a hyper- surface type toric log Calabi-Yau space X, compute its E1 term explicitly in terms of tropical degeneration data and Jacobian rings and prove its degeneration at E2 under mild assumptions. We prove the basechange of the affine Hodge groups and deduce itfor the logarithmic Hodge groups in low dimensions. As an application, we prove a mirror symmetry duality in dimension two and four involving the ordinary Hodge numbers, the stringy Hodge numbers and the affine Hodge numbers. The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau-Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main results shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf. This exchange can be anticipated from an analysis of Hochschild homology of the relevant categories arising in homological mirror symmetry in this case; we also conjecture that a similar, but different, exchange of dimensions arises from Hochschild cohomology, relating the cohomology of sheaves of polyvector fields on the hypersurface to the cohomology of the critical locus.