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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We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective DM toric stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex torus. Hypersurfaces with every Newton polytope can be obtained. Our proof has the following ingredients. Using Mikhalkin-Viro patchworking, we compute the skeleton of the hypersurface. The result matches the [FLTZ] skeleton and is naturally realized as a Legendrian in the cosphere bundle of a torus. By [GPS1, GPS2, GPS3], we trade wrapped Fukaya categories for microlocal sheaf theory. 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. Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-K "ahler setting, we can then deduce a finiteness statement for monodromy acting on @math , once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the geometry of hyper-K "ahler manifolds, we prove that a finite monodromy projective degeneration of hyper-K "ahler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts' theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain geometric constructions of hyper-K "ahler manifolds (e.g. Debarre--Voisin or Laza--Sacc a--Voisin). In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-K "ahler manifolds. 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. Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but squarefree. From the hypersurface f=0 one may construct a number of other subschemes Y by extracting prime components, taking intersections, taking unions, and iterating this procedure. We prove that if the number of solutions to f=0 in ^n is not a multiple of p, then all these intersections in ^n_ just described are reduced. (If this holds for infinitely many p, then it holds over as well.) More specifically, there is a_Frobenius splitting_ on ^n_ compatibly splitting all these subschemes Y . We determine when a Gr "obner degeneration f_0=0 of such a hypersurface f=0 is again such a hypersurface. Under this condition, we prove that compatibly split subschemes degenerate to compatibly split subschemes, and stay reduced. Our results are strongest in the case that f's lexicographically first term is i=1 ^n x_i. Then for all large p, there is a Frobenius splitting that compatibly splits f's hypersurface and all the associated Y . The Gr "obner degeneration Y' of each such Y is a reduced union of coordinate spaces (a Stanley-Reisner scheme), and we give a result to help compute its Gr "obner basis. We exhibit an f whose associated Y include Fulton's matrix Schubert varieties, and recover much more easily the Gr "obner basis theorem of [Knutson-Miller '05]. We show that in Bott-Samelson coordinates on an opposite Bruhat cell X^v_ in G B, the f defining the complement of the big cell also has initial term i=1 ^n x_i, and hence the Kazhdan-Lusztig subvarieties X^v_ w degenerate to Stanley-Reisner schemes. This recovers, in a weak form, the main result of [Knutson '08].