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.
In the symplectic-topological setting, Mikhalkin @cite_5 constructed a degeneration of a projective algebraic hypersurface using a triangulation of its Newton polytope to provide a higher-dimensional pair-of-pants'' decomposition. He further identified a stratified torus fibration over the spine of the corresponding amoeba. This viewpoint was first applied to homological mirror symmetry ( HMS'') by Abouzaid @cite_11 . Mikhalkin's construction and perspective inform the current work greatly, even though our route from HMS is a bit top-down.'' We describe it here.
@cite_5 @cite_11
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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. We prove a homological mirror symmetry equivalence between an @math -brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a @math -brane category for a mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the @math -model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces. 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. Abstract It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants . Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n -dimensional pair-of-pants is diffeomorphic to CP n minus n +2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n -dimensional polyhedral complex as its base and a real n -torus as its fiber. The base accommodates the geometric genus of a hypersurface V . Its homotopy type is a wedge of h n , o ( V ) spheres S n .