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.
A skeleton for Fermat hypersurfaces was described by Deligne in [pp. 88--90] Deligne , and this skeleton is visible in our own in a manner described in Remark . Our skeleta'' are different than the skeleta'' that appear in nonarchimedean geometry @cite_10 @cite_15 , but @math plays a similar role in both constructions. It would be interesting to study this resemblance further.
@cite_15 @cite_10
2154785235 1974286590
In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of the Homological Mirror conjecture and the geometric Strominger-Yau-Zaslow conjecture. In particular, we glue from “flat pieces” an analytic K3 surface. As a byproduct of our approach we obtain an action of an arithmetic subgroup of the group SO(1, 18) by piecewise-linear transformations on the two-dimensional sphere S 2 equipped with naturally defined singular affine structure. 0 Introduction 293 1 Piecewise RS -linear spaces 298 2 R-colored polysimplicial sets 308 3 R-colored polysimplicial sets of length l 313 4 The skeleton of a nondegenerate pluri-stable formal scheme 327 5 A colored polysimplicial set associated with a nondegenerate poly-stable fibration 336 6 p-Adic analytic and piecewise linear spaces 346 7 Strong local contractibility of smooth analytic spaces 355 8 Cohomology with coefficients in the sheaf of constant functions 362