Skeleta of Affine Hypersurfaces

Abstract

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.

41 pages, 3 figure

References in corpus (6)

• A categorification of Morelli's theorem
• The Coherent-Constructible Correspondence for Toric Deligne-Mumford Stacks
• Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces
• Ribbon Graphs and Mirror Symmetry I
• Polytopes and Skeleta
• Affine structures and non-archimedean analytic spaces

Cited by in corpus (12)

• Mirror symmetry for very affine hypersurfaces
• Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
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• Mirror symmetry for the Landau-Ginzburg A-model $M={\mathbb{C}}^n,W=z_1\cdots z_n$
• Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes
• Phase tropical hypersurfaces
• An Introduction to Hodge Structures
• Perverse Curves and Mirror Symmetry
• Real loci in (log-) Calabi-Yau manifolds via Kato-Nakayama spaces of toric degenerations
• A Lagrangian filling for every cluster seed
• Topological torus fibrations on Calabi--Yau manifolds via Kato--Nakayama spaces
• Variation of GIT and Variation of Lagrangian Skeletons I: Flip and Flop