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Combinatorial patchworking: back from tropical geometry

Erwan Brugallé, Lucía López de Medrano, Johannes Rau

Abstract

We show that, once translated to the dual setting of convex triangulations of lattice polytopes, results and methods from previous tropical works by Arnal-Renaudineau-Shaw, Renaudineau-Shaw, Renaudineau-Rau-Shaw, and Jell-Rau-Shaw extend to non-convex triangulations. So, while the translation of Viro's patchworking method to the setting of tropical hypersurfaces has inspired several tremendous developments over the last two decades, we return to the the original polytope setting in order to generalize and simplify some results regarding the topology of $T$-submanifolds of real toric varieties.

Combinatorial patchworking: back from tropical geometry

Abstract

We show that, once translated to the dual setting of convex triangulations of lattice polytopes, results and methods from previous tropical works by Arnal-Renaudineau-Shaw, Renaudineau-Shaw, Renaudineau-Rau-Shaw, and Jell-Rau-Shaw extend to non-convex triangulations. So, while the translation of Viro's patchworking method to the setting of tropical hypersurfaces has inspired several tremendous developments over the last two decades, we return to the the original polytope setting in order to generalize and simplify some results regarding the topology of -submanifolds of real toric varieties.
Paper Structure (22 sections, 24 theorems, 153 equations, 7 figures)

This paper contains 22 sections, 24 theorems, 153 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Betti numbers of $T$-manifolds
  3. Hodge numbers, Poincaré duality, and Heredity
  4. Real phase structures
  5. Preliminaries
  6. Poset homology
  7. The cosheaves $\mathcal{F}_p^k$
  8. Homology of non-singular toric varieties
  9. Homology of $\mathcal{F}_p^k$
  10. Heredity
  11. Poincaré duality
  12. Geometric realization of $\Xi$
  13. The barycentric subdivision
  14. Cohomology and Mayer-Vietoris
  15. The cap product
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\Delta\subset \mathbb{R}^n$ be a non-singular lattice polytope, let $\Gamma$ be a unimodular triangulation of $\Delta$, and let $\varepsilon$ be a sign distribution on $\Delta$. Then one has

Figures (7)

  • Figure 1: Example of patchworking of $6\Delta_2$; the sign distribution $\varepsilon$ takes value 0 on $\bullet$-points , and 1 on $\bullet$-points.
  • Figure 2: A $T$-line in $\mathbb{R}\mathbf{P}^3$.
  • Figure 3: The spectral sequence for the filtration $F_i = \bigoplus_{r,s \; : \; s \leqslant i} E^0_{r,s}$
  • Figure 4: The spectral sequence for the filtration $F'_i = \bigoplus_{r,s \; : \; r \leqslant i} E^0_{r,s}$
  • Figure 5: The subdivision of $\Delta_2$ induced by the cells $C(F,\sigma)$.
  • ...and 2 more figures

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Definition 1.7
  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3: For the attention of tropical experts
  • ...and 34 more