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Tropical contractions to integral affine manifolds with singularities

Yuto Yamamoto

Abstract

We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. One can associate two types of tropical spaces with it. One is a tropical variety obtained by tropicalization. The other one is an integral affine manifold with singularities, which arises as the dual intersection complex of the toric degeneration. In this article, we show that the latter is contained in the former as a subset, and construct an integral affine contraction map from the former to the latter. We also show that the contraction preserves tropical cohomology groups, and sends the eigenwave to the radiance obstruction.

Tropical contractions to integral affine manifolds with singularities

Abstract

We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. One can associate two types of tropical spaces with it. One is a tropical variety obtained by tropicalization. The other one is an integral affine manifold with singularities, which arises as the dual intersection complex of the toric degeneration. In this article, we show that the latter is contained in the former as a subset, and construct an integral affine contraction map from the former to the latter. We also show that the contraction preserves tropical cohomology groups, and sends the eigenwave to the radiance obstruction.

Paper Structure

This paper contains 26 sections, 59 theorems, 392 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Tropical geometry
  3. Convex geometry
  4. Integral affine structures with singularities
  5. Radiance obstructions
  6. Tropical (co)homology groups and wave groups
  7. Tropical toric varieties and tropicalizations
  8. Stable intersections of tropical hypersurfaces
  9. Local models of tropical contractions
  10. Local models of IAMS
  11. Constructions of local models of tropical contractions
  12. Examples of local models of tropical contractions
  13. Proof of th:local-cont.2
  14. Proof of th:local-cont.3 and th:local-cont.4
  15. Proof of th:local-cont.1
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $B$ be a quasi-simple IAMS. Then for any point $x \in B$, there exists a neighborhood $U_x \subset B$ of $x$, a local model of tropical contractions $\delta \colon V \to U$, and a homeomorphism from $U_x$ to $U$, which is an integral isomorphism on the complement of the singular loci. Furthermor

Figures (8)

  • Figure 3.1: The tropical hypersurface $X(f_1)$ and the subset $B$
  • Figure 3.2: The polyhedral complex $\mathscr{P}$, its subdivision $\widetilde{\mathscr{P}}$, and the subsets $W_{v_1}^\circ, W_{v_2}^\circ, W_{\xi}^\circ$
  • Figure 3.3: The subsets $V_{v_1} \cap \sigma, V_{v_2} \cap \sigma, V_{\xi} \cap \sigma$ and the contractions $\delta_{v_1}, \delta_{v_2}, \delta_{\xi}$
  • Figure 3.4: A contraction of a tropical curve of valence $4$ to a line.
  • Figure 5.1: A tropical contraction of the tropical hypersurface $X(f_1)$
  • ...and 3 more figures

Theorems & Definitions (155)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Proposition 2.9
  • ...and 145 more