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Lefschetz section theorems for tropical hypersurfaces

Charles Arnal, Arthur Renaudineau, Kristin Shaw

Abstract

We establish variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of tropical hypersurfaces of toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which are compact or contained in $\mathbb{R}^n$ are torsion free. We prove a relationship between the coefficients of the $χ_y$ genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellular chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropical hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge-Deligne numbers of their complex counterparts.

Lefschetz section theorems for tropical hypersurfaces

Abstract

We establish variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of tropical hypersurfaces of toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which are compact or contained in are torsion free. We prove a relationship between the coefficients of the genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellular chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropical hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge-Deligne numbers of their complex counterparts.

Paper Structure

This paper contains 8 sections, 21 theorems, 140 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tropical toric varieties
  4. Tropical hypersurfaces
  5. Tropical homology
  6. Tropical Lefschetz hyperplane section theorem
  7. The tropical homology of hypersurfaces is torsion free
  8. Betti numbers of tropical homology and Hodge numbers

Key Result

Theorem 1.1

Let $X$ be a non-singular and combinatorially ample tropical hypersurface of an $n+1$ dimensional non-singular tropical toric variety $Y$. Then the map induced by inclusion is an isomorphism when $p+q < n$ and a surjection when $p +q = n$. If additionally, the tropical hypersurface $X$ has full-dimensional Newton polytope and the pair $(Y, X)$ is a cellular pair, then the map induced by inclusion

Figures (4)

  • Figure 1: The tropical projective plane ${\mathbb{T}P}^2$ on the left and its normal fan on the right.
  • Figure 2: The standard tropical hyperplane in ${\mathbb R}^3$ on the left its closure in the tropical toric variety described in Example on the right.
  • Figure 3: The tropical line $X$ in ${\mathbb{T}P}^2$ from Example .
  • Figure 4: A depiction of the polyhedral complexes $\gamma^o$ for two faces $\gamma$ from Example

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • Corollary 1.11
  • ...and 60 more