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Matroids and the integral Hodge conjecture for abelian varieties

Philip Engel, Olivier de Gaay Fortman, Stefan Schreieder

Abstract

We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic threefold. This disproves the integral Hodge conjecture for abelian varieties and shows that very general cubic threefolds are not stably rational. Our proof is motivated by tropical geometry; it relies on multivariable Mumford constructions, monodromy considerations, and the combinatorial theory of matroids.

Matroids and the integral Hodge conjecture for abelian varieties

Abstract

We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic threefold. This disproves the integral Hodge conjecture for abelian varieties and shows that very general cubic threefolds are not stably rational. Our proof is motivated by tropical geometry; it relies on multivariable Mumford constructions, monodromy considerations, and the combinatorial theory of matroids.

Paper Structure

This paper contains 61 sections, 65 theorems, 219 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Integral Hodge conjecture for abelian varieties
  3. Cubic threefolds
  4. Algebraic curves on matroidal families
  5. Obstruction and combinatorial results
  6. Outline
  7. Outline of the proof of Theorem \ref{['thm:reduction-to-combinatorics-intro']}
  8. Outline of the proof of Theorem \ref{['thm:splitting-in-cographic=cographic-intro']}
  9. Acknowledgements
  10. Preliminaries
  11. Conventions
  12. Schemes and analytic spaces
  13. Linear algebra
  14. Graphs
  15. Cell complexes
  16. ...and 46 more sections

Key Result

Theorem 1.1

Let $(X,\Theta)$ be a very general principally polarized abelian variety of dimension $g\geq 4$. Let $Z\in {\rm CH}^c(X)$ be an algebraic cycle of codimension $2\leq c\leq g-1$. Then $[Z]=m\cdot [\Theta]^{c}/c!\in H^{2c}(X,{\mathbb Z})$ with $m$ even.

Figures (3)

  • Figure 1: Left: Cube $\Gamma(c_p)$ of the dual complex $\Gamma(Y_p)$. Right: Cuboid refinement $\Gamma(c_p')$ of this cube, corresponding to the fiber of the blowup $Y_p'\to Y_p$. Map $\varphi_p\colon \Gamma(c_p')\to \Gamma(c_p)$ is depicted by arrows.
  • Figure 2: Left: Cube $\Gamma(c_p)$ of $\Gamma(Y_p)$. Middle: Dual complex $\Gamma(c_p')$ in the fiber $Y_p'\to Y_p$ of the resolution with colorless (grey) diagonal edge. Right: Subdivision $\hat{\Gamma}^1(c_p')$ of $\Gamma^1(c_p')$ and its coloring. Map $\varphi_p\colon \hat{\Gamma}^1(c_p')\to \Gamma^1(c_p)$ of colored graphs is depicted by arrows.
  • Figure 3: Depiction of the map $h\colon [0,2^2]^2\to [0,2^2]^2$ (i.e., $\ell=j=|S|=2$).

Theorems & Definitions (194)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 184 more