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Abelian varieties with no power isogenous to a Jacobian

Olivier de Gaay Fortman, Stefan Schreieder

TL;DR

The paper proves that for a very general curve ${X}$ of genus ${g\ge4}$ (including very general hyperelliptic cases), any isogeny ${\varphi:(JX)^k\to J C_1\times\cdots\times J C_n}$ to a product of Jacobians forces ${k=n}$ and ${C_i\simeq X}$; in particular, no positive power of a very general principally polarized abelian variety of dimension ${g\ge4}$, nor the intermediate Jacobian of a very general cubic threefold, is isogenous to a product of Jacobians. The argument blends degeneration to nodal, especially hyperelliptic, curves with intricate lattice-theoretic and Hodge-theoretic techniques: moving extension classes, Gauß maps, and variations of Hodge structure are combined with Kneser-Laguerre-type classification of integral inner product spaces and Lu–Zuo type Arakelov inequalities to constrain polarizations on powers. The results yield instances of the Coleman–Oort conjecture and provide progress on the integral Hodge conjecture for curve classes by ruling out decompositions into Jacobians in broad families. Overall, the work establishes a rigidity phenomenon for powers of Jacobians in high dimensions and highlights deep interplays between degeneration geometry, lattice theory, and Hodge theory in abelian varieties and Torelli-type questions.

Abstract

For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a product of Jacobians of curves. This confirms some cases of the Coleman-Oort conjecture. We further deduce from our results some progress on the question whether the integral Hodge conjecture fails for such abelian varieties.

Abelian varieties with no power isogenous to a Jacobian

TL;DR

The paper proves that for a very general curve of genus (including very general hyperelliptic cases), any isogeny to a product of Jacobians forces and ; in particular, no positive power of a very general principally polarized abelian variety of dimension , nor the intermediate Jacobian of a very general cubic threefold, is isogenous to a product of Jacobians. The argument blends degeneration to nodal, especially hyperelliptic, curves with intricate lattice-theoretic and Hodge-theoretic techniques: moving extension classes, Gauß maps, and variations of Hodge structure are combined with Kneser-Laguerre-type classification of integral inner product spaces and Lu–Zuo type Arakelov inequalities to constrain polarizations on powers. The results yield instances of the Coleman–Oort conjecture and provide progress on the integral Hodge conjecture for curve classes by ruling out decompositions into Jacobians in broad families. Overall, the work establishes a rigidity phenomenon for powers of Jacobians in high dimensions and highlights deep interplays between degeneration geometry, lattice theory, and Hodge theory in abelian varieties and Torelli-type questions.

Abstract

For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a product of Jacobians of curves. This confirms some cases of the Coleman-Oort conjecture. We further deduce from our results some progress on the question whether the integral Hodge conjecture fails for such abelian varieties.
Paper Structure (40 sections, 64 theorems, 196 equations)

This paper contains 40 sections, 64 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Outline of the argument
  3. Theorem \ref{['theorem:maintheorem1:hyperelliptic']} implies Theorem \ref{['thm:cor:maintheorem2']}.
  4. Sketch of the proof of Theorem \ref{['theorem:maintheorem1:hyperelliptic']}.
  5. Conventions
  6. Acknowledgements
  7. Preliminaries
  8. Isogenies between powers of abelian schemes and cohomology
  9. Gauß maps.
  10. Extensions of abelian varieties
  11. Extension classes of nodal curves
  12. Graph homology and extension classes.
  13. Extension classes of nodal hyperelliptic curves.
  14. Nodal degeneration and vanishing cycles
  15. Degenerations of hyperelliptic curves
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $Z \subset {\mathcal{M}}_g$ be an irreducible subvariety containing the hyperelliptic locus and let $X$ be a curve that defines a very general point in $Z$. Assume that there is an isogeny for some positive integers $k$ and $n$ and some smooth projective connected curves $C_1,\dots ,C_n$ of positive genus. If $g \geq 4$, then $k = n$ and there is an isomorphism $C_i \simeq X$ for each $i \in

Theorems & Definitions (151)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 141 more