On the Chow ring of very general abelian varieties and a question of Pirola

Technical Summary of "On the Chow ring of very general abelian varieties and a question of Pirola" by Claire Voisin

Problem Statement
The paper addresses two interconnected problems in the arithmetic and geometry of abelian varieties and Jacobians.

1. Chow Ring Structure: The author investigates the set of divisors $D \in \text{CH}^1(A)_{\text{hom}}$ on a very general abelian variety $A$ of dimension $g \geq 4$ such that the self-intersection $D^2$ vanishes in $\text{CH}^2(A)$. Previous work established that for $g \geq 3$, the locus of such divisors has dimension 0, but the question remained whether these divisors are necessarily torsion points.
2. Pirola's Question: The paper seeks to characterize the rational sections of the Kummer fibration $K = J/\pm \text{Id} \to \mathcal{M}_4$, where $J \to \mathcal{M}_4$ is the universal Jacobian fibration over the moduli space of genus 4 curves. Specifically, it addresses a conjecture by Pirola stating that any such rational section must be a multiple of the Griffiths-Pirola normal function $\gamma_P$, which arises from the difference of the two trigonal divisors on a genus 4 curve.

Methodology
The proofs rely heavily on the theory of Griffiths infinitesimal invariants of normal functions, extended to higher codimensional cycles. The core strategy involves analyzing the vanishing of these invariants to deduce global properties of sections.

1. Infinitesimal Invariants and Spectral Sequences: The author utilizes the infinitesimal invariant $\delta_Z(m)$ associated with a cycle $Z$. For a divisor $D$, the invariant of the square $D^2$ is related to the square of the invariant of $D$, i.e., $\delta_{D^2} = (\delta_D)^2$. The paper establishes the degeneracy of specific spectral sequences (Proposition 2.5) to rigorously define these invariants for families of abelian varieties.
2. Algebraic Analysis of the Square Map: A central technical component is the analysis of the map $M^2$ induced by the square operation on the space of infinitesimal invariants.
   • For the moduli space of abelian varieties $\mathcal{A}_g$ ($g \geq 4$), the space of invariants is identified with a quotient of $\text{Hom}(V, \text{Sym}^2 V)$.
   • For the moduli space of curves $\mathcal{M}_4$, the space is identified with a quotient involving $V_2 = \text{Sym}^2 V / q$, where $q$ is the quadratic form defining the canonical embedding of a genus 4 curve.
   • Proposition 3.1 proves that if the square of an invariant vanishes (or satisfies a specific proportionality relation), the invariant itself must be of rank $\leq 1$ (or proportional to the Griffiths-Pirola invariant). This relies on linear algebraic arguments involving restrictions to 2-dimensional subspaces and the geometry of the Grassmannian.
3. Ax-Shanuel Type Arguments: Once a local lift of a normal function is shown to have a rank 1 differential (or a specific algebraic structure), the author employs results of Ax-Shanuel type (citing [6], [1], [2]) and properties of monodromy groups (specifically that the image is Zariski dense in the symplectic group). These arguments force the normal function to be horizontal (i.e., its derivative vanishes), which implies the section is torsion.
4. Descent and Cohomology: For the Kummer fibration problem, the author uses the fact that while a section $\gamma$ is only defined up to sign, its square $D_\gamma^2$ is well-defined. By comparing $D_\gamma^2$ with the square of the Griffiths-Pirola section $D_{\gamma_P}^2$ (using Theorem 4.1 which states the Chow group of the generic fiber is 1-dimensional), the author reduces the problem to the linear relation of infinitesimal invariants derived in Section 3.

Key Results

• Theorem 1.2:

  • (i) For a very general abelian variety $A$ of dimension $g \geq 4$, if a divisor $D \in \text{Pic}^0(A)$ satisfies $D^2 = 0$ in $\text{CH}^2(A)$, then $D$ is a torsion point.
  • (ii) The same result holds for a very general Jacobian of a genus 4 curve.
  • Corollary: This implies that the set of points $x \in A$ such that $({x} - \{0_A\})^{*2} = 0$ in $\text{CH}^0(A)$ consists exactly of the torsion points.

• Theorem 1.8 (Pirola's Conjecture):

  • All rational sections of the Kummer fibration $K = J/\pm \text{Id} \to \mathcal{M}_4$ are rational multiples of the Griffiths-Pirola normal function $\gamma_P$.
  • Specifically, any rational section $\gamma$ satisfies $N\gamma = N'\gamma_P$ for integers $N, N'$ with $N \neq 0$.

Significance and Claims
The paper claims to resolve the question of whether the vanishing of the square of a divisor in the Chow ring of a very general abelian variety implies the divisor is torsion, a question left open for dimensions $g \geq 4$. The author notes that while the result is likely true for all $g \geq 4$, the proof currently requires $g \geq 4$ (and specifically $g=4$ for the Jacobian case) due to the specific linear algebraic properties of the infinitesimal invariants in these dimensions.

Furthermore, the paper provides a definitive answer to Pirola's question regarding the uniqueness of the Griffiths-Pirola section. It establishes that the "extra" rational sections one might expect on the Kummer fibration (which is a quotient of the Jacobian) do not exist; the only rational sections are those generated by the canonical geometric construction involving the two trigonal pencils of genus 4 curves.

The work connects the study of the Chow ring structure (specifically the Pontryagin powers of 0-cycles) with the theory of normal functions and their infinitesimal invariants, demonstrating that the non-vanishing of these invariants for the Griffiths-Pirola section is the key obstruction to the existence of other rational sections. The author explicitly states that the method used for Theorem 1.2 could potentially be extended to study higher powers $D^k = 0$, leading to a conjecture that for $g \geq 2k$, the only divisors with $D^k=0$ are torsion divisors.