Chow ring and gonality of general abelian varieties

Voisin, Claire

Annales Henri Lebesgue, Tome 1 (2018), pp. 313-332

We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building on the work of Pirola, we show that very general abelian varieties of dimension $g$ have (covering) gonality at least $f\left(g\right)$, where $f\left(g\right)$ grows like $\log g$. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties $A$ of dimension $g$, e.g., if $g\ge 2k-1$, the set of divisors $D\in {\mathrm{Pic}}^0\left(A\right)$ such that $D^k=0$ in ${\mathrm{CH}}^k\left(A\right)$ is at most countable.

Nous étudions la gonalité des variétés abéliennes ainsi que leurs orbites de zéro-cycles pour l’équivalence rationnelle. Nous montrons que l’orbite d’un zéro-cycle de degré $k$ est de dimension au plus $k-1$. En développant des idées de Pirola, nous montrons qu’une variété abélienne très générale a une gonalité au moins égale à $f\left(g\right)$, où $f\left(g\right)$ croît comme $\log g$. Ceci répond à une question posée par Bastianelli, De Poi, Ein, Lazarsfeld et B. Ullery. Nous obtenons aussi des résultats sur l’anneau de Chow des variétés abéliennes $A$ de dimension $g$ ; par exemple, si $g\ge 2k-1$, l’ensemble des diviseurs $D\in {\mathrm{Pic}}^0\left(A\right)$ tels que $D^k=0$ dans ${\mathrm{CH}}^k\left(A\right)$ est au plus dénombrable.

Classification : 14K12, 14C25
Keywords: Abelian varieties, covering gonality, zero-cycles, Chow ring

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