| Résume | Same access code. Send a mail to O. Debarre or F. Han or A. Höring to get it.
I will discuss research done in collaboration with J.C. Naranjo and G.P. Pirola on the subsets V_k (A) of a complex abelian variety A consisting of the points x ∈ A such that the zero-cycle {x} − {0_A } is k-nilpotent with respect to the Pontryagin product in CH_0 (A). These sets were introduced by Voisin in Chow rings and gonality of general abelian varieties, Ann. H. Lebesgue, I (2018). She showed that dim V_k (A) ≤ k − 1 and this dimension is zero for a general abelian variety of dimension at least 2k − 1.
We study in particular the locus V_{g,2} in the moduli space of abelian varieties of dimension g with a fixed polarization of the A for which V_2 (A) is positive dimensional and we prove
Theorem Let g ≥ 3 and consider an irreducible subvariety Y ⊂ V g,2 such that for a very general A ∈ Y there is a curve in V_2 (A) generating A. Then dim Y ≤ 2g − 1. The hyperelliptic locus shows that this bound is sharp. |
