Divisors in the moduli space of Debarre-Voisin varieties

Vladimiro Benedetti (1) , Jieao Song (2)

1 I2M - Institut de Mathématiques de Marseille
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

Let σ be a non-zero alternating 3-form on a 10-dimensional complex vector space V. One can define several associated degeneracy loci: the Debarre-Voisin variety X_6 in Gr(6, 10), the Peskine variety X_1 in P^9, and the hyperplane section X_3 in Gr(3, 10). Their interest stems from the fact that the Debarre-Voisin varieties form a locally complete family of projective hyperkähler fourfolds of K3^[2]-type. We prove that when smooth, the varieties X_6, X_1, and X_3 share one same integral Hodge structure, and that X_1 and X_3 both satisfy the integral Hodge conjecture in all degrees. This is obtained as a consequence of a detailed analysis of the geometry of these varieties along three divisors in the moduli space. On one of the divisors, an associated K3 surface S of degree 6 can be constructed geometrically and the Debarre-Voisin fourfold is shown to be isomorphic to a moduli space of twisted sheaves on S, in analogy with the case of cubic fourfolds containing a plane.

Domaines: Géométrie algébrique [math.AG]

Vladimiro Benedetti, Jieao Song. Divisors in the moduli space of Debarre-Voisin varieties. 2021. ⟨hal-03259123⟩