Some divisors in the moduli space of Debarre-Voisin varieties

Abstract

Let $V_{10}$ be a 10-dimensional complex vector space and let $\sigma \in {\bigwedge }^3{V}_{10}^{\vee }$ be a non-zero alternating 3-form. One can define several associated degeneracy loci: the Debarre-Voisin hyperkähler variety $X_6^{\sigma }\subset \mathrm{Gr}(6,V_{10})$, the Peskine variety $X_1^{\sigma }\subset \mathbf{P}(V_{10})$, and the hyperplane section $X_3^{\sigma }\subset \mathrm{Gr}(3,V_{10})$. We prove that when smooth, the varieties $X_6^{\sigma }$, $X_1^{\sigma }$, and $X_3^{\sigma }$ share one common integral Hodge structure, and that $X_1^{\sigma }$ and $X_3^{\sigma }$ 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.

52 pages; revised version

References in corpus (1)

• Geometry of orbit closures for the representations associated to gradings of Lie algebras of types $E_7$

Cited by in corpus (1)

• A special Debarre-Voisin fourfold