Divisors in the moduli space of Debarre-Voisin varieties

2021/06/12 by Vladimiro Benedetti, Benedetti, Vladimiro, Jieao Song +1 · 1 citation
Mathematics · #14C30 #14J28 #14J42 #14J45 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2106.06859

openalex publication_date 2021/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V₁₀ be a 10-dimensional complex vector space and let σ∈\bigwedge³V₁₀^\vee be a non-zero alternating 3-form. One can define several associated degeneracy loci: the Debarre-Voisin variety X₆^σ\subsetGr(6,V₁₀), the Peskine variety X₁^σ\subsetP(V₁₀), and the hyperplane section X₃^σ⊂ Gr(3,V₁₀). 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 X₆^σ, X₁^σ, and X₃^σ share one same integral Hodge structure, and that X₁^σ and X₃^σ 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.

Citations

• Geometry of orbit closures for the representations associated to\n gradings of Lie algebras of types E₇

Cited by

• Peskine sixfolds and Debarre-Voisin fourfolds with associated cubic fourfolds

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