  Let $V_{10}$ be a 10-dimensional complex vector space and let
$\sigma\in\bigwedge^3V_{10}^\vee$ be a non-zero alternating 3-form. One can
define several associated degeneracy loci: the Debarre-Voisin 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
$X_3^\sigma\subset \mathrm{Gr}(3,V_{10})$. Their interest stems from the fact
that the Debarre-Voisin varieties form a locally complete family of projective
hyperk\"ahler fourfolds of $\mathrm{K3}^{[2]}$-type. We prove that when smooth,
the varieties $X_6^\sigma$, $X_1^\sigma$, and $X_3^\sigma$ share one same
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.
