Compact Tori Associated to Hyperkähler Manifolds of Kummer Type

Dedicato alla piccola Mia. For $X$ a hyperkähler manifold of Kummer type, let $J^3(X)$ be the intermediate Jacobian associated to $H^3(X)$. We prove that $H^2(X)$ can be embedded into $H^2(J^3(X))$. We show that there exists a natural smooth quadric $Q(X)$ in the projectivization of $H^3(X)$, such that Gauss–Manin parallel transport identifies the set of projectivizations of $H^{2,1}(Y)$, for $Y$ a deformation of $X$, with an open subset of a linear section of $Q^{+}(X)$, one component of the variety of maximal linear subspaces of $Q(X)$. We give a new proof of a result of Mongardi restricting the action of monodromy on $H^2(X)$. Lastly, we show that if $X$ is projective, then $J^3(X)$ is an abelian fourfold of Weil type.