Compact tori associated to hyperkaehler manifolds of Kummer type

Abstract

Let X be a hyperkaehler manifold of Kummer type. We study the 4 dimensional intermediate Jacobian J^3(X) constructed out of the 3rd cohomology of X, motivated by the desire to understand the Kuga-Satake torus associated to X. We prove that there is an embedding of Hodge structures of H^2(X) into H^2(J^3(X)). It follows that if X is projective with polarization L, the projective Kuga-Satake torus of (X,L) is isogenous to the fourth power of J^3(X). By studying the cohomology ring of X, we find that there is a natural smooth quadric Q(X) in the projectivization of H^3(X), with a natural choice of one of the two irreducible components of the variety parametrizing maximal linear subspaces of Q(X). Let Q^{+}(X) be the chosen irreducible component; then it is a smooth quadric in the projectivization of S^{+}(X), one of the two spinor representations of Spin H^3(X), where H^3(X) is equipped with the quadratic form whose zero locus is Q(X). 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). This amounts to an effective description of the intermediate Jacobian J^3(X) in terms of the H.S. on H^2(X). A simple consequence is a new proof of a result of Mongardi giving (necessary) conditions for 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.