Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space \mathcal{A}_{6} of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general principally polarized abelian variety of dimension 6 is a Prym–Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_{6} lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_{6}-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge–Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym–Tyurin map from the Hurwitz space to \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.

As in the case of irreducible holomorphic symplectic manifolds, the period domain Compl of compact complex tori of even dimension 2n contains twistor lines. These are special 2-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a generic chain of twistor lines. Furthermore, we show that twistor lines are holomorphic submanifolds of Compl, of degree 2 in the Plücker embedding of Compl.

A Hodge structure V of weight k on which a CM field acts defines, under certain conditions, a Hodge structure of weight k − 1, its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces. We use our results to prove the existence of a Kuga-Satake correspondence for certain cubic 4-folds.

No abstract

No abstract