The uniformization of the moduli space of principally polarized abelian 6-folds

Abstract

Abstract Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space 𝒜 6 \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 E_{6} lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E 6 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 𝒜 6 \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.