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

AbstractStarting from a beautiful idea of Kanev, we construct a uniformization of the moduli space?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 theE6E_{6}lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of suchE6E_{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?6\mathcal{A}_{6}in the terms of syzygies of the Abel–Prym–Tyurin curve