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

TL;DR

This paper constructs a uniformization of the moduli space of principally polarized abelian 6-folds using curves and monodromy data, revealing geometric and algebraic properties of related Hurwitz spaces and Prym-Tyurin varieties.

Contribution

It introduces a new uniformization approach for A_6 via E_6-covers and analyzes the geometry of associated Hurwitz spaces and Prym-Tyurin varieties.

Findings

Proved the canonical class of the Hurwitz space is big.

Provided a geometric description of Hodge-Hurwitz eigenbundles.

Described the ramification divisor of the Prym-Tyurin map in terms of syzygies.

Abstract

Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space A_6 of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general ppav 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 A_6 in the terms of syzygies of the Abel-Prym-Tyurin curve.