Hodge classes on the moduli space of W(E_6)-covers and the geometry of A_6

TL;DR

This paper studies the Hodge classes on the Hurwitz space of W(E_6)-covers and explores their implications for the geometry and intersection theory of the moduli space A_6 of abelian 6-folds, providing new insights and proofs.

Contribution

It determines the 25 Hodge classes on the Hurwitz space associated with W(E_6) and offers an elementary proof of the uniformization of A_6 via Prym-Tyurin varieties.

Findings

Identified the 25 Hodge classes corresponding to irreducible representations of W(E_6)

Connected Hodge classes to intersection theory on A_6

Provided an elementary proof of the uniformization of A_6

Abstract

In previous work we showed that the Hurwitz space of W(E_6)-covers of the projective line branched over 24 points dominates via the Prym-Tyurin map the moduli space A_6 of principally polarized abelian 6-folds. Here we determine the 25 Hodge classes on the Hurwitz space of W(E_6)-covers corresponding to the 25 irreducible representations of the Weyl group W(E_6). This result has direct implications to the intersection theory of the toroidal compactification A_6. In the final part of the paper, we present an alternative, elementary proof of our uniformization result on A_6 via Prym-Tyurin varieties of type W(E_6).