Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 4

Special issue celebrating the work of Herb Clemens

Guest Editor: Ron Donagi

Hodge classes on the moduli space of $W(E_6)$-covers and the geometry of $\mathcal{A}_6$

Pages: 1211 – 1263

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a1

Authors

Valery Alexeev (Department of Mathematics, University of Georgia, Athens, Ga., U.S.A.)

Ron Donagi (Department of Mathematics, University of Pennsylvania, Philadelphia, Pa., U.S.A.)

Gavril Farkas (Institut für Mathematik, Humboldt-Universität zu Berlin, Germany)

Elham Izadi (Department of Mathematics, University of California, San Diego, Calif., U.S.A.)

Angela Ortega (Institut für Mathematik, Humboldt-Universität zu Berlin, Germany)

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 $\mathcal{A}_6$ 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 $\overline{\mathcal{A}}_6$. In the final part of the paper, we present an alternative, elementary proof of our uniformization result on $\mathcal{A}_6$ via Prym–Yurin varieties of type $W(E_6)$.

Received 25 July 2021

Received revised 7 April 2022

Accepted 20 April 2022

Published 25 October 2022