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

Valery Alexeev, Ron Donagi, Gavril Farkas, Elham Izadi, Angela Ortega

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).

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

Paper Structure

This paper contains 27 sections, 30 theorems, 137 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The Weyl group of $E_6$ and the uniformization of ${\mathcal{A}}_6$.
  3. The group $\bm{W(E_6)}$ and its representations
  4. Maximal subgroups of $W(E_6)$
  5. Three versions of compactified Hurwitz spaces of $W(E_6)$-covers
  6. The dominance of the Prym-Tyurin map
  7. Boundary divisors on the Hurwitz space
  8. Properties of the rational map $\bm{PT}$
  9. The Hodge classes $\bm{\lambda, \lambda^{(-5)}, \lambda^{(+1)}}$
  10. Twenty five fundamental Hodge bundles on ${\overline{\rm Hur}}$.
  11. ${\overline{\rm Hur}}$ as a moduli space of Galois admissible covers
  12. Computation of Hodge classes on ${\overline{\mathcal{H}}}$.
  13. Prym-Tyurin varieties via Galois covers
  14. Computing the 25 fundamental Hodge classes
  15. The Weyl-Petri divisor and the ramification of the Prym-Tyurin map
  16. ...and 12 more sections

Key Result

Theorem 1.1

The class of the $(-5)$-Hodge eigenbundle on $\widetilde{\rm Hur}$ is given by the following formula:

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Definition 2.7
  • ...and 54 more