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

Abstract

Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space \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 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 \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

“…Proposition 2.14(iv)), so G acts by biholomorphic covering maps of the finite, étale, holomorphic covering map p ′ . By [44, Théorème 5.1 (1)] the group G acts by covering automorphisms of the finite, étale morphism of algebraic varieties…”

“…Besides the well-developed theory of Prym varieties not much is known in this direction. Recently Alexeev, Donagi, Farkas, Izadi and Ortega proved in [1] that every sufficiently general principally polarized abelian variety of dimension 6 is isomorphic to a Prym-Tyurin variety of a cover of P 1 of degree 27, branched in 24 points, with monodromy group W (E 6 ) ⊂ S 27 . The unirationality of the moduli spaces of three-dimensional abelian varieties A 3 (1, 1, d) and A 3 (1, d, d) with d ≤ 4 was proved in [28,29] by means of families of simply ramified covers of elliptic curves of degree d branched in 6 points.…”

“…He worked with the following type of families of étale morphisms to P 1 . The variety S is an étale cover of an open subset of P (n) 1 , the families are étale Galois covers p ′ : X ′ → P 1 × S \ B with Galois group isomorphic to G, where B is the preimage of the universal divisor A ⊂ P 1 × P (n) 1 . One of his results is that the Hurwitz space H G n (P 1 , y 0 ) is a fine moduli variety for the category of this particular type of families of pointed étale G-morphisms to (P 1 , y 0 ) [10, Théorème 6 and § 7.1].…”

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

“…Since it has been shown in [ADFIO,Theorem 6.17] that the Hodge class λ on Hur can be expressed in terms of boundary divisors, Theorem 1.1 can be rewritten using only D 0 , D syz and D azy and one has the following identity on Hur:…”

“…where L = π * O P 1 (1) ∈ W 1 27 (C). By [ADFIO,Theorem 9.2], the map μ(L) is injective for a general point of Hur. Furthermore, it factors through the (+1)-eigenspace, that is, one has a map…”