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

2015/07/21 by Valery Alexeev, Alexeev, Valery, Ron Donagi +7
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1507.05710

openalex publication_date 2015/07/21 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Starting from a beautiful idea of Kanev, we construct a uniformization of the\nmoduli space A₆ of principally polarized abelian 6-folds in terms of curves\nand monodromy data. We show that the general ppav of dimension 6 is a\nPrym-Tyurin variety corresponding to a degree 27 cover of the projective line\nhaving monodromy the Weyl group of the E₆ lattice. Along the way, we establish\nnumerous facts concerning the geometry of the Hurwitz space of such E₆-covers,\nincluding: (1) a proof that the canonical class of the Hurwitz space is big,\n(2) a concrete geometric description of the Hodge-Hurwitz eigenbundles with\nrespect to the Kanev correspondence and (3) a description of the ramification\ndivisor of the Prym-Tyurin map from the Hurwitz space to A₆ in the terms of\nsyzygies of the Abel-Prym-Tyurin curve.\n

Related

• Cycles on the Moduli Space of Abelian Varieties
• The fibres of the Prym map of étale cyclic coverings of degree 7
• Polarizations of Prym varieties for Weyl groups via abelianization
• Moduli of (1,7)-polarized abelian surfaces via syzygies
• Optimality of the Prym-Tyurin construction for A₆