Topological methods in moduli theory

TL;DR

This paper explores the topology and moduli spaces of K(H,1) varieties, including Bagnera-de Franchis and Inoue type varieties, analyzing their deformation classes, automorphisms, and Galois actions to deepen understanding of their geometric and arithmetic properties.

Contribution

It introduces new results on the moduli spaces of K(H,1) varieties, especially surfaces isogenous to a product, and studies Galois actions on these moduli spaces.

Findings

Moduli spaces of algebraic curves and Abelian varieties are rational K(H,1)'s.

New results on the deformation classes of Inoue type varieties.

Galois automorphisms can alter the fundamental group of surfaces, affecting their moduli components.

Abstract

One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)'s, through Teichmueller theory. The main thrust of the paper is to show how in the case of K(H,1)'s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Loenne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphisms of order different from 2 there is a surface S such that the Galois conjugate surface of S has fundamental group not isomorphic to the one of S.