Topological methods in moduli theory
Fabrizio Catanese (Universitaet Bayreuth)

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…
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