On Hyperelliptic Manifolds

Abstract

Hyperelliptic surfaces arise classically in the Enriques-Kodaira classification of compact complex surfaces as the surfaces S, which are uniquely determined through the invariants kod(S) = 0, p_g(S) = 1, q(S) = 0 and 12K_S \equiv 0. Due to the work of Enriques-Severi and Bagnera-de Franchis, these surfaces are very well understood and are all isomorphic to the quotient of an Abelian surface A by a non-trivial finite group G, which acts freely on A and contains no translations. They showed that A is isogenous to a product of two elliptic curves, which allows an explicit classification of hyperelliptic surfaces. In particular, hyperelliptic surfaces are always projective. In the '90s, Herbert Lange studied higher-dimensional analogues of hyperelliptic surfaces and in 1999, he published an article, which is dedicated to the classification of projective hyperelliptic threefolds. As it turns out, Lange's classification is incomplete, and in collaboration with Fabrizio Catanese, we describe the missing case(s) of Lange's classification. More precisely, we prove the existence of a unique complete $2$-dimensional hyperelliptic threefolds A/D_4, where D_4 is the dihedral group of order 8. Motivated by the 3-dimensional case, we decided to investigate in this thesis the case of dimension 4 in more detail as well. Using group-theoretic methods, we work out the list of exactly those abstract finite groups, which admit an embedding in the group of biholomorphic self-maps of some Abelian fourfold A in such a way that the image contains no translations and acts freely on A. We will say that such a group is associated with a hyperelliptic fourfold. The question if there exist complete families of hyperelliptic threefolds (or, more generally, hyperelliptic manifolds of arbitrary dimension), which do not contain a projective manifold, remained open in Lange's article. This is studied in more detail in this thesis: we show, together with Fabrizio Catanese and Benoit Claudon, that every hyperelliptic manifold admits arbitrarily small deformations which are projective. Furthermore, we discuss in detail a special case of this result, namely the case, in which the group action on the complex torus is rigid: in this case, we construct explicitly a polarization on the complex torus coming from a direct sum of Hodge structures on CM-fields. This is a result obtained by Torsten Ekedahl around 1999.%%%%In der Enriques-Kodaira Klassifikation kompakter komplexer Flachen treten minimale hyperelliptische Flachen klassisch als diejenigen Flachen S auf, die durch die Invarianten kod(S) = 0, p_g(S) = 0, q(S) = 1 und 12K_S \equiv 0 eindeutig festgelegt sind. Durch die Arbeit von Enriques-Severi und Bagnera-de Franchis sind diese Flachen sehr gut verstanden und sind allesamt isomorph zu Quotienten einer abelschen Flache A nach der Wirkung einer nicht-trivial endlichen Gruppe G, die frei auf A operiert und keine Translationen enthalt. Es lasst sich zeigen, dass A isogen zu einem Produkt zweier elliptischer Kurven ist, was eine explizite Klassifikation…