The Classification of Hyperelliptic Groups in Dimension 4 (Lecture Notes of the Unione Matematica Italiana, 32)

This book explores the geometry of hyperelliptic manifolds, a higher-dimensional generalization of classical hyperelliptic surfaces.Hyperelliptic surfaces, historically classified by Enriques, Severi, and Bagnera-de Franchis, are compact complex surfaces with Kodaira dimension zero, geometric genus zero, and irregularity one.Moreover, their canonical divisor K is torsion: indeed, 12K is trivial.This monograph extends these ideas to complex tori of arbitrary dimension, quotienting complex tori by finite groups acting freely and without translations.Focusing on the classification of hyperelliptic manifolds, the book presents new results in dimension four, completing a key step that had remained largely unexplored.Using methods from group theory, representation theory, and computer algebra, it identifies all finite groups that admit free and translation-free actions on four-dimensional complex tori.The work also investigates the torsion order of the canonical divisor for hyperelliptic man