The Classification of Hyperelliptic Groups in Dimension 4

This book explores the geometry of hyperelliptic manifolds, a higher-dimensional generalization of classical hyperelliptic surfaces.

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 manifolds in dimension at most five. The text includes detailed proofs, some of which are complemented by the computer algebra system GAP. The book also highlights connections with related topics such as Iitaka’s conjecture, and complex Bieberbach groups, situating hyperelliptic manifolds within broader contexts in algebraic geometry. Designed for researchers interested in group actions on complex tori, this monograph provides both a comprehensive reference and a roadmap for further exploration. By combining classical theory with modern computational methods, it offers new insights into the structure and classification of higher-dimensional hyperelliptic manifolds.