Kodaira fibrations and beyond: methods for moduli theory

TL;DR

This survey explores Kodaira fibrations, their topological and algebraic properties, and recent advances in moduli theory, including new results on Galois coverings, deformations, and special algebraic surfaces.

Contribution

It provides a comprehensive overview of methods in moduli theory related to Kodaira fibrations, including recent results and open questions in the field.

Findings

New results on Galois coverings and deformations with Ingrid Bauer

Answer to Fujita's long-standing question on Variation of Hodge Structures

Discussion of algebraic surfaces like BCDH and Inoue-type surfaces

Abstract

Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. The paper contains some new results but is mostly a survey paper, dealing with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (new results obtained with Ingrid Bauer) projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance the BCDH surfaces, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces.