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On Kodaira fibrations with invariant cohomology

Corey Bregman

Geometry & Topology 25 (2021) 2385–2404
Abstract

A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant –cohomology in degree 1, proving that if the dimension of the holomorphic invariants is 1 or 2, then X admits a branch covering over a product of curves inducing an isomorphism on rational cohomology in degree 1. We also study the class of Kodaira fibrations possessing a holomorphic section, and demonstrate that having a section imposes no restriction on possible monodromies.

Keywords
Kodaira fibration, surface bundle, Albanese variety, monodromy
Mathematical Subject Classification 2010
Primary: 14D06, 14H15, 32Q15
Secondary: 14F40, 14J29, 20F34, 57M50
References
Publication
Received: 18 April 2019
Revised: 7 August 2020
Accepted: 8 September 2020
Published: 3 September 2021
Proposed: Jim Bryan
Seconded: Dan Abramovich, Benson Farb
Authors
Corey Bregman
Department of Mathematics and Statistics
University of Southern Maine
Portland, ME
United States
https://sites.google.com/view/cbregman