On Kodaira fibrations with invariant cohomology

Bregman, Corey

doi:10.2140/gt.2021.25.2385

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ISSN (electronic): 1364-0380

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On Kodaira fibrations with invariant cohomology

Corey Bregman

Geometry & Topology 25 (2021) 2385–2404

DOI: 10.2140/gt.2021.25.2385

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 $ℚ$ –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

Volume 25, issue 5 (2021)