#### Volume 25, issue 5 (2021)

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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 $ℚ$–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