On Kodaira fibrations with invariant cohomology

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 $\mathbb{Q}$–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.