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 with nontrivial invariant –cohomology in degree , proving that if the dimension of the holomorphic invariants is or , then admits a branch covering over a product of curves inducing an isomorphism on rational cohomology in degree . We also study the class of Kodaira fibrations possessing a holomorphic section, and demonstrate that having a section imposes no restriction on possible monodromies.
"On Kodaira fibrations with invariant cohomology." Geom. Topol. 25 (5) 2385 - 2404, 2021. https://doi.org/10.2140/gt.2021.25.2385