A Note on the Fundamental Group of Kodaira Fibrations

Abstract

AbstractThe fundamental group π of a Kodaira fibration is, by definition, the extension of a surface group $\Pi_b$ by another surface group $\Pi _g$, i.e. $$1 \rightarrow \Pi_g \rightarrow \pi \rightarrow \Pi_b \rightarrow 1.$$ Conversely, Catanese (2017) inquires about what conditions need to be satisfied by a group of that sort in order to be the fundamental group of a Kodaira fibration. In this short note we collect some restrictions on the image of the classifying map $m \colon \Pi_b \to \Gamma_g$ in terms of the coinvariant homology of $\Pi_g$. In particular, we observe that if π is the fundamental group of a Kodaira fibration with relative irregularity g−s, then $g \leq 1+ 6s$, and we show that this effectively constrains the possible choices for π, namely that there are group extensions as above that fail to satisfy this bound, hence it cannot be the fundamental group of a Kodaira fibration. A noteworthy consequence of this construction is that it provides examples of symplectic 4-manifolds that fail to admit a Kähler structure for reasons that eschew the usual obstructions.