Double Kodaira fibrations with small signature

TL;DR

This paper explores the construction of Kodaira fibrations with small signature, focusing on ramified covers of product curves and analyzing monodromy actions, leading to a classification of automorphisms on low-genus curves.

Contribution

It introduces new methods for constructing low-signature Kodaira fibrations with multiple fibrations and classifies automorphisms on curves of genus up to 9.

Findings

Constructed examples of low-signature Kodaira fibrations with multiple fibrations

Analyzed monodromy actions in ramified covers of product curves

Classified automorphisms on curves of genus at most 9

Abstract

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fibre bundles. They are known to have positive signature divisible by $4$. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyse by studying the monodromy action for bundles of punctured curves. As a by-product we obtain a classification of all fix-point-free automorphisms on curves of genus at most $9$.