The arithmeticity of a Kodaira Fibration is determined by its universal cover

TL;DR

This paper demonstrates that the algebraic definability of a Kodaira fibration's surface over a number field is solely determined by the biholomorphic class of its universal cover.

Contribution

It establishes a direct link between the universal cover's biholomorphic class and the arithmetic property of the Kodaira fibration.

Findings

The algebraic structure over a number field depends only on the universal cover.

Universal covers determine the arithmetic nature of Kodaira fibrations.

The result connects complex geometry with number theory in algebraic surfaces.

Abstract

Let $S \to C$ be a Kodaira fibration. Here we show that whether or not the algebraic surface $S$ is defined over a number field depends only on the biholomorphic class of its universal cover.