Fundamental Group of some Genus-2 Fibrations and Applications

TL;DR

This paper investigates the fundamental group structure of genus-2 fibered surfaces and confirms the Shafarevich Conjecture on their universal covers' holomorphic convexity.

Contribution

It proves the fundamental group is nearly a product of the base curve and an elliptic curve, and verifies the Shafarevich Conjecture for certain genus-2 fibered surfaces.

Findings

Fundamental group is almost isomorphic to a product of base and elliptic curve groups.

Confirmed holomorphic convexity of universal covers for specific genus-2 fibrations.

Provides new insights into the topology and complex geometry of genus-2 fibered surfaces.

Abstract

We will prove that given a genus-2 fibration $f: X \rightarrow C$ on a smooth projective surface $X$ such that $b_1(X)=b_1(C)+2$, the fundamental group of $X$ is almost isomorphic to $\pi_1(C) \times \pi_1(E)$, where $E$ is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces $X$ with genus-2 fibration $X\rightarrow C$ such that $b_1(X)>b_1(C)$.