Fibrations of genus two on complex surfaces

TL;DR

This paper explores the local geometric structure of genus two fibrations on complex surfaces, showing that the singular fiber's data uniquely determines the nearby fibration up to a smooth holomorphic equivalence.

Contribution

It provides a detailed geometric description of neighborhoods of singular fibers in genus two fibrations and demonstrates the general applicability of the method to higher genus cases.

Findings

Singular fiber data determines the local fibration structure.

The method applies broadly to fibrations of arbitrary genus.

Neighborhoods are classified up to transversely holomorphic diffeomorphism.

Abstract

We consider fibrations of genus 2 over complex surfaces. The purpose of this paper is primarily to provide a geometric description of the possible structures of the fibration on a neighborhood of a singular fiber. In particular it is shown that the "geometric data" of the singular fiber determines the fibration on its neighborhood up to a transversely holomorphic $C^{\infty}$-diffeomorphism. The method employed is quite flexible and it applies to good extent to fibrations of arbitrary genus.