Fibrations on four-folds with trivial canonical bundles

TL;DR

This paper classifies fibrations of four-dimensional manifolds with trivial canonical bundles into six classes based on holonomy, constructing examples in five classes and proving non-existence in the sixth, with a focus on Jacobian genus two fibrations.

Contribution

It provides a comprehensive classification of abelian surface fibrations on four-folds with trivial canonical bundles, including explicit constructions and non-existence results.

Findings

Constructed fibrations in five holonomy classes

Proved non-existence in the sixth class

Classified fibrations with Jacobian genus two fibers

Abstract

Four-folds with trivial canonical bundles are divided into six classes according to their holonomy group. We consider examples that are fibred by abelian surfaces over the projective plane. We construct such fibrations in five of the six classes, and prove that there is no such fibration in the sixth class. We classify all such fibrations whose generic fibre is the Jacobian of a genus two curve.