Relative Chow-Kuenneth decompositions for morphisms of threefolds

TL;DR

This paper proves that any nonconstant morphism of a threefold admits a relative Chow-Kuenneth decomposition, providing new insights into the structure of threefolds and their decompositions.

Contribution

It introduces the first general proof of relative Chow-Kuenneth decompositions for morphisms of threefolds, extending previous results and offering new conditions for absolute decompositions.

Findings

Any nonconstant morphism of a threefold admits a relative Chow-Kuenneth decomposition.

Provides sufficient conditions for threefolds to have an absolute Chow-Kuenneth decomposition.

Improves existing theorems for cases where the image is a surface or a curve.

Abstract

We show that any nonconstant morphism of a threefold admits a relative Chow-Kuenneth decomposition. As a corollary we get sufficient conditions for threefolds to admit an absolute Chow-Kuenneth decomposition. In case the image of the morphism is a surface, this implies another proof of a theorem on the absolute Chow-Kuenneth decomposition for threefolds satisfying a certain condition, which was obtained by the first author with P. L. del Angel. In case the image is a curve, this improves in the threefold case a theorem obtained by the second author where the singularity of the morphism was assumed isolated and the condition on the general fiber was stronger.