Maximal Chow constant and cohomologically constant fibrations

TL;DR

This paper introduces and constructs maximal fibrations that are Chow constant and cohomologically constant, exploring their properties and applications to classifying singularities and understanding Chow groups of fibers.

Contribution

It defines and constructs maximal Chow constant and cohomologically constant fibrations, advancing the understanding of birationally simple fibrations and their applications.

Findings

Constructed maximal Chow constant fibrations.

Constructed maximal cohomologically constant fibrations.

Applied results to classify singularities of cones.

Abstract

Motivated by the study of rationally connected fibrations (and the MRC quotient) we study different notions of birationally simple fibrations. We say a fibration of smooth projective varieties is Chow constant if pushforward induces an isomorphism on the Chow group of 0-cycles. Likewise we say a fibration is cohomologically constant if pullback induces an isomorphism on holomorphic p-forms for all p. Our main result is the construction of maximal Chow constant and cohomologically constant fibrations. The paper is largely self contained and we prove a number of basic properties of these fibrations. One application is to the classification of "rationalizations of singularities of cones." We also consider consequences for the Chow groups of the generic fiber of a Chow constant fibration.