  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.
