Zero-cycles in families of rationally connected varieties

TL;DR

This paper investigates the behavior of zero-cycles in families of rationally connected varieties, establishing isomorphisms of Chow groups under certain conditions and extending results to higher Chow groups.

Contribution

It generalizes Kollár's result by showing isomorphisms of Chow groups for zero-cycles in families with separably rationally connected fibers and extends to higher Chow groups.

Findings

Restriction of zero cycles induces Chow group isomorphism

Extension of results to higher Chow groups

Conjectures proposed for non-smooth cases

Abstract

We study zero-cycles in families of rationally connected varieties. We show that for a smooth projective scheme over a henselian discrete valuation ring the restriction of relative zero cycles to the special fiber induces an isomorphism on Chow groups if the special fiber is separably rationally connected. We further extend this result to certain higher Chow groups and develop conjectures in the non-smooth case. Our main results generalise a result of Koll\'ar [31].