One Cycles on Rationally Connected Varieties

TL;DR

This paper proves that on certain rationally connected varieties, the first Chow group is generated by rational curves, and applies this to solve a question about Hodge classes on rationally connected 3-folds, linking it to the Tate conjecture.

Contribution

It establishes that the first Chow group of separably rationally connected varieties is generated by rational curves, including Fano complete intersections with index at least 2.

Findings

First Chow group generated by rational curves on these varieties

Solved a question of Burt Totaro about Hodge classes on 3-folds

Connected the problem to the Tate conjecture for surfaces over finite fields

Abstract

All curves on a separably rationally connected variety are rationally equivalent to a (non-effective) integral sum of rational curves, hence the first Chow group is generated by rational curves. Applying the same techniques, we also proved that the first Chow group of all separably rationally connected Fano complete intersections with index at least 2 is generated by lines. As a consequence, a question of Professor Burt Totaro about integral Hodge classess on rationally connected 3-folds is solved, and positive answer to the question for general n-fold due to Professor J\'anos Koll\'ar will follow from the Tate conjecture for surfaces over finite fields.