One Cycles on Rationally Connected Varieties

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.