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.
Cite
@article{arxiv.1209.4342,
title = {One Cycles on Rationally Connected Varieties},
author = {Zhiyu Tian and Hong R. Zong},
journal= {arXiv preprint arXiv:1209.4342},
year = {2019}
}
Comments
14 pages, final version with minor typos corrected and some new remarks, to appear in Compositio Mathematica, comments are welcome! arXiv admin note: text overlap with arXiv:1207.0575 by other authors