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One-cycles on rationally connected varieties

Published online by Cambridge University Press:  10 March 2014


Zhiyu Tian
Affiliation:
Department of Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA email tian@caltech.edu
Hong R. Zong
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA email rzong@math.princeton.edu
Corresponding

Abstract

We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersection of index at least 2 is generated by lines. As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.


Type
Research Article
Copyright
© The Author(s) 2014 

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