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One-cycles on rationally connected varieties

Tian, Zhiyu and Zong, Hong R. (2014) One-cycles on rationally connected varieties. Compositio Mathematica, 150 (3). pp. 396-408. ISSN 0010-437X.

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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.

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Additional Information:© 2014 Foundation Compositio Mathematica. Received 22 January 2013, accepted in final form 13 May 2013, published online 10 March 2014. We thank Professor Burt Totaro for introducing the question to us by his enlightening lectures, correcting many mistakes in the first draft of this paper, and helping us to form the final argument, Professor Claire Voisin for the argument of reducing rational equivalence to algebraic equivalence and Remark 6.4, Professor János Kollár for his constant support for the second named author and enlightening comments on the proof, Professor Jason Starr for helpful discussions about rational curves on Fano complete intersections, and Professor Chenyang Xu for very encouraging comments during the project.
Subject Keywords:rationally connected varieties, algebraic cycles
Classification Code:2010 Mathematics Subject Classification: 14M22 (primary), 14C25 (secondary).
Record Number:CaltechAUTHORS:20140606-142158094
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:46135
Deposited By: Tony Diaz
Deposited On:06 Jun 2014 23:09
Last Modified:06 Jun 2014 23:09

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