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One-cycles on rationally connected varieties

Part of: Cycles and subschemes Special varieties

Published online by Cambridge University Press:  10 March 2014

Zhiyu Tian  and
Hong R. Zong
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
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Abstract

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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.

Keywords

rationally connected varietiesalgebraic cycles

MSC classification

Secondary: 14M22: Rationally connected varieties 14C25: Algebraic cycles
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 3 , March 2014 , pp. 396 - 408
Copyright
© The Author(s) 2014 

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