Algebraic cycles on smooth varieties fibred by varieties with small Chow groups

Details Algebraic cycles on smooth varieties fibred by varieties with small Chow groups Algebraic cycles on smooth varieties fibred by varieties with small Chow groups

2012-03-12

arxiv.org/abs/1203.2650v1

Mathematics

Algebraic Geometry

20 pages

Scientific paper

We study the Chow groups of smooth complex varieties $X$ which are fibred by varieties with small Chow groups, e.g. varieties fibred by low degree complete intersections or cellular varieties. As an application, we give new examples of smooth projective varieties satisfying conjectures on algebraic cycles such as the Hodge conjecture, the Lefschetz standard conjecture, Kimura's finite-dimensionality conjecture and Murre's conjectures. If $X$ is fibred by cellular varieties over a curve, then we show that $X$ is Kimura finite-dimensional. If $X$ is fibred by cubic fivefolds over a curve, then we show that $X$ satisfies Murre's conjectures as well as the standard conjectures.

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