Chow-Künneth decomposition for 3- and 4-folds fibred by varieties with small Chow group of zero-cycles

Details Chow-Künneth decomposition for 3- and 4-folds fibred by varieties with small Chow group of zero-cycles Chow-Künneth decomposition for 3- and 4-folds fibred by varieties with small Chow group of zero-cycles

2011-11-15

arxiv.org/abs/1111.3658v1

Mathematics

Algebraic Geometry

29 pages

Scientific paper

Let $f:X \r S$ be a dominant morphism from a smooth projective variety $X$ to a smooth projective variety $S$ of dimension less or equal than 2 over a field $k$ with general fibre having trivial Chow group of zero-cycles. For example, $X$ could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that $X$ has dimension less or equal than 4. Then, we prove that $X$ has a self-dual Murre decomposition. Moreover we prove that the motivic Lefschetz conjecture holds for $X$ and hence so does the Lefschetz standard conjecture. We also give new examples of threefolds of general type which are Kimura finite dimensional, new examples of fourfolds of general type having a self-dual Murre decomposition, as well as new examples of varieties with finite degree three unramified cohomology.

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