Transcendence degree of zero-cycles and the structure of Chow motives

Sergey Gorchinskiy ; Vladimir Guletskiĭ

Open Mathematics, Tome 10 (2012), p. 559-568 / Harvested from The Polish Digital Mathematics Library

Résumé

In the paper we introduce a transcendence degree of a zero-cycle on a smooth projective variety X and relate it to the structure of the motive of X. In particular, we show that in order to prove Bloch’s conjecture for a smooth projective complex surface X of general type with p g = 0 it suffices to prove that one single point of a transcendence degree 2 in X(ℂ), over the minimal subfield of definition k ⊂ ℂ of X, is rationally equivalent to another single point of a transcendence degree zero over k. This can be of particular interest in the context of Bloch’s conjecture for those surfaces which admit a concrete presentation, such as Mumford’s fake surface, see [Mumford D., An algebraic surface with K ample, (K 2) = 9, p g = q = 0, Amer. J. Math., 1979, 101(1), 233–244].

Détail

Publié le : 2012-01-01

Zbl 1244.14006

EUDML-ID : urn:eudml:doc:269676

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