Abstract
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].
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Gorchinskiy, S., Guletskiĭ, V. Transcendence degree of zero-cycles and the structure of Chow motives. centr.eur.j.math. 10, 559–568 (2012). https://doi.org/10.2478/s11533-011-0130-z
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DOI: https://doi.org/10.2478/s11533-011-0130-z