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

Sergey Gorchinskiy; Vladimir Guletskiĭ

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 559-568
  • ISSN: 2391-5455

Abstract

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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].

How to cite

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Sergey Gorchinskiy, and Vladimir Guletskiĭ. "Transcendence degree of zero-cycles and the structure of Chow motives." Open Mathematics 10.2 (2012): 559-568. <http://eudml.org/doc/269676>.

@article{SergeyGorchinskiy2012,
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].},
author = {Sergey Gorchinskiy, Vladimir Guletskiĭ},
journal = {Open Mathematics},
keywords = {Algebraic cycles; Rational equivalence; Motives; Balanced correspondence; Generic cycle; Minimal field of definition; Transcendence degree; Bloch’s conjecture; Rational curve; algebraic cycles; rational equivalence; motives; balanced correspondence; generic cycle; minimal field of definition; transcendence degree; Bloch's conjecture; rational curve},
language = {eng},
number = {2},
pages = {559-568},
title = {Transcendence degree of zero-cycles and the structure of Chow motives},
url = {http://eudml.org/doc/269676},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Sergey Gorchinskiy
AU - Vladimir Guletskiĭ
TI - Transcendence degree of zero-cycles and the structure of Chow motives
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 559
EP - 568
AB - 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].
LA - eng
KW - Algebraic cycles; Rational equivalence; Motives; Balanced correspondence; Generic cycle; Minimal field of definition; Transcendence degree; Bloch’s conjecture; Rational curve; algebraic cycles; rational equivalence; motives; balanced correspondence; generic cycle; minimal field of definition; transcendence degree; Bloch's conjecture; rational curve
UR - http://eudml.org/doc/269676
ER -

References

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