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

• Autorzy: Sergey Gorchinskiy , Vladimir Guletskiĭ
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

Algebraic cycles; Rational equivalence; Motives; Balanced correspondence; Generic cycle; Minimal field of definition; Transcendence degree; Bloch’s conjecture; Rational curve

Kategorie tematyczne

• 14C15: (Equivariant) Chow groups and rings; motives
• 14C25: Algebraic cycles

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 559-568

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Sergey Gorchinskiy

• Steklov Mathematical Institute

autor

Vladimir Guletskiĭ

• University of Liverpool, Mathematics & Oceanography Building

Bibliografia

• [1] Barbieri Viale L., Balanced varieties, In: Algebraic K-theory and its Applications, Trieste, September 1–19, 1997, World Scientific, River Edge, 1999, 298–312
• [2] Barlow R., Rational equivalence of zero cycles for some more surfaces with p g = 0, Invent. Math., 1985, 79(2), 303–308 http://dx.doi.org/10.1007/BF01388975
• [3] Bloch S., Lectures on Algebraic Cycles, Duke Univ. Math. Ser., 4, Duke University, Durham, 1980
• [4] Bloch S., Kas A., Lieberman D., Zero cycles on surfaces with p g = 0, Compositio Math., 1976, 33(2), 135–145
• [5] Bloch S., Srinivas V., Remarks on correspondences and algebraic cycles, Amer. J. Math., 1983, 105(5), 1235–1253 http://dx.doi.org/10.2307/2374341
• [6] Gorchinskiy S., Guletskii V., Non-trivial elements in the Abel-Jacobi kernels on higher dimensional varieties, preprint available at http://arxiv.org/abs/1009.1431
• [7] Guletskiı V., Pedrini C., Finite-dimensional motives and the conjectures of Beilinson and Murre, K-Theory, 2003, 30(3), 243–263
• [8] Inose H., Mizukami M., Rational equivalence of 0-cycles on some surfaces of general type with p g = 0, Math. Ann., 1979, 244(3), 205–217 http://dx.doi.org/10.1007/BF01420343
• [9] Kahn B., Murre J.P., Pedrini C., On the transcendental part of the motive of a surface, In: Algebraic Cycles and Motives, 2, London Math. Soc. Lecture Note Ser., 344, Cambridge University Press, Cambridge, 2007, 143–202
• [10] Kimura S.-I., Chow groups are finite dimensional, in some sense, Math. Ann., 2005, 331(1), 173–201 http://dx.doi.org/10.1007/s00208-004-0577-3
• [11] Kollár J., Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer, Berlin, 1996
• [12] Mumford D., An algebraic surface with K ample, (K 2) = 9, p g = q = 0, Amer. J. Math., 1979, 101(1), 233–244 http://dx.doi.org/10.2307/2373947
• [13] Nesterenko Y., Modular functions and transcendence problems, C. R. Acad. Sci. Paris Sér. I Math., 1996, 322(10), 909–914
• [14] Prasad G., Yeung S.-K., Fake projective planes, Invent. Math., 2007, 168(2), 321–370 http://dx.doi.org/10.1007/s00222-007-0034-5
• [15] Reid M., Campedelli versus Godeaux, In: Problems in the Theory of Surfaces and their Classification, Cortona, October 10–15, 1988, Sympos. Math., 32, Academic Press, London, 1991, 309–365
• [16] Scholl A., Classical motives, In: Motives, Seattle, July 20–August 2, 1991, Proc. Sympos. Pure Math., 55(1), American Mathematical Society, Providence, 1994, 163–187
• [17] Voisin C., Sur les zéro-cycles de certaines hypersurfaces munies d’un automorphisme, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1992, 19(4), 473–492

Identyfikatory

DOI

10.2478/s11533-011-0130-z