Abstract
A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove a similar statement for Chow groups of arbitrary codimension, provided the variety satisfies the Lefschetz standard conjecture.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
It is somewhat frustrating that it is not known unconditionally whether (P1) implies (P2), i.e. without assuming the generalized Hodge conjecture. Apparently Esnault, Srinivas and Viehweg had claimed to prove this in an earlier version of their paper, but the argument was found to be incomplete [5, remark 2].
References
Arapura, D.: Motivation for Hodge cycles. Adv. Math. 207, 762–781 (2006)
Bloch, S.: On an argument of Mumford in the theory of algebraic cycles. In: Beauville, A. (ed.) Géométrie algébrique. Sijthoff and Noordhoff, Angers (1979)
Bloch, S.: Lectures on algebraic cycles. Duke Univ Press, Durham (1980)
Bloch, S., Srinivas, V.: Remarks on correspondences and algebraic cycles. Am. J. Math. 105(5), 1235–1253 (1983)
Esnault, H., Srinivas, V., Viehweg, E.: Decomposability of Chow groups implies decomposability of cohomology. Astérisque 218, 227–242 (1993) (Journées de Géométrie Algébrique d’Orsay, Juillet 1992)
Fulton, W.: Intersection Theory. Springer, Berlin Heidelberg New York (1984)
Kahn, B., Murrem, J.P., Pedrini, C.: On the transcendental part of the motive of a surface. In: Algebraic Cycles and Motives, vol. 2, volume 344 of London Math. Soc. Lecture Note Ser., pp. 143—202. Cambridge Univ. Press, Cambridge (2007)
Kleiman, S.: Algebraic Cycles and the Weil Conjectures. In: Dix exposés sur la cohomologie des schémas, North–Holland Amsterdam, pp. 359—386 (1968)
Kleiman, S.: The standard conjectures. In: Jannsen, U. et al. (ed.) Motives, Proceedings of Symposia in Pure Mathematics, vol. 55, Part 1 (1994)
Laterveer, R.: Yet another version of Mumford’s theorem. Archiv Math. 104, 125–131 (2015)
Laterveer, R.: On a multiplicative version of Bloch’s conjecture. Beiträge zur Algebra und Geometrie
Mumford, D.: Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ. 9(2), 195–204 (1969)
Murre, J.: On a conjectural filtration on the Chow groups of an algebraic variety. Indag. Math. (N.S.) 4, 177—201 (1993)
Murre, J., Nagel, J., Peters, C.: Lectures on the theory of pure motives. Amer. Math. Soc. University Lecture Series, vol. 61 (2013)
Scholl, A.: Classical Motives. In: Jannsen, U. et al. (ed.) Motives, Proceedings of Symposia in Pure Mathematics, vol. 55, Part 1 (1994)
Tankeev, S.: On the standard conjecture of Lefschetz type for complex projective threefolds. II. Izvestiya Math. 75(5), 1047–1062 (2011)
Vial, C.: Algebraic cycles and fibrations. Documenta Math. 18, 1521–1553 (2013)
Vial, C.: Projectors on the intermediate algebraic Jacobians. N. Y. J. Math. 19, 793–822 (2013)
Voisin, C.: Chow Rings, Decomposition of the Diagonal, and the Topology of Families. Princeton University Press, Princeton and Oxford (2014)
Acknowledgments
This note is the fruit of the Strasbourg 2014–2015 “groupe de travail” based on the monograph [19]. I want to thank all the participants of this groupe de travail for the very pleasant and stimulating atmosphere, and their interesting lectures. Furthermore, many thanks to Yasuyo, Kai and Len for providing a wonderful working environment at home in Schiltigheim. Special thanks to Kai for all the bicycle trips made together this summer.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jens Funke and Ulf Kühn.
Rights and permissions
About this article
Cite this article
Laterveer, R. On a multiplicative version of Mumford’s theorem. Abh. Math. Semin. Univ. Hambg. 86, 89–96 (2016). https://doi.org/10.1007/s12188-016-0121-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-016-0121-x