Abstract
Let X be a smooth cubic hypersurface, and let F be the Fano variety of lines on X. We establish a relation between the Chow motives of X and F. This relation implies in particular that if X has finite-dimensional motive (in the sense of Kimura), then F also has finite-dimensional motive. This proves finite-dimensionality for motives of Fano varieties of cubics of dimension 3 and 5, and of certain cubics in other dimensions.
Résumé
Soit X une hypersurface cubique lisse, et soit F la variété de Fano paramétrant les droites contenues dans X. On établit une relation entre les motifs de Chow de X et de F. Cette relation implique le fait que F a motif de dimension finie (au sens de Kimura) à condition que X a motif de dimension finie. En particulier, si X est une cubique lisse de dimension 3 ou 5, alors F a motif de dimension finie.
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References
André, Y.: Motifs de dimension finie (d’après S.-I. Kimura, P. O’Sullivan,...), Séminaire Bourbaki 2003/2004, Astérisque 299 Exp. No. 929, viii, 115–145,
Brion, M.: Log homogeneous varieties. In: Actas del XVI Coloquio Latinoamericano de Algebra, Revista Matemática Iberoamericana, Madrid 2007, arXiv: math/0609669
de Cataldo, M., Migliorini, L.: The Chow groups and the motive of the Hilbert scheme of points on a surface. J. Algebra 251(2), 824–848 (2002)
Deligne, P.: La conjecture de Weil pour les surfaces \(K3\). Invent. Math. 15, 206–226 (1972)
Diaz, H.: The motive of the Fano surface of lines, arXiv:1602.06403v1
Fulton, W.: Intersection theory. Springer-Verlag Ergebnisse der Mathematik, Berlin (1984)
Galkin, S., Shinder, E.: The Fano variety of lines and rationality problem for a cubic hypersurface, arXiv:1405.5154
Guletskiĭ, V., Pedrini, C.: The Chow motive of the Godeaux surface, in: Algebraic Geometry, a volume in memory of Paolo Francia (M.C. Beltrametti et alii, editors), Walter de Gruyter, Berlin New York (2002)
Hirschowitz, A., Iyer, J.: Hilbert schemes of fat r-planes and the triviality of Chow groups of complete intersections. In: Vector bundles and complex geometry, Contemp. Math. 522, Amer. Math. Soc., Providence (2010)
Ivorra, F: Finite dimensional motives and applications (following S.-I. Kimura, P. O’Sullivan and others). In: Autour des motifs, Asian-French summer school on algebraic geometry and number theory, Volume III, Panoramas et synthèses, Société mathématique de France (2011)
Iyer, J.: Murre’s conjectures and explicit Chow-Künneth projectors for varieties with a nef tangent bundle. Trans. Am. Math. Soc. 361, 1667–1681 (2008)
Iyer, J.: Absolute Chow-Künneth decomposition for rational homogeneous bundles and for log homogeneous varieties. Michigan Math. J. 60(1), 79–91 (2011)
Jannsen, U.: Motivic sheaves and filtrations on Chow groups. In: Jannsen et alii, U. (ed.), Proceedings of Symposia in Pure Mathematics Vol. 55, Part 1 (1994)
Jannsen, U.: On finite-dimensional motives and Murre’s conjecture. In: Nagel, J., Peters, C. (eds.) Cycles and motives, Cambridge University Press, Cambridge (2007)
Kimura, S.: Chow groups are finite dimensional, in some sense. Math. Ann. 331, 173–201 (2005)
Laterveer, R.: A family of cubic fourfolds with finite-dimensional motive, preprint
Lewis, J.: Cylinder homomorphisms and Chow groups. Math. Nachr. 160, 205–221 (1993)
Murre, J.: On a conjectural filtration on the Chow groups of an algebraic variety, parts I and II. Indag. Math. 4, 177–201 (1993)
Murre, J., Nagel, J., Peters, C.: Lectures on the theory of pure motives, Am. Math. Soc. University Lecture Series 61, Providence (2013)
Otwinowska, A.: Remarques sur les groupes de Chow des hypersurfaces de petit degré, C. R. Acad. Sci. Paris Série I Math. 329, no. 1, 51–56 (1999)
Pedrini, C.: On the finite dimensionality of a \(K3\) surface. Manuscripta Mathematica 138, 59–72 (2012)
Pedrini, C., Weibel, C.: Some surfaces of general type for which Bloch’s conjecture holds, to appear. In: Period Domains, Algebraic Cycles, and Arithmetic, Cambridge Univ. Press (2015)
Scholl, T.: Classical motives. In: Motives (U. Jannsen et alii, editors), Proceedings of Symposia in Pure Mathematics Vol. 55, Part 1 (1994)
Shioda, T.: The Hodge conjecture for Fermat varieties. Math. Ann. 245, 175–184 (1979)
Vial, C.: Projectors on the intermediate algebraic Jacobians. N. Y. J. Math. 19, 793–822 (2013)
Vial, C.: Remarks on motives of abelian type, to appear in Tohoku Math. J
Vial, C.: Niveau and coniveau filtrations on cohomology groups and Chow groups. Proc. LMS 106(2), 410–444 (2013)
Vial, C.: Chow-Künneth decomposition for \(3\)- and \(4\)-folds fibred by varieties with trivial Chow group of zero-cycles. J. Alg. Geom. 24, 51–80 (2015)
Voisin, C.: Remarks on zero-cycles of self-products of varieties. In: Moduli of vector bundles, Proceedings of the Taniguchi Congress (M. Maruyama, editor), Marcel Dekker New York Basel Hong Kong (1994)
Voisin, C.: On the universal \(CH_0\) group of cubic hypersurfaces, to appear in Journal Eur. Math. Soc
Voisin, C.: Bloch’s conjecture for Catanese and Barlow surfaces. J. Differ. Geom. 97, 149–175 (2014)
Voisin, C.: Chow rings, decomposition of the diagonal, and the topology of families. Princeton University Press, Princeton (2014)
Xu, Z.: Algebraic cycles on a generalized Kummer variety, arXiv:1506.04297v1
Acknowledgments
This note is a protracted after-effect of the Strasbourg 2014-2015 groupe de travail based on the monograph [32]; thanks to all the participants for the pleasant and stimulating atmosphere. Many thanks to Yasuyo, Kai and Len for not being there when I work, and for being there when I don’t.
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Laterveer, R. A remark on the motive of the Fano variety of lines of a cubic. Ann. Math. Québec 41, 141–154 (2017). https://doi.org/10.1007/s40316-016-0070-x
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DOI: https://doi.org/10.1007/s40316-016-0070-x