Abstract
This article is about motives of quadric bundles. In the case of odd dimensional fibers and where the basis is of dimension two we give an explicit relative Chow–Künneth decomposition. This relative Chow–Künneth decomposition shows that the motive of the quadric bundle is isomorphic to the direct sum of Tate twists of the motive of the base and of the Prym motive of a double cover of the discriminant. In particular this is a refinement with \({\mathbb{Q}}\)—coefficients of a result of Beauville concerning the cohomology and the Chow groups of an odd dimensional quadric bundle over \({\mathbb{P}^{2}}\). This relative Chow–Künneth decomposition induces an absolute Chow–Künneth decomposition which satisfies parts of Murre’s conjectures. This article is a generalization of an article of Nagel and Saito on conic bundles (Nagel and Saito in Int Math Res Not IMRN 16, 2978–3001, 2009).
Similar content being viewed by others
References
Beauville A.: Variétés de Prym et jacobiennes intermediaires. Ann. Sci. Ecole Norm. Sup.(4) 10, 309–391 (1977)
Beilinson A., Bernstein J., Deligne P.: Faisceaux pervers. Asterisque 100, 5–171 (1982)
Beltrametti, M., Francia, P.: Conic bundles on nonrational surfaces. Algebraic geometry—open problems (Ravello, 1982), 34–89, Lecture Notes in Mathematics, vol 997, Springer, Berlin (1983)
Chernousov V., Gille S., Merkurjev A.: Motivic decomposition of isotropic homogeneous varieties. Duke Math. J. 126, 137–159 (2005)
Corti A., Hanamura M.: Motivic decomposition and intersection Chow groups. I. Duke Math. J. 103, 459–522 (2000)
del Angel P.L., Müller-Stach S.: Motives of uniruled threefolds. Compos. Math. 112, 1–16 (1998)
del Angel P.L., Müller-Stach S.: On Chow motives of threefolds. Trans. Am. Math. Soc. 352, 1623–1633 (2000)
Deninger C., Murre J.P.: Motivic decomposition of abelian schemes and Fourier transform. J. Reine Angew. Math. 422, 201–219 (1991)
Fulton W.: Intersection Theory. Second Edition. A Series of Modern Surveys in Mathematics. Springer, Berlin (1998)
Gordon B.B., Hanamura M., Murre J.P.: Relative Chow–Künneth projection for modular varieties. J. Reine Angew. Math. 558, 1–14 (2003)
Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, NewYork (1978)
Jannsen, U.: Motivic sheaves and filtration on Chow groups. In: Seattle Conference on Motives 1991, AMS Proceedings of Symposium Pure Math 55, AMS Providence, RI, pp. 245–302 (1994)
Karpenko N.: Cohomology of relative cellular spaces and isotropic flag varieties. St. Petersb. Math. J. 12(1), 1–50 (2001)
Köck B.: Chow motif and higher Chow theory of G/P. Manuscripta Math. 70, 363–372 (1991)
Müller-Stach, S., Peters, C.: Transcendental aspects of algebraic cycles. in: Proceedings of the Summer School held in Grenoble, June 18–July 6, 2001. London Mathematical Society Lecture Note Series, vol 313. Cambridge University Press, Cambridge (2004)
Müller-Stach S., Saito M.: Relative Chow–Küenneth decompositions for morphisms of threefolds. J. Reine Angew. Math. 666, 141–161 (2012)
Murre J.P.: On the motive of an algebraic surface. J. Reine Angew. Math. 409, 190–204 (1990)
Murre J.P.: On a conjectural filtration on the Chow groups of an algebraic variety I and II. Indag. Mathem. 4, 177–201 (1993)
Murre, J.P., Nagel, J., Peters, C.: Lectures on the Theory of Pure Motives. AMS University lecture series, vol 61 (2013)
Nagel J, Saito M: Relative Chow-Künneth decompositions for conic bundles and Prym varieties. Int. Math. Res. Not. IMRN 16, 2978–3001 (2009)
Shermenev A.M.: The motive of an abelian variety. Funk. Anal. 8, 47–53 (1974)
Terasoma T.: Complete intersections of hypersurfaces the Fermat case and the quadric case. Jpn. J. Math. 14(2), 329–384 (1988)
Vial C.: Chow–Künneth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles. J. Algebraic Geom. 24, 51–80 (2015)
Vial C.: Algebraic cycles and fibrations. Doc. Math. 18, 1521–1553 (2013)
Vial C.: Projectors on the algebraic intermediate jacobians. NY J. Math. 19, 793–822 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bouali, J. Motives of quadric bundles. manuscripta math. 149, 347–368 (2016). https://doi.org/10.1007/s00229-015-0783-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-015-0783-1