Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T21:03:11.097Z Has data issue: false hasContentIssue false
  • English
  • Français

Cylindrical homomorphisms and Lawson homology

Published online by Cambridge University Press:  08 June 2010

Mircea Voineagu
Mircea Voineagu
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA, voineagu@usc.edu
Get access

Abstract

We use the cylindrical homomorphism and a geometric construction introduced by J. Lewis to study the Lawson homology groups of certain hypersurfaces X ⊂ ℙn + 1 of degree d ℙ n + 1. As an application, we compute the rational semi-topological K-theory of generic cubics of dimensions 5, 6 and 8 and, using the Bloch-Kato conjecture, we prove Suslin's conjecture for these varieties. Using generic cubic sevenfolds, we show that there are smooth projective varieties such that the lowest nontrivial step in their s-filtration is infinitely generated and undetected by the Abel-Jacobi map.

Keywords

Lawson homologymotivic cohomologysemi-topological K-theory
Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 1 , August 2011 , pp. 135 - 168
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abd99.Abdulali, S.. Abelian varieties of type III and the Hodge conjecture. Internat. J. Math., 10(6):667–675, 1999.CrossRefGoogle Scholar
AC94.Albano, A. and Collino, A.. On the Griffiths group of the cubic sevenfold. Math. Ann., 299(4):715–726, 1994.CrossRefGoogle Scholar
BM79.Bloch, S. and Murre, J.-P.. On the Chow group of certain types of Fano threefolds. Compositio Math., 39(1):47–105, 1979.Google Scholar
Bor90.Borcea, C.. Deforming varieties of k-planes of projective complete intersections. Pacific J. Math., 143(1):25–36, 1990.CrossRefGoogle Scholar
BS83.Bloch, S. and Srinivas, V.. Remarks on correspondences and algebraic cycles. Amer. J. Math., 105(5):1235–1253, 1983.CrossRefGoogle Scholar
ELV97.Esnault, H., Levine, M., and Viehweg, E.. Chow groups of projective varieties of very small degree. Duke Math. J., 87(1):29–58, 1997.CrossRefGoogle Scholar
FG93.Friedlander, E.-M. and Gabber, O.. Cycle spaces and intersection theory. In Topological methods in modern mathematics (Stony Brook, NY, 1991), pages 325–370. Publish or Perish, Houston, TX, 1993.Google Scholar
FHW04.Friedlander, E.-M., Haesemeyer, C., and Walker, M.-E.. Techniques, computations, and conjectures for semi-topological K-theory. Math. Ann., 330(4):759–807, 2004.CrossRefGoogle Scholar
FL92.Friedlander, E.-M. and Lawson, B.-H.. A theory of algebraic cocycles. Ann. of Math. (2), 136(2):361–428, 1992.CrossRefGoogle Scholar
FL97.Friedlander, E.-M. and Lawson, B.-H. Duality relating spaces of algebraic cocycles and cycles. Topology, 36(2):533–565, 1997.CrossRefGoogle Scholar
FM94a.Friedlander, E.-M. and Mazur, B.. Correspondence homomorphisms for singular varieties. Ann. Inst. Fourier (Grenoble), 44(3):703–727, 1994.CrossRefGoogle Scholar
FM94b.Friedlander, E.-M. and Mazur, B.. Filtrations on the homology of algebraic varieties. Mem. Amer. Math. Soc., 110(529):x+110, 1994. With an appendix by Daniel Quillen.Google Scholar
Fri91.Friedlander, E.-M.. Algebraic cycles, Chow varieties, and Lawson homology. Compositio Math., 77(1):55–93, 1991.Google Scholar
Fri95.Friedlander, E.-M.. Filtrations on algebraic cycles and homology. Ann. Sci. École Norm. Sup. (4), 28(3):317–343, 1995.CrossRefGoogle Scholar
Fri00.Friedlander, E.-M.. Relative Chow correspondences and the Griffiths group. Ann. Inst. Fourier (Grenoble), 50(4):1073–1098, 2000.CrossRefGoogle Scholar
Ful84.Fulton, W.. Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1984.CrossRefGoogle Scholar
FW02.Friedlander, E.-M. and Walker, M.-E.. Semi-topological K-theory using function complexes. Topology, 41(3):591–644, 2002.CrossRefGoogle Scholar
FW05.Friedlander, E.-M. and Walker, M.-E.. Semi-topological K-theory. In Handbook of K-theory. Vol. 1, 2, pages 877–924. Springer, Berlin, 2005.CrossRefGoogle Scholar
HL.Hu, W. and Li, L.. Lawson homology, morphic cohomology and Chow motives. arXiv: math.AG/0711.0383.Google Scholar
Hu.Hu, W.. Some birational invariants defined by Lawson homology. arXiv:math.AG/0511722.Google Scholar
Law89.Lawson, B.-H.. Algebraic cycles and homotopy theory. Ann. of Math. (2), 129(2):253–291, 1989.CrossRefGoogle Scholar
Lew88.Lewis, J.-D.. The cylinder correspondence for hypersurfaces of degree n in Pn. Amer. J. Math., 110(1):77–114, 1988.CrossRefGoogle Scholar
Lew93.Lewis, J.-D.. Cylinder homomorphisms and Chow groups. Math. Nachr., 160:205–221, 1993.CrossRefGoogle Scholar
Lew99.Lewis, James D.. A survey of the Hodge conjecture, volume 10 of CRM Monograph Series. American Mathematical Society, Providence, RI, second edition, 1999. Appendix B by B. Brent Gordon.Google Scholar
LF92.Lima-Filho, P.. Lawson homology for quasiprojective varieties. Compositio Math., 84(1):1–23, 1992.Google Scholar
Nor93.Nori, M.-V.. Algebraic cycles and Hodge-theoretic connectivity. Invent. Math., 111(2):349–373, 1993.CrossRefGoogle Scholar
Pet00.Peters, C.. Lawson homology for varieties with small Chow groups and the induced filtration on the Griffiths groups. Math. Z., 234(2):209–223, 2000.CrossRefGoogle Scholar
Ros.Rost, M.. On the basic correspondence of a splitting variety. in preparation.Google Scholar
SV00.Suslin, A. and Voevodsky, V.. Bloch-Kato conjecture and motivic cohomology with finite coefficients. In The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), volume 548 of NATO Sci. Ser. C Math. Phys. Sci., pages 117–189. Kluwer Acad. Publ., Dordrecht, 2000.Google Scholar
Voe.Voevodsky, V.. On motivic cohomology with ℤ/l coefficients. preprint.Google Scholar
Voi03.Voisin, C.. Hodge theory and complex algebraic geometry. II, volume 77 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2003. Translated from the French by Leila Schneps.CrossRefGoogle Scholar
Voi08.Voineagu, M.. Semi-topological K-theory of certain projective varieties. Journal of Pure and Applied Algebra, 212(8):1960–1983, 2008.CrossRefGoogle Scholar
Wei.Weibel, C.. Patching the norm residue isomorphism theorem. preprint.Google Scholar