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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Harnack-Thom theorem for higher cycle groups and Picard varieties

Author: Jyh-Haur Teh
Journal: Trans. Amer. Math. Soc. 360 (2008), 3263-3285
MSC (2000): Primary 14C25, 14P25; Secondary 55Q52, 55N35
Published electronically: November 28, 2007
MathSciNet review: 2379796
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Abstract: We generalize the Harnack-Thom theorem to relate the ranks of the Lawson homology groups with $ \mathbb{Z}_2$-coefficients of a real quasiprojective variety with the ranks of its reduced real Lawson homology groups. In the case of zero-cycle group, we recover the classical Harnack-Thom theorem and generalize the classical version to include real quasiprojective varieties. We use Weil's construction of Picard varieties to construct reduced real Picard groups, and Milnor's construction of universal bundles to construct some weak models of classifying spaces of some cycle groups. These weak models are used to produce long exact sequences of homotopy groups which are the main tool in computing the homotopy groups of some cycle groups of divisors. We obtain some congruences involving the Picard number of a nonsingular real projective variety and the rank of its reduced real Lawson homology groups of divisors.

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  • 1. D. J. Benson, Representations and cohomology II, Cambridge studies in adv. math., 31, Cambridge, 1991. MR 1156302 (93g:20099)
  • 2. A. Degtyarev and V. Kharlamov, Topological properties of real algebraic varieties: Rokhlin's way, Russian Math. Surveys 55 (2000), no. 4, 735-814. MR 1786731 (2001h:14077)
  • 3. E. Friedlander, Algebraic cycles, Chow varieties, and Lawson homology, Compositio Math. 77 (1991), 55-93. MR 1091892 (92a:14005)
  • 4. E. Friedlander, Filtration on algebraic cycles and homology, Annales Scientifiques de l'Ecole Normale Superieure Ser. 4, 28 no. 3 (1995), p. 317-343. MR 1326671 (96i:14004)
  • 5. Y. Félix, S. Halperin, J. Thomas, Rational homotopy theory, Springer, 2000.
  • 6. E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428. MR 1185123 (93g:14013)
  • 7. E. Friedlander and H.B. Lawson, Moving algebraic cycles of bounded degree, Invent. Math. 132 (1998), 91-119. MR 1618633 (99k:14011)
  • 8. E. Friedlander and H.B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), 533-565. MR 1415605 (97k:14007)
  • 9. D. A. Gudkov, The toplogy of real projective algebraic varieties, Russian Math. Surveys 29, no. 4, (1974), 3-79. MR 0399085 (53:2936)
  • 10. H.B. Lawson, Algebraic cycles and homotopy theory, Annals of Math. 129 (1989), 253-291. MR 986794 (90h:14008)
  • 11. H.B. Lawson, Spaces of algebraic cycles, Surveys. Diff. Geo., 1995, Vol 2, 137-213. MR 1375256 (97m:14006)
  • 12. P. Lima-Filho, Lawson homology for quasi- projective varieties, Compositio Math. 84 (1992), 1-23. MR 1183559 (93j:14007)
  • 13. P. Lima-Filho, The topological group structure of algebriac cycles, Duke Math. J., 75 (1994), no. 2, 467-491. MR 1290199 (95i:14010)
  • 14. P. Lima-Filho, Completions and fibrations for topological monoids, Trans. AMS, 340 (1993), no. 1, 127-147. MR 1134758 (94a:55009)
  • 15. H.B. Lawson, P. Lima-Filho, M. Michelsohn, Algebraic cycles and the classical groups I: real cyclces, Topology, 42, (2003) 467-506. MR 1941445 (2003m:14013)
  • 16. P. Samuel, Méthodes d'algèbre abstraite en géométrie algébrique, Ergebnisse der math., Springer-Verlag, 1955. MR 0072531 (17:300b)
  • 17. Norman E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133-152. MR 0210075 (35:970)
  • 18. Jyh-Haur Teh, A homology and cohomology theory for real projective varieties, preprint in, math.AG/0508238.
  • 19. R. Thom, ``Sur l'homologie des varietés algèbriques réelles'', Diff. and comb. topology, Princeton Univ. Press, Princeton-New York 1965, pp. 255-265. MR 0200942 (34:828)
  • 20. Andre Weil, On Picard varieties, Amer. J. Math., Vol. 74, No. 4, 865-894. MR 0050330 (14:314e)

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Additional Information

Jyh-Haur Teh
Affiliation: Department of Mathematics, National Tsing Hua University of Taiwan, No. 101, Kuang Fu Road, Hsinchu, 30043, Taiwan

Keywords: Harnack-Thom theorem, algebraic cycles, Lawson homology, homotopy groups, Picard varieties, classifying spaces
Received by editor(s): May 9, 2006
Received by editor(s) in revised form: September 20, 2006
Published electronically: November 28, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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