Harnack-Thom theorem for higher cycle groups and Picard varieties
HTML articles powered by AMS MathViewer
- by Jyh-Haur Teh PDF
- Trans. Amer. Math. Soc. 360 (2008), 3263-3285 Request permission
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.References
- D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR 1156302
- A. I. Degtyarev and V. M. Kharlamov, Topological properties of real algebraic varieties: Rokhlin’s way, Uspekhi Mat. Nauk 55 (2000), no. 4(334), 129–212 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 4, 735–814. MR 1786731, DOI 10.1070/rm2000v055n04ABEH000315
- Eric M. Friedlander, Algebraic cycles, Chow varieties, and Lawson homology, Compositio Math. 77 (1991), no. 1, 55–93. MR 1091892
- Eric M. Friedlander, Filtrations on algebraic cycles and homology, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 3, 317–343. MR 1326671
- Y. Félix, S. Halperin, J. Thomas, Rational homotopy theory, Springer, 2000.
- Eric M. Friedlander and H. Blaine Lawson Jr., A theory of algebraic cocycles, Ann. of Math. (2) 136 (1992), no. 2, 361–428. MR 1185123, DOI 10.2307/2946609
- Eric M. Friedlander and H. Blaine Lawson, Moving algebraic cycles of bounded degree, Invent. Math. 132 (1998), no. 1, 91–119. MR 1618633, DOI 10.1007/s002220050219
- Eric M. Friedlander and H. Blaine Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), no. 2, 533–565. MR 1415605, DOI 10.1016/0040-9383(96)00011-0
- D. A. Gudkov, The topology of real projective algebraic varieties, Uspehi Mat. Nauk 29 (1974), no. 4(178), 3–79 (Russian). Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ, II. MR 0399085
- H. Blaine Lawson Jr., Algebraic cycles and homotopy theory, Ann. of Math. (2) 129 (1989), no. 2, 253–291. MR 986794, DOI 10.2307/1971448
- H. Blaine Lawson Jr., Spaces of algebraic cycles, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 137–213. MR 1375256
- Paulo Lima-Filho, Lawson homology for quasiprojective varieties, Compositio Math. 84 (1992), no. 1, 1–23. MR 1183559
- Paulo Lima-Filho, The topological group structure of algebraic cycles, Duke Math. J. 75 (1994), no. 2, 467–491. MR 1290199, DOI 10.1215/S0012-7094-94-07513-3
- Paulo Lima-Filho, Completions and fibrations for topological monoids, Trans. Amer. Math. Soc. 340 (1993), no. 1, 127–147. MR 1134758, DOI 10.1090/S0002-9947-1993-1134758-4
- H. Blaine Lawson, Paulo Lima-Filho, and Marie-Louise Michelsohn, Algebraic cycles and the classical groups. I. Real cycles, Topology 42 (2003), no. 2, 467–506. MR 1941445, DOI 10.1016/S0040-9383(02)00018-6
- P. Samuel, Méthodes d’algèbre abstraite en géométrie algébrique, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 4, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955 (French). MR 0072531
- N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. MR 210075
- Jyh-Haur Teh, A homology and cohomology theory for real projective varieties, preprint in Arxiv.org, math.AG/0508238.
- René Thom, Sur l’homologie des variétés algébriques réelles, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 255–265 (French). MR 0200942
- André Weil, On Picard varieties, Amer. J. Math. 74 (1952), 865–894. MR 50330, DOI 10.2307/2372230
Additional Information
- Jyh-Haur Teh
- Affiliation: Department of Mathematics, National Tsing Hua University of Taiwan, No. 101, Kuang Fu Road, Hsinchu, 30043, Taiwan
- Email: jyhhaur@math.nthu.edu.tw
- Received by editor(s): May 9, 2006
- Received by editor(s) in revised form: September 20, 2006
- Published electronically: November 28, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 3263-3285
- MSC (2000): Primary 14C25, 14P25; Secondary 55Q52, 55N35
- DOI: https://doi.org/10.1090/S0002-9947-07-04432-7
- MathSciNet review: 2379796