Blowup formulas and smooth birational invariants
Author:
Zhaohu Nie
Journal:
Proc. Amer. Math. Soc. 137 (2009), 25292539
MSC (2000):
Primary 14F43, 14E99
Published electronically:
March 20, 2009
MathSciNet review:
2497464
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Abstract: We prove that the blowup formula for the singular homology of a complex smooth projective variety with a smooth center respects two natural filtrations, namely the topological and the geometric filtrations. This then enables us to establish some smooth birational invariants.
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 Abramovich, D.; Karu, K.; Matsuki, K.; Wł odarczyk, J. Torification and factorization of birational maps. J. Amer. Math. Soc. 15 (2002), no. 3, 531572 (electronic). MR 1896232 (2003c:14016)
 [AK1]
 Arapura, D.; Kang, S.J. Functoriality of the coniveau filtration. Canad. Math. Bull. 50 (2007), no. 2, 161171. MR 2317438 (2008f:14018)
 [AK2]
 Arapura, D.; Kang, S.J. Coniveau and the Grothendieck group of varieties. Michigan Math. J. 54 (2006), no. 3, 611622. MR 2280497 (2007k:14011)
 [Bl]
 Bloch, S. Algebraic cycles and the Beĭlinson conjectures. The Lefschetz centennial conference, Part I (Mexico City, 1984), 6579, Contemp. Math., 58, Amer. Math. Soc., Providence, RI, 1986. MR 860404 (88e:14006)
 [DT]
 Dold, A.; Thom, R. Quasifaserungen und unendliche symmetrische Produkte (German). Ann. of Math. (2) 67 (1958), 239281. MR 0097062 (20:3542)
 [EV]
 Esnault, H. and Viehweg, E. DeligneBeĭ linson cohomology. Beĭlinson's conjectures on special values of functions, 4391, Perspect. Math., 4, Academic Press, Boston, MA, 1988. MR 944991 (89k:14008)
 [FG]
 Friedlander, E. M.; Gabber, O. Cycle spaces and intersection theory. Topological methods in modern mathematics (Stony Brook, NY, 1991), 325370, Publish or Perish, Houston, TX, 1993. MR 1215970 (94j:14010)
 [FM]
 Friedlander, E. M.; Mazur, B. Filtrations on the homology of algebraic varieties. With an appendix by Daniel Quillen. Mem. Amer. Math. Soc. 110 (1994), no. 529, x+110 pp. MR 1211371 (95a:14023)
 [H1]
 Hu, W. Birational invariants defined by Lawson homology. To appear in Int. J. Pure Appl. Math. arXiv:math/0511722.
 [H2]
 Hu, W. The Generalized Hodge conjecture for cycles and codimension two algebraic cycles. arXiv:math/0511725.
 [H3]
 Hu, W. Some relations between the topological and geometric filtration for smooth projective varieties. arXiv:math/0603203.
 [L1]
 Lawson, H. B., Jr. Algebraic cycles and homotopy theory. Ann. of Math. (2) 129 (1989), no. 2, 253291. MR 986794 (90h:14008)
 [Le]
 Lewis, J. D. A survey of the Hodge conjecture. Second edition. Appendix B by B. Brent Gordon. CRM Monograph Series, 10. American Mathematical Society, Providence, RI, 1999. MR 1683216 (2000a:14010)
 [LF]
 LimaFilho, P. Lawson homology for quasiprojective varieties. Compositio Math. 84 (1992), no. 1, 123. MR 1183559 (93j:14007)
 [Ma]
 Manin, Ju. I. Correspondences, motifs and monoidal transformations (Russian). Mat. Sb. (N.S.) 77 (119) 1968 475507. MR 0258836 (41:3482)
 [Pe]
 Peters, C. Lawson homology for varieties with small Chow groups and the induced filtration on the Griffiths groups. Math. Z. 234 (2000), no. 2, 209223. MR 1765879 (2001f:14042)
 [Ro]
 Roberts, Joel. Chow's moving lemma. Algebraic geometry, Oslo, 1970 (Proc. Fifth Nordic Summer School in Math.), pp. 8996. WoltersNoordhoff, Gröningen, 1972. MR 0382269 (52:3154)
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Additional Information
Zhaohu Nie
Affiliation:
Department of Mathematics, Penn State Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601
Email:
znie@psu.edu
DOI:
http://dx.doi.org/10.1090/S0002993909098724
PII:
S 00029939(09)098724
Keywords:
Lawson homology,
topological filtration,
geometric filtration,
blowup formula,
birational invariants
Received by editor(s):
October 1, 2007
Received by editor(s) in revised form:
September 30, 2008
Published electronically:
March 20, 2009
Communicated by:
Ted Chinburg
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
