Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Blow-up formulas and smooth birational invariants


Author: Zhaohu Nie
Journal: Proc. Amer. Math. Soc. 137 (2009), 2529-2539
MSC (2000): Primary 14F43, 14E99
DOI: https://doi.org/10.1090/S0002-9939-09-09872-4
Published electronically: March 20, 2009
MathSciNet review: 2497464
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the blow-up 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.


References [Enhancements On Off] (What's this?)

  • [AKMW] Abramovich, D.; Karu, K.; Matsuki, K.; Wł odarczyk, J. Torification and factorization of birational maps. J. Amer. Math. Soc. 15 (2002), no. 3, 531-572 (electronic). MR 1896232 (2003c:14016)
  • [AK1] Arapura, D.; Kang, S.-J. Functoriality of the coniveau filtration. Canad. Math. Bull. 50 (2007), no. 2, 161-171. MR 2317438 (2008f:14018)
  • [AK2] Arapura, D.; Kang, S.-J. Coniveau and the Grothendieck group of varieties. Michigan Math. J. 54 (2006), no. 3, 611-622. MR 2280497 (2007k:14011)
  • [Bl] Bloch, S. Algebraic cycles and the Beĭlinson conjectures. The Lefschetz centennial conference, Part I (Mexico City, 1984), 65-79, 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), 239-281. MR 0097062 (20:3542)
  • [EV] Esnault, H. and Viehweg, E. Deligne-Beĭ linson cohomology. Beĭlinson's conjectures on special values of $ L$-functions, 43-91, 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), 325-370, 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 $ 1$-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, 253-291. 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] Lima-Filho, P. Lawson homology for quasiprojective varieties. Compositio Math. 84 (1992), no. 1, 1-23. MR 1183559 (93j:14007)
  • [Ma] Manin, Ju. I. Correspondences, motifs and monoidal transformations (Russian). Mat. Sb. (N.S.) 77 (119) 1968 475-507. 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, 209-223. MR 1765879 (2001f:14042)
  • [Ro] Roberts, Joel. Chow's moving lemma. Algebraic geometry, Oslo, 1970 (Proc. Fifth Nordic Summer School in Math.), pp. 89-96. Wolters-Noordhoff, Gröningen, 1972. MR 0382269 (52:3154)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14F43, 14E99

Retrieve articles in all journals with MSC (2000): 14F43, 14E99


Additional Information

Zhaohu Nie
Affiliation: Department of Mathematics, Penn State Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601
Email: znie@psu.edu

DOI: https://doi.org/10.1090/S0002-9939-09-09872-4
Keywords: Lawson homology, topological filtration, geometric filtration, blow-up 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.

American Mathematical Society