Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T03:11:53.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Chern classes of blow-ups

Published online by Cambridge University Press:  04 August 2009

PAOLO ALUFFI
PAOLO ALUFFI*
Affiliation:
Mathematics Department, Florida State University, Tallahassee FL 32306, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the standard argument for the nonsingular case, involving Riemann–Roch without denominators. The new approach relies on the explicit computation of an ideal, and a mild generalization of a well-known formula for the normal bundle of a proper transform ([8, B·6·10]).

We also discuss alternative, very short proofs of the standard formula in some cases: an approach relying on the theory of Chern–Schwartz–MacPherson classes (working in characteristic 0), and an argument reducing the formula to a straightforward computation of Chern classes for sheaves of differential 1-forms with logarithmic poles (when the center of the blow-up is a complete intersection).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 2 , March 2010 , pp. 227 - 242
Copyright
Copyright © Cambridge Philosophical Society 2009

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

REFERENCES

[1]Aluffi, P.Differential forms with logarithmic poles and Chern–Schwartz–MacPherson classes of singular varieties. C. R. Acad. Sci. Paris Sér. I Math. 329 (7) (1999), 619–624.Google Scholar
[2]Aluffi, P.Modification systems and integration in their Chow groups. Selecta Math. (N.S.) 11 (2) (2005), 155–202.Google Scholar
[3]Aluffi, P.Limits of Chow groups, and a new construction of Chern–Schwartz–MacPherson classes. Pure Appl. Math. Q. 2 (4) (2006), 915–941.Google Scholar
[4]Aluffi, P. and Brasselet, J.-P.Une nouvelle preuve de la concordance des classes définies par M.-H. Schwartz et par R. MacPherson. Bull. Soc. Math. France 136 (2) (2008), 159–166.Google Scholar
[5]Aluffi, P. and Marcolli, M.Feynman graphs of banana motives. Comm. in Number Theory and Physics 3 (2009), 1–57.CrossRefGoogle Scholar
[6]Andreas, B. and Curio, G.On discrete twist and four-flux in n=1 heterotic/f-theory compactifications. Adv. Theor. Math. Phys. 3 (1999), 1325.CrossRefGoogle Scholar
[7]Brasselet, J.-P. and Schwartz, M.-H. Sur les classes de Chern d'un ensemble analytique complexe. In Caractéristique d'Euler-Poincaré, Astérisque 83, 93–147. (Soc. Math. France, 1981).Google Scholar
[8]Fulton, W.Intersection Theory (Springer-Verlag 1984).Google Scholar
[9]Geiges, H. and Pasquotto, F.A formula for the Chern classes of symplectic blow-ups. J. Lond. Math. Soc. (2) 76 (2), (2007), 313–330.CrossRefGoogle Scholar
[10]Lascu, A. T. and Scott, D. B.An algebraic correspondence with applications to projective bundles and blowing up Chern classes. Ann. Mat. Pura Appl. (4), 102 (1975), 1–36.Google Scholar
[11]Lascu, A. T. and Scott, D. B.Un polynôme invariant par l'éclatement d'un intersection complète. C. R. Acad. Sci. Paris Sér. A-B, 282 (15):Aii (1976), A789–A792.Google Scholar
[12]Lascu, A. T. and Scott, D. B.A simple proof of the formula for the blowing up of Chern classes. Amer. J. Math. 100 (2) (1978), 293–301.Google Scholar
[13]MacPherson, R. D.Chern classes for singular algebraic varieties. Ann. of Math. (2), 100 (1974), 423–432.Google Scholar
[14]Porteous, I. R.Blowing up Chern classes. Proc. Camb. Phil. Soc. 56 (1960), 118–124.Google Scholar
[15]Segre, B.Dilatazioni e varietà canoniche sulle varietà algebriche. Ann. Mat. Pura Appl. (4), 37 (1954), 139–155.Google Scholar
[16]Todd, John ArthurBirational transformations with a fundamental surface. Proc. London Math. Soc. (2), 47 (1941), 81–100.Google Scholar