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Transactions of the American Mathematical Society

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Chern classes for singular hypersurfaces


Author: Paolo Aluffi
Journal: Trans. Amer. Math. Soc. 351 (1999), 3989-4026
MSC (1991): Primary 14C17, 32S60
DOI: https://doi.org/10.1090/S0002-9947-99-02256-4
Published electronically: February 8, 1999
MathSciNet review: 1697199
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Abstract: We prove a formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety as a variation on another definition of the homology Chern class of singular varieties, introduced by W. Fulton; and we discuss the relation between these classes and others, such as Mather's Chern class and the $\mu $-class we introduced in previous work.


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  • [Aluffi1] P. Aluffi, Singular schemes of hypersurfaces, Duke Math. J. 80 (1995), 325-351. MR 97b:14057
  • [Aluffi2] P. Aluffi, MacPherson's and Fulton's Chern Classes of Hypersurfaces, I.M.R.N. (1994), 455-465. MR 96d:14004
  • [A-F] P. Aluffi, C. Faber, A remark on the Chern class of a tensor product, Manu. Math. 88 (1995), 85-86. MR 96e:14002
  • [B-S] J.-P. Brasselet, M.-H. Schwartz, Sur les classes de Chern d'un ensemble analytique complexe, Astérisque 82-83 (1981), 93-147. MR 83h:32011
  • [BDK] J.-L. Brylinski, A. Dubson, M. Kashiwara, Formule de l'indice pour les Modules Holonomes et obstruction d'Euler locale, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 573-576. MR 83a:32010
  • [BFM] P. Baum, W. Fulton, R. MacPherson, Riemann-Roch for singular variety, Publ. Math. I.H.E.S. 45 (1975), 101-145. MR 54:317
  • [BMM] J. Briançon, P. Maisonobe, M. Merle, Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), 531-550. MR 95h:32043
  • [Fulton] W. Fulton, Intersection Theory, Springer Verlag, 1984. MR 85k:14004
  • [Ginsburg] V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), 327-402. MR 87j:32030
  • [Kennedy] G. Kennedy, MacPherson's Chern classes of singular algebraic varieties, Comm. in Algebra 18 (1990), 2821-2839. MR 91h:14010
  • [Kwieci\'{n}ski] M. Kwieci\'{n}ski, Sur le transformé de Nash et la construction du graphe de MacPherson, Thèse, Université de Provence (1994).
  • [Lê-Mebkhout] Lê Dung Tráng and Z. Mebkhout, Variétés caractéristiques et variétés polaires, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 129-132. MR 84g:32018
  • [MacPherson] R. MacPherson, Chern classes for singular algebraic varieties, Annals of Math. 100 (1974), 423-432. MR 50:13587
  • [Massey] D. Massey, Numerical invariants of perverse sheaves, Duke Math. J. 73 (1994), 307-369. MR 95e:32045
  • [Nobile] A. Nobile, Some properties of the Nash blowing-up, Pacific J. of Math. 60 (1975), 297-305. MR 53:13217
  • [Parusi\'{n}ski1] A. Parusi\'{n}ski, A generalization of the Milnor number, Math. Ann. 281 (1988), 247-254. MR 89k:32023
  • [Parusi\'{n}ski2] A. Parusi\'{n}ski, Limits of tangent spaces to fibres and the $w_{f}$ condition, Duke Math. J. 72 (1993), 99-108. MR 94i:32059
  • [Sabbah] C. Sabbah, Quelque remarques sur la géométrie des espaces conormaux, Astérisque 130 (1985), 161-192. MR 87f:32031
  • [Schwartz] M.-H. Schwartz, Classes caractéristiques définies par une stratification d'une variété analytique complexe, C. R. Acad. Sci. Paris 260 (1965), 3262-3264, 3535-3537. MR 32:1727; MR 35:3707
  • [Silvotti] R. Silvotti, On a conjecture of Varchenko, Invent. Math. 126 (1996), 235-248. MR 98d:32038
  • [Suwa] T. Suwa, Classes de Chern des intersections complètes locales, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 67-70. MR 97m:14003

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Additional Information

Paolo Aluffi
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306
Email: aluffi@math.fsu.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02256-4
Received by editor(s): June 3, 1997
Published electronically: February 8, 1999
Additional Notes: Supported in part by NSF grant DMS-9500843
Article copyright: © Copyright 1999 American Mathematical Society

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