×

A remark on the Chern classes of local complete intersections. (English) Zbl 0902.32017

Let \(X\) be an \(n\)-dimensional compact analytic variety with isolated singularities \(p_1, \ldots, p_r \in X.\) Assume additionally that \(X\) is a strong local complete intersection in the sense [D. Lehmann and T. Suwa, J. Differ. Geom. 42, 165-192 (1995; Zbl 0844.32007)]. The authors show that the difference between the Chern-Mather and Fulton-Johnson’s Chern classes of \(X\) is equal to \((-1)^{n+1}\sum_{i=1}^r m_n(X,p_i),\) where \(m_n(X,p_i)\) is the \(n\)-th polar multiplicity of \(X\) at the singular point \(p_i\), \(i=1,\dots, r\). The prove is based essentially on results of T. Suwa [C. R. Acad. Sci., Paris, Sér. I 324, No. 1, 67-70 (1997)] and L. Ernström [Duke Math. J. 76, 1-21 (1994; Zbl 0831.32016)].

MSC:

32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14M10 Complete intersections
19L10 Riemann-Roch theorems, Chern characters
13H15 Multiplicity theory and related topics
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19E08 \(K\)-theory of schemes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. Aluffi: Singular schemes of hypersurfaces. Duke Math. J., 80, 325-351 (1995). · Zbl 0876.14028 · doi:10.1215/S0012-7094-95-08014-4
[2] P. Aluffi: Chera classes for singular hypersurfaces (1996) (preprint). · Zbl 0972.57015 · doi:10.1090/S0002-9947-99-02256-4
[3] J.-P. Brasselet and M.-H. Schwartz : Sur les classes de Chern d’un ensemble analytigue complexe, Caract6ristique d’Euler-Poincare. Society Mathe-matique de France, Asterisque 82-83, 93-147 (1981). · Zbl 0471.57006
[4] A. Dubson : Classes caracteristiques des varietes singulieres. C.R. Acad. Sci. Paris, 287, 237-240 (1978). · Zbl 0387.14005
[5] L. Ernstrom : Duality for the local Euler obstruction with applications to real and complex singularities. Ph. D. Thesis, MIT (1993).
[6] L. ErnstrOm : Topological Radon transforms and the local Euler obstruction. Duke Math. J., 76, 1-21 (1994). · Zbl 0831.32016 · doi:10.1215/S0012-7094-94-07601-1
[7] W. Fulton : Intersection Theory. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1984). · Zbl 0541.14005
[8] W. Fulton and K. Johnson : Canonical classes on singular varieties. Manuscripta Math., 32, 381-389 (1980). · Zbl 0451.14001 · doi:10.1007/BF01299611
[9] T. Gaffney : Multiplicities and equisingularity of ICIS qerms. Invent. Math., 123, 209-220 (1996). · Zbl 0846.32024 · doi:10.1007/s002220050022
[10] G.-M. Greuel: Der GauB-Manin Zusammenhang isolierter Singularitaten von vollstandigen Durch-schnitten. Math. Ann., 214, 235-266 (1975). · Zbl 0285.14002 · doi:10.1007/BF01352108
[11] H. Hamm: Lokale topologische Eigenschaften komplexer Raume. Math. Ann., 191, 235-252 (1971). ] 12 · Zbl 0214.22801 · doi:10.1007/BF01578709
[12] M. Kasjiiwara: Index theorem for a maximally overdetermined system of linear differential equations. Proc. Japan Acad., 49A, 803-804 (1973). · Zbl 0305.35073 · doi:10.3792/pja/1195519148
[13] D. Lehmann and T. Suwa: Residues of holomor-phic vector fields relative to singular invariant subvarieties. J. Differential Geom., 42, 165-192 (1995). · Zbl 0844.32007
[14] D.-T. Le: Calculation of Milnor number of isolated singularity of complete intersection. Funct. Anal. Appl., 8, 127-131 (1974). · Zbl 0351.32007 · doi:10.1007/BF01078597
[15] D.-T. Le and B. Teissier: Varietes polaires locales et classes de Chern des varietes singulieres. Ann. of Math., 114, 457-491 (1981). JSTOR: · Zbl 0572.14002 · doi:10.2307/1971299
[16] E. Looijenga: Isolated Singular Points on Complete Intersections. London Mathematical Society Lecture Note, ser. 77, Cambridge Univ. Press, Cambridge, London, New York, New Rochelle, Meldourne, Sydney (1984). · Zbl 0552.14002
[17] R. MacPherson : Chern classes for singular algebraic varieties. Ann. of Math., 100, 423-432 (1974). JSTOR: · Zbl 0311.14001 · doi:10.2307/1971080
[18] J. Seade and T. Suwa: An adjunction formula for local complete intersections (1996) (preprint). · Zbl 0918.32019 · doi:10.1142/S0129167X98000324
[19] T. Suwa: Classes de Chern des intersections completes locales. C.R. Acad. Sci. Paris, 324, 67-70 (1996). · Zbl 0957.14034 · doi:10.1016/S0764-4442(97)80105-X
[20] B. Teissier: Varietes polaires II, Multipliers polaires, sections planes, et conditions de Whitney. Algebraic Geometry, Proceedings, La Rabida, 1981, Lecture Notes Math., no. 961, Springer-Verlag, Berlin, Heidelberg, New York, pp. 314-491 (1982). · Zbl 0585.14008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.