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A generalization of the Milnor number. (English) Zbl 0617.32012

Let M be an n-dimensional connected complex manifold and v be a holomorphic section of a holomorphic line bundle L over M. Take a connected component Y of the zero set X of v and any holomorphic connection \(D=D'+{\bar \partial}\) on L. Then Y is a connected component of the zero set of D’v. Take a small neighbourhood U of Y. The obstruction number, denotes by \(\mu\) (X,Y), to extending D’v restricted to Fr(U) onto U does not depend on D and U and generalizes the notion of Milnor number of an isolated singularity. Using this number the author establishes a formula describing the behaviour of Milnor number under blowing-ups.
For M compact he introduces \(\mu\) (X) as the sum of \(\mu\) (X,Y) taken over all connected components of Sing X and proves that it can be written in terms of Chern numbers and the Euler characteristic which generalizes the well-known formula for the Euler characteristic of a submanifold.

MSC:

32S05 Local complex singularities
32S30 Deformations of complex singularities; vanishing cycles
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
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References:

[1] Amann, H., Weiss, S.: On the uniqueness of the topological degree. Math. Z.130, 39–54 (1973) · Zbl 0249.55004 · doi:10.1007/BF01178975
[2] Greub, W., Halperin, S., Vanstone, R.: Connections, curvature, and cohomology. New York London: Academic Press 1972 · Zbl 0322.58001
[3] Hirzebruch, F.: Topological methods in algebraic geometry. New York: Springer 1966 · Zbl 0138.42001
[4] Lê, D.T., Teissier, B.: Cycles évanescents, section planes et conditions de Whitney. II. Proc. Symp. Pure Math.40, 65–103 (1983) · Zbl 0532.32003
[5] Łojasiewicz, S.: Ensembles semi-analytiques. Inst. Hautes Etud. Sci. Preprint (1965) · Zbl 0241.32005
[6] Łojasiewicz, S.: Triangulation of semi-analytic sets. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.18, 449–474 (1964) · Zbl 0128.17101
[7] Milnor, J.: Singular points of complex hypersurfaces. Ann. of Math. Studies, No. 61. Princeton: Princeton Univ. Press 1968 · Zbl 0184.48405
[8] Pham, F.: Courbes discriminantes des singularités planes d’ordre 3. Astérisque7–8, 363–391 (1973)
[9] Sullivan, D.: Combinatorial invariants of analytic spaces. Proc. Liverpool Singularities Sympos. I. (Lect. Notes Math., Vol. 192, pp. 165–168). Berlin Heidelberg New York: Springer 1971 · Zbl 0227.32005
[10] Teissier, B.: Cycles évanescents, section planes et conditions de Whitney. Astérisque7–8, 285–362 (1973) · Zbl 0295.14003
[11] Teissier, B.: Variété polaires II. (Lect. Notes Math., Vol. 961, pp. 314–491). Berlin Heidelberg New York: Springer 1982
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