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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.

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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