A generalization of the Milnor number (Q1821909)

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.