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Residue formulas for meromorphic functions on surfaces. (English) Zbl 0933.32044

The paper is about a special case of Baum-Bott residues formula of a singular holomorphic foliation on a compact complex surface \(X\). This formula is described when the foliation is defined by the differential \(d\varphi\) of a meromorphic function \(\varphi\) on \(X\) (leaves are the level sets of \(\varphi\), and singularities of the foliation are the critical and indeterminacy points of \(\varphi\)). In this setting, Baum-Bott residues are computed in terms of the zero and pole divisors of \(d\varphi\). This is applied to the case where \(X\) is the complex projective plane \(\text{ P}^2\) and \(\varphi\) is defined by a complex polynomial \(f\) of two variables, obtaining a nice formula of D. T. Lê, which involves Baum-Bott residues, Milnor numbers at critical points of \(f\), and the degree of \(f\). Applications to other compactifications of the complex plane arising from polynomials are also considered.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
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References:

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