## 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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