Holomorphic foliations by curves on \(\mathbb P^3\) with non-isolated singularities. (English) Zbl 1129.32018

The paper under review studies singular holomorphic foliations \(\mathcal F\) by curves on \(\mathbb{CP}^3\) with the following properties: – the singular set is a union of a finite number of isolated singularities \(p_1,\dots,p_q\in\mathbb{CP}^3\) and a regular holomorphic curve \(C\subset\mathbb{CP}^3\); – the latter curve \(C\) is special, i.e., the blowing up along \(C\) yields a foliation with isolated singularities. Each foliation \(\mathcal F\) under consideration is defined by a polynomial vector field in any affine chart. The degree of the foliation is the maximal degree of the corresponding vector field minus one. The multiplicity of the foliation along the curve \(C\) is defined by using the restriction of the corresponding vector field to a local transversal section to \(C\). This is the minimal power of the lower term of the latter restriction.
The main result (Theorem 1.1) expresses the sum of the multiplicities \(\mu(\mathcal F,p_j)\) of the isolated singularities via the genus and the degree of the singularity curve \(C\), the degree of the foliation and its multiplicity along the curve \(C\). A small perturbation of the foliation \(\mathcal F\) in the class of foliations of the same degree destroys the singularity curve \(C\) and replaces it by a finite number of isolated singularities. Theorem 1.1 implies an explicit formula for the sum of their multiplicities.
The proof of Theorem 1.1 is based on the Baum-Bott’s formula [P. F. Baum and R. Bott, Essays Topol. Relat. Top., Mem. ded. Georges de Rham, 29–47 (1970; Zbl 0193.52201)]. The sum of the multiplicities \(\mu(\mathcal F,p_j)\) is expressed via Chern classes of appropriate vector bundles (on \(\mathbb{CP}^3\), its blowing up and the exceptional divisor) associated to the foliation \(\mathcal F\). The latter Chern classes are calculated by using Porteous’ theorem [I. R. Porteous, Proc. Camb. Philos. Soc. 56, 118–124 (1960; Zbl 0166.16701)], which relates the Chern classes of the tangent bundle of a manifold and that of its blowing up.


32S65 Singularities of holomorphic vector fields and foliations
37F75 Dynamical aspects of holomorphic foliations and vector fields
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