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Irreducible components of the space of holomorphic foliations. (English) Zbl 0811.58006

A codimension one holomorphic foliation with singularities on a complex manifold \(M\) is given by a family \(\omega_ \alpha\) of holomorphic non- identically vanishing 1-forms defined on an open cover \(\{U_ \alpha\}\), satisfying the Frobenius conditions \(\omega_ \alpha \wedge d \omega_ \alpha \equiv 0\) and the cocycle condition \(\omega_ \alpha \equiv \lambda_{\alpha \beta} \omega_ \beta\), i.e., it is given by an element \(\omega \in H^ 0(M, \Omega^ 1(L))\) where \(L\) is a holomorphic line bundle. If moreover \(M\) is a projective manifold of dimension \(\geq 3\) and \(H^ 1 (M,\mathbb{C}) = 0\), then the author proves that any integrable deformation of a generic logarithmic foliation represented by a section \(\omega = f_ 1 \dots f_ k \sum \lambda_ i df_ i/f_ i\) \((k \geq 3)\) is again of this type. It follows that the set \(\text{Log}(M, \{L_ i\})\) of all these foliations \(\omega\) (where \(f_ i \in H^ 0(M, {\mathcal O}(L_ i))\), \(L_ i\) are ample line bundles) is an irreducible component of the variety \(\text{Fol}(M, L_ 1 \otimes \cdots \otimes L_ k)\) of all foliations represented by sections of \(T^* M \otimes L_ 1 \otimes \cdots \otimes L_ k\).
Reviewer: J.Chrastina (Brno)

MSC:

58A17 Pfaffian systems
32S65 Singularities of holomorphic vector fields and foliations
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References:

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