# On the leaf spaces of singular holomorphic foliations and multiplicities on leaves

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30.05.2001

vi, 106 p

Thèse de doctorat: Université de Fribourg, 2001

English French We study the leaf space X/F of regular or singular holomorphic foliations F on a complex manifold X. Using the theory of analytic cycles, we give certain solutions to the two following problems: First Leaf space problem Find sufficient conditions which imply that the leaf space of a foliation with leaves everywhere admits a canonical complex structure. Second Leaf space Problem What can be done if F does not have leaves everywhere, or if it has leaves everywhere but X/F is not a complex space? First we study the First Leaf space problem for regular foliations. We define the notion of the topological multiplicity µt(L) for some leaves L of the foliation. Then we define a mapping ζF :G(F) → Zd(X). Here G(F) is the set of those leaves for which the topological multiplicity is well-defined and Zd(X) is the space of the analytic cycles of dimension d (d is the dimension of the foliation) with the topology of Barlet. More precisely, ζF associates to each x ∈ G(F) the cycle µt(Lx)Lx. In theorem 6.1.1 we prove the following equivalences: X/F is a complex space ⇐⇒ G(F) = X and ζF is continuous ⇐⇒ There exists a continuous and F-invariant mapping ϕ:X → Zd(X) such that for each x ∈ X, the support of ϕ(x) is equal to Lx. ⇐⇒ The canonical mapping X → Zd(X) that associates to each x the leaf passing through x is continuous ( Zd(X) is the set of d-dimensional analytic subsets of X; the topology of Zd(X) is the Barlet topology defined in §4.4). In theorem 6.2.1 we generalize this result for certain singular holomorphic foliations that have leaves everywhere. Then we explain how the leaf space can be interpreted as a subspace of Zd(X). Finally, we use an example of Hirzebruch to illustrate theorem 6.1.1. In the last part we give a partial solution of the second problem for a particular type of foliations. We consider regular or singular holomorphic foliations for which there exists an open, dense and F-saturated subset C of X such that C/F is a complex space. Under certain conditions on such foliations we construct a generalisation of the leaf space: the meromorphic leaf space Z(F). In a first step, we associate a meromorphic equivalence relation MF to each foliation F of the above type. Then we use the theory of Grauert and Siebert on meromorphic equivalence relations to define Z(F) as the meromorphic quotient of MF. Using the theorem of Grauert-Siebert on meromorphic equivalence relations, we find a condition which implies that Z(F) has a complex structure (see theorem 7.2.5). In particular cases, we give other characterisations of Z(F). We conclude with some examples that illustrate some phenomena which can appear.
Faculty
Faculté des sciences et de médecine
Department
Département de Mathématiques
Language
• English
Classification
Mathematics
License
License undefined
Identifiers
Persistent URL
https://folia.unifr.ch/unifr/documents/299878
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