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Cohomologie de Dolbeault feuilletée de certaines laminations complexes

Abstract : In this thesis, we are interested in computing the foliated Dolbeault cohomology groups H0∗L (M) for some complex laminations. This amounts to solving the problem of the ∂ along the leaves ∂Lα = ω. (Here M is a metric space or a differentiable manifold if L is a foliation F.) Three situations were considered explicitly.1. Let M = Ω be an open set of C×R equipped with the foliation F whose leaves are the sections Ωt = {z ∈ C(z, t) ∈ Ω}; we say that F is the canonical foliation of Ω. Under certain conditions on Ω and growth conditions on the foliated form ω, we show that the equation ∂Fα = ω has a solution.2. Let (αn)n≥1 be a sequence of real numbers, strictly increasing with α1 = −1 and converging to 1. In C × R we consider the points A = (0, 1) and An = (0, αn) for n ≥ 1. For all n ≥ 1, let Sn be the sphere of C × R with a diameter segment [AnA] and E the union of all these spheres. Then E is a compact and connected subset of C × R. Let γ : E −→ E the homeomorphism defined by γ(w,u) = (ρn(w),u), where (w,u) ∈ Sn and ρn is the rotation in C with angle 2πn. The suspension of γ gives rise to a complex lamination L whose leaves are all equivalent Riemann surfaces isomorphic to C∗. For This example we show that the vector space H01 (L) is zero.3. Consider the manifold M = C × Rn \ {(0, 0)} (the coordinates of a point are denoted (z,t)) endowed with the complex foliation F defined by the differential system dt1 = • • • = dn = 0. The diffeomorphism γ : (z, t) ∈ M −→ (λz, λt) ∈ M (where 0 < λ < 1) acts on M freely and properly ; moreover it is an automorphism of the complex foliation F ; then F induces on the quotient M = M/γ (which is diffeomorphic to S n+1 × S1) a complex foliation F by Riemann surfaces. All leaves are isomorphic to C except one of them which is an elliptic curve. We show that the vector spaces H00 F (M) and H01F (M) of foliated Dolbeault cohomology are isomorphic to C.
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Submitted on : Thursday, October 10, 2013 - 11:33:08 AM
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Rochdi Ben Charrada. Cohomologie de Dolbeault feuilletée de certaines laminations complexes. Autre. Université de Valenciennes et du Hainaut-Cambresis; Université de Sfax (Tunisie), 2013. Français. ⟨NNT : 2013VALE0010⟩. ⟨tel-00871710⟩

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