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Métriques naturelles associées aux familles de variétés Kahlériennes compactes

Abstract : In this thesis we consider families π:X→S of compact Kähler manifolds with zero first Chern class over a smooth base S. We construct a relative complexified Kähler cone p:K→S over the base of deformations. Then we prove the existence of natural hermitian metrics on the total spaces K and X x _SK that generalize the classical Weil--Petersson metrics associated to polarized families of such manifolds. As a byproduct we obtain a Riemannian metric on the Kähler cone of any compact Kähler manifold. We obtain an expression of its curvature tensor via an embedding of the Kähler cone into the space of hermitian metrics on the manifold. We also prove that if the manifolds in our family have trivial canonical bundle, then our generalized Weil--Petersson metric is the curvature form of a positive holomorphic line bundle. We then give some examples and applications.
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Gunnar Thor Magnusson. Métriques naturelles associées aux familles de variétés Kahlériennes compactes. Mathématiques générales [math.GM]. Université de Grenoble, 2012. Français. ⟨NNT : 2012GRENM080⟩. ⟨tel-00849096⟩

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