# Métriques naturelles associées aux familles de variétés Kahlériennes compactes

Abstract : In this thesis we consider families $pi : cc X to S$ of compact K"ahler manifolds with zero first Chern class over a smooth base $S$. We construct a relative complexified K"ahler cone $p : cc K to S$ over the base of deformations. Then we prove the existence of natural hermitian metrics on the total spaces $cc K$ and $cc X times_S cc K$ that generalize the classical Weil--Petersson metrics associated to polarized families of such manifolds. As a byproduct we obtain a Riemannian metric on the K"ahler cone of any compact K"ahler manifold. We obtain an expression of its curvature tensor via an embedding of the K"ahler 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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### Citation

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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