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Standard general relativity is based on Riemannian manifolds.

However, the simplest extension of Riemannian manifolds seems to be Kahler manifolds, which have a complex (hermitian) structure, a symplectic structure, a riemannian structure, all these structures being compatible. So my question is :

What is the "simplest" theory, extending general relativity, based on Kahler manifolds ?

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Kähler manifolds are no "extension" of Riemannian ones! Every Kähler manifold is Riemannian, but not every Riemannian manifold is Kähler. So we are speaking about a (small) subset of Riemannian manifolds, for which the default general relativity-formalism naturally exists. I think I didn't understand your question :) –  Tobias Diez Nov 6 '12 at 9:52
    
@altertoby: What I mean is that the compatible complex hermitian, symplectic and riemannian structures of Kahler manifolds, are an extension of the simple riemannian structure of Riemannian manifoldS. So I am interesting with theories which make use of all these compatible structures of Kahler manifolds, and not only the Riemannian structure part. –  Trimok Nov 6 '12 at 11:09
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