# Kähler manifold which is not algebraic

Can someone provide examples of Kähler manifolds which are not algebraic?

This question came to my mind seeing the post of Andrea Ferretti.

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generic complex tori in complex dimension 2 or higher. MR

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You mean those which are not Abelian Varieties? Thanks! –  Anweshi Jan 29 '10 at 20:55
yes that is what I meant.MR –  Mohan Ramachandran Jan 29 '10 at 21:04
ramachndran ==> ramachandran? –  Anweshi Jan 30 '10 at 14:32

In (complex) dimension greater or equal with 2 there are complex tori which are not algebraic. A criterion for projectivity is that of Riemann (see http://en.wikipedia.org/wiki/Abelian_variety#Riemann_conditions). Also there are K3 surfaces which are not algebraic.

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It doesn't in itself give specific examples, but the theoretical answer to the question "When is a compact Kahler manifold algebraic?" is given by the Kodaira embedding theorem. A nice exposition is given in Richard Wells' Differential Analysis on Complex Manifolds. Indeed, the KET is the crescendo to which the entire book builds.

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Thanks. This is helpful. As it must have been obvious from the question, I knew dear nothing about Kahler manifold beyond their definition, and an awareness of their importance. –  Anweshi Jan 29 '10 at 21:36

You might want to take a look at this previous MO question.

There, I mentioned Voisin's results disproving Kodaira's conjecture (every Kahler manifold is deformation equivalent to a projective manifold).

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