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I know some examples that first Chern class has not sign(negative, positive or zero). But I am looking for a necessary and sufficient condition that first Chern class has sign.

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What do you mean by "has sign"? Do you mean "is negative"? –  Misha Jun 15 '12 at 23:31
    
I revised my question –  Hassan Jolany Jun 15 '12 at 23:34
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Positive in the sense of Kahler geometry? You can look up Kodaira's embedding theorem. But I'm not sure if that's what you're after. –  Donu Arapura Jun 15 '12 at 23:44
    
Dear Donu, I mean on Kahler manifolds with complex dimension n –  Hassan Jolany Jun 16 '12 at 0:51
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up vote 10 down vote accepted

The first Chern class of a Kaehler manifold is the cohomology class of the Ricci form. A sufficient condition that the first Chern class is positive (negative) is that the Ricci curvature is positive (negative). See Aubin's book Nonlinear Analysis on Manifolds: Monge-Ampere Equations. A sufficient condition for the first Chern class being nonnegative is that you find a holomorphic section of the anticanonical bundle which is nonzero on a dense open set. A necessary condition for the first Chern class to be positive (negative) is that the integral of the Ricci form is positive (negative) over all compact complex curves in the manifold.

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sounds nice, Thanks Ben –  Hassan Jolany Jun 16 '12 at 10:51
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