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I know some examples of compact complex manifolds whose first Chern class does not have a definite sign (is neither negative, nor positive nor zero on all complex curves). I am looking for a necessary and sufficient condition that the first Chern class has a definite sign.

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    $\begingroup$ What do you mean by "has sign"? Do you mean "is negative"? $\endgroup$
    – Misha
    Jun 15, 2012 at 23:31
  • $\begingroup$ I revised my question $\endgroup$
    – user21574
    Jun 15, 2012 at 23:34
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    $\begingroup$ 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. $\endgroup$ Jun 15, 2012 at 23:44
  • $\begingroup$ Dear Donu, I mean on Kahler manifolds with complex dimension n $\endgroup$
    – user21574
    Jun 16, 2012 at 0:51

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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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