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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.

edited Jun 15 2012 at 23:33

asked Jun 15 2012 at 22:58

Hassan Jolany
1,165●1●2●12

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What do you mean by "has sign"? Do you mean "is negative"? – Misha Jun 15 2012 at 23:31

I revised my question – Hassan Jolany Jun 15 2012 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 2012 at 23:44

Dear Donu, I mean on Kahler manifolds with complex dimension n – Hassan Jolany Jun 16 2012 at 0:51

Ben McKay · Accepted Answer · 2012-06-16 06:39:01Z UTC

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.

First chern class

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