Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|cite|improve this question
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 海桑乔朗丽 Jun 15 '12 at 23:34
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 海桑乔朗丽 Jun 16 '12 at 0:51

1 Answer 1

up vote 11 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.

share|cite|improve this answer
sounds nice, Thanks Ben – Hassan Jolany 海桑乔朗丽 Jun 16 '12 at 10:51

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.