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.



When is the first chern class of a Kaehler manifold positive/negative?
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](https://books.google.com/books?id=9lnlBwAAQBAJ&lpg=PP1&pg=PA140#v=onepage&q&f=false). 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.


