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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 is that the Ricci curvature is positive, etc. 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 is that the integral of the Ricci form is positive over all compact complex curves in the manifold.