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Let $X$ be a complex, projective, nonsingular variety. We also understand it as a Kähler Manifold. My question now is, when people say $c_1(X) < 0$, what exactly do they mean? Let me elaborate. In this paper, it is said that Yau's inequality

$$ (-1)^n c_1^n \le (-1)^n \frac{2(n+1)}{n} c_2 c_1^{n-2} $$

holds under the condition that $c_1(X) < 0$. I would have thought that this is equivalent to $K_X$ being effective. In the original paper, Yau requires $X$ to have ample canonical class, however. Now, I am wondering: For the above equality to hold, do I need $K_X$ to be ample or does it suffice for $K_X$ to be effective?

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You should read $c_1(X)<0$ as saying that the first Chern class of $T_X$ is negative, or the line bundle $K_X$ is positive, in the sense of curvature. But positive line bundles are ample line bundles. This fact is sometimes called the Kodaira Embedding Theorem. See for example p. 181 of Griffiths-Harris, Principles of Algebraic Geometry.

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In the case of surfaces the inequality is true EDIT: with the only exception of surfaces ruled over a curve of genus $>1$. It has beeen proven by Miyaoka [On the Chern numbers of surfaces of general type. Invent. Math. 42 (1977), 225–237] for surfaces of general type. For surfaces not of general type it can be proven easily by looking at the classification.

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  • $\begingroup$ With the exception of ruled surfaces of genus $g > 1$, I suppose. $\endgroup$
    – Jesko Hüttenhain
    Commented Nov 5, 2011 at 12:04
  • $\begingroup$ @Jesko: you are right, thank you. I will edit the answer. $\endgroup$
    – rita
    Commented Nov 5, 2011 at 14:37

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