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](http://www.math.cuhk.edu.hk/~kwchan/MYIneq.pdf), 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](http://www.pnas.org/content/74/5/1798.full.pdf), 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?



Condition on the canonical divisor for Yau Inequality - effective or ample?
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.


