Condition on the canonical divisor for Yau Inequality - effective or ample? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:08:15Z http://mathoverflow.net/feeds/question/80061 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80061/condition-on-the-canonical-divisor-for-yau-inequality-effective-or-ample Condition on the canonical divisor for Yau Inequality - effective or ample? Jesko Hüttenhain 2011-11-04T15:59:19Z 2011-11-05T14:39:31Z <p>Let $X$ be a complex, projective, nonsingular variety. We also understand it as a Kähler Manifold. My question now is, when people say $c_1(X) &lt; 0$, what exactly do they mean? Let me elaborate. In <a href="http://www.math.cuhk.edu.hk/~kwchan/MYIneq.pdf" rel="nofollow">this paper</a>, it is said that Yau's inequality</p> <p>$$(-1)^n c_1^n \le (-1)^n \frac{2(n+1)}{n} c_2 c_1^{n-2}$$</p> <p>holds under the condition that $c_1(X) &lt; 0$. I would have thought that this is equivalent to $K_X$ being effective. In the <a href="http://www.pnas.org/content/74/5/1798.full.pdf" rel="nofollow">original paper</a>, Yau requires $X$ to have <b>ample</b> 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?</p> http://mathoverflow.net/questions/80061/condition-on-the-canonical-divisor-for-yau-inequality-effective-or-ample/80070#80070 Answer by Jack Huizenga for Condition on the canonical divisor for Yau Inequality - effective or ample? Jack Huizenga 2011-11-04T17:09:31Z 2011-11-04T17:15:45Z <p>You should read $c_1(X)&lt;0$ as saying that the first Chern class of $T_X$ is <em>negative</em>, or the line bundle $K_X$ is <em>positive</em>, 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.</p> http://mathoverflow.net/questions/80061/condition-on-the-canonical-divisor-for-yau-inequality-effective-or-ample/80076#80076 Answer by rita for Condition on the canonical divisor for Yau Inequality - effective or ample? rita 2011-11-04T18:02:15Z 2011-11-05T14:39:31Z <p>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. </p>