First chern class - MathOverflow most recent 30 from http://mathoverflow.net2013-05-21T06:05:03Zhttp://mathoverflow.net/feeds/question/99755http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/99755/first-chern-classFirst chern classHassan Jolany2012-06-15T22:58:23Z2012-06-16T06:39:01Z
<p>I know some examples that first Chern class has not sign(negative, positive or zero). But I am looking for a necessary and sufficient condition that first Chern class has sign. </p>
http://mathoverflow.net/questions/99755/first-chern-class/99774#99774Answer by Ben McKay for First chern classBen McKay2012-06-16T06:22:15Z2012-06-16T06:39:01Z<p>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 <a href="http://www.google.ie/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CFwQFjAE&url=http%253A%252F%252Fbooks.google.com%252Fbooks%252Fabout%252FNonlinear_Analysis_on_Manifolds_Monge_Am.html%253Fid%253DNcLiTzEwFbcC&ei=1CXcT5exDIeZhQfb4ZCXCg&usg=AFQjCNFWV_PlxirqqHua2IZDJBXm1nBWKg&sig2=6JcmG_kKO40JAkrLwCwykQhttp%3A//www.google.ie/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CFwQFjAE&url=http%253A%252F%252Fbooks.google.com%252Fbooks%252Fabout%252FNonlinear_Analysis_on_Manifolds_Monge_Am.html%253Fid%253DNcLiTzEwFbcC&ei=1CXcT5exDIeZhQfb4ZCXCg&usg=AFQjCNFWV_PlxirqqHua2IZDJBXm1nBWKg&sig2=6JcmG_kKO40JAkrLwCwykQ" rel="nofollow">Nonlinear Analysis on Manifolds: Monge-Ampere Equations</a>. 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.</p>