First chern class - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T06:05:03Z http://mathoverflow.net/feeds/question/99755 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99755/first-chern-class First chern class Hassan Jolany 2012-06-15T22:58:23Z 2012-06-16T06:39:01Z <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#99774 Answer by Ben McKay for First chern class Ben McKay 2012-06-16T06:22:15Z 2012-06-16T06:39:01Z <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&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=5&amp;ved=0CFwQFjAE&amp;url=http%253A%252F%252Fbooks.google.com%252Fbooks%252Fabout%252FNonlinear_Analysis_on_Manifolds_Monge_Am.html%253Fid%253DNcLiTzEwFbcC&amp;ei=1CXcT5exDIeZhQfb4ZCXCg&amp;usg=AFQjCNFWV_PlxirqqHua2IZDJBXm1nBWKg&amp;sig2=6JcmG_kKO40JAkrLwCwykQhttp%3A//www.google.ie/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=5&amp;ved=0CFwQFjAE&amp;url=http%253A%252F%252Fbooks.google.com%252Fbooks%252Fabout%252FNonlinear_Analysis_on_Manifolds_Monge_Am.html%253Fid%253DNcLiTzEwFbcC&amp;ei=1CXcT5exDIeZhQfb4ZCXCg&amp;usg=AFQjCNFWV_PlxirqqHua2IZDJBXm1nBWKg&amp;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>