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 , etc. (negative). See Aubin's book Nonlinear Analysis on Manifolds: Monge-Ampere Equations. 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.
|
3 | added 80 characters in body | ||
|
|
||||
|
|
2 | added 38 characters in body; added 122 characters in body | ||
|
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 is that the Ricci curvature is positive, etc. See Aubin's book Nonlinear Analysis on Manifolds: Monge-Ampere Equations. Another 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 is that the integral of the Ricci form is positive over all compact complex curves in the manifold. |
||||
|
|
1 |
|
||
|
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 is that the Ricci curvature is positive, etc. See Aubin's book Nonlinear Analysis on Manifolds: Monge-Ampere Equations. Another sufficient condition is that you find a holomorphic section of the anticanonical bundle which is nonzero on a dense open set. |
||||
