Specifc Line Bundle over Complex Manifold implies Kahler? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:24:05Z http://mathoverflow.net/feeds/question/70672 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70672/specifc-line-bundle-over-complex-manifold-implies-kahler Specifc Line Bundle over Complex Manifold implies Kahler? Matt Fahrad 2011-07-18T20:54:39Z 2011-12-16T09:22:08Z <p>Let $L$ be a holomorphic line bundle on complex manifold $X$, such that it admits a hermitian structure whose Chern connection has positive curvature. Is $X$ then Kahler?</p> http://mathoverflow.net/questions/70672/specifc-line-bundle-over-complex-manifold-implies-kahler/70686#70686 Answer by diverietti for Specifc Line Bundle over Complex Manifold implies Kahler? diverietti 2011-07-18T23:14:59Z 2011-12-16T09:22:08Z <p>As Dima said, it is much more: in fact it is projective. But let me give you some more insights on this kind of questions. </p> <p>I shall give you the definition of four different classes of compact complex manifolds.</p> <ol> <li>Projective manifold: closed submanifold of some complex projective space.</li> <li>Moishezon manifold: compact complex manifold such that the field of meromorphic functions on it has transcendence degree equal to its complex dimension.</li> <li>(Compact) Kähler manifold: compact complex manifold carrying a Kähler form, that is a closed positive smooth (1,1)-form.</li> <li>Manifold in the Fujiki class ($\mathcal C$): compact complex manifold bimeromorphic to a compact Kähler manifold.</li> </ol> <p>A Moishezon manifold can be shown to be bimeromorphic to a projective manifold, so that -in some sense- Moishezon manifolds are with respect to projective manifolds as manifolds in the Fujiki class ($\mathcal C$) are with respect to Kähler manifolds.</p> <p>It turns out, that one can characterize these four classes in terms of cohomological properties (these characterizations reflect again this relation between projective-Moishezon and Kähler-Fujiki). Here is the characterization for you:</p> <ol> <li>A compact complex manifold is projective if and only if it carries a (1,1) rational cohomology class which can be represented by a positive (1,1)-form (or equivalently if it carries a positive hermitian line bundle). This is the content of Kodaira's embedding theorem.</li> <li>A compact complex manifold is Kähler if and only if it carries a (1,1) real cohomology class which can be represented by a positive (1,1)-form. This is almost the definition.</li> <li>A compact complex manifold is Moishezon if and only if it carries a (1,1) rational cohomology class which can be represented by a (1,1) Kähler current, that is a (1,1)-closed positive current which is bounded from below by a (non necessarily closed) smooth positive (1,1)-form (or equivalently if it carries a big line bundle).</li> <li>A compact complex manifold is in the Fujiki class ($\mathcal C$) if and only if it carries a (1,1) real cohomology class which can be represented by a (1,1) Kähler current. This is the content of a theorem by Demailly-Paun. </li> </ol>