Reference Request: de Rham vs. Dolbeault - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:41:59Z http://mathoverflow.net/feeds/question/102983 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102983/reference-request-de-rham-vs-dolbeault Reference Request: de Rham vs. Dolbeault Alex Gavrilov 2012-07-24T06:49:49Z 2012-07-24T08:05:15Z <p>Hi everyone. I need the following statement: </p> <p>For a Kahler manifold $X$, the natural map <code>$H^n(X,\mathbb{C})\to H^n(X,\mathcal{O})$</code> (from the sheaf extension) coincides with the Hodge projection $\Pr_{0,n}$, up to the de Rham isomorphism and the Dolbeault isomorphism. </p> <p>Does anybody know a good reference?</p> <p>P.S. Surely there must be a reference. I am much less interested in proofs: I think I know one. </p> http://mathoverflow.net/questions/102983/reference-request-de-rham-vs-dolbeault/102985#102985 Answer by diverietti for Reference Request: de Rham vs. Dolbeault diverietti 2012-07-24T07:45:37Z 2012-07-24T08:05:15Z <p>There are lots of references. Mainly every textbook which treats Hodge theory. Try to look at:</p> <ul> <li>Voisin: Hodge theory and complex algebraic geometry. I</li> <li>Huybrechts: Complex geometry</li> <li>Wells: Differential analysis on complex manifolds</li> <li>Griffiths, Harris: Principles of algebraic geometry</li> </ul> <p>There, you will find mainly the proof in the case $n=2$, which is used to prove the Lefschez theorem on $(1,1)$-classes. The general case is a straightforward adaptation of that argument.</p>