Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Hi everyone. I need the following statement:

For a Kahler manifold $X$, the natural map $H^n(X,\mathbb{C})\to H^n(X,\mathcal{O})$ (from the sheaf extension) coincides with the Hodge projection $\Pr_{0,n}$, up to the de Rham isomorphism and the Dolbeault isomorphism.

Does anybody know a good reference?

P.S. Surely there must be a reference. I am much less interested in proofs: I think I know one.

share|improve this question
add comment

1 Answer

There are lots of references. Mainly every textbook which treats Hodge theory. Try to look at:

  • Voisin: Hodge theory and complex algebraic geometry. I
  • Huybrechts: Complex geometry
  • Wells: Differential analysis on complex manifolds
  • Griffiths, Harris: Principles of algebraic geometry

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.

share|improve this answer
1  
Yes, but to write in a paper "The general case is a straightforward adaptation of that" is not a good manner, is it? And the proofs in Griffiths&Harris and Voisin are rather specific for $n=2$. (They may be generalized, to be sure, but I wouldn't say it is straightforward. Do not remember about other books). Do not take me wrong, but it doesn't look like a very good reference. –  Alex Gavrilov Jul 24 '12 at 13:29
    
The point is rather that you didn't say you needed this reference for a paper you are writing. In this case, sincerely, you can just state that fact as well-known. No referee would protest! –  diverietti Jul 25 '12 at 7:03
    
Yes, perhaps you are right. Anyway, it won't do any harm: if the referee insists, then I can write my own proof (with the reference to Griffiths&Harris for a special case). Of course I shoud have made my purpose clear from the beginning, so we could avoid this bit of confusion. However, I slill hope that someone may give me a reference for a complete proof, which is why I do not accept your answer. Do not mind this. By the way, I have Griffiths&Harris on my bookshelf. –  Alex Gavrilov Jul 26 '12 at 13:29
    
Of course I don't mind! If I find a complete reference I'll tell you! –  diverietti Jul 26 '12 at 16:24
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.