## Reference Request: de Rham vs. Dolbeault

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.

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## 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.

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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 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 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 at 13:29
Of course I don't mind! If I find a complete reference I'll tell you! – diverietti Jul 26 at 16:24