2
$\begingroup$

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.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
11
$\begingroup$

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.

Improve this answer
$\endgroup$
4
  • 1
    $\begingroup$ 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. $\endgroup$
    – Alex Gavrilov
    Commented Jul 24, 2012 at 13:29
  • $\begingroup$ 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! $\endgroup$
    – diverietti
    Commented Jul 25, 2012 at 7:03
  • $\begingroup$ 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. $\endgroup$
    – Alex Gavrilov
    Commented Jul 26, 2012 at 13:29
  • $\begingroup$ Of course I don't mind! If I find a complete reference I'll tell you! $\endgroup$
    – diverietti
    Commented Jul 26, 2012 at 16:24

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .