TITLE: Question about Hodge number
QUESTION [12 upvotes]: Hi. I am studying Hodge theory on Kahler manifolds. 
I have several questions.

Is Hodge number a topological invariant? (I mean, is it independent of the choice of 
Kahler structure?)
If the question 1 is true, then is there any variation formula of Hodge numbers on blowing up(down)? (along Kahler submanifolds) --- please let me know the reference. 
I read Huybrechts's book "complex geometry". In there, Hodge-index theorem is explained in 
case of Kahler surfaces. Is there Hodge-index theorem in higher dimensional cases?
Is there a symplectic version of Hodge-Riemann bilnear relation? 

I am sorry to ask much questions. 
Thank you in advance.

REPLY [6 votes]: For a Kaehler surface, the Hodge numbers are topological invariants.
By Hodge Index Theorem, the signature of the Poincare pairing is equal
to 2h^{2,0} +2 - h^{1,1}, hence h^{2,0} is a topological invariant,
and for the rest of the numbers it is obvious.
For dimension 3, I think there are counterexamples.