Question about Hodge number 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. 
 A: Proposition. For compact complex manifolds of dimension $1$ (a.k.a. complex Riemann surfaces) the Hodge numbers are topological invariants.
Proof. The Hodge numbers are determined by the genus, which happens to be a topological invariant.
In a more general setting it is true that
Theorem. Let $f:X\to B$ be a family of complex manifolds and assume that $X_0$ is Kähler for some $0\in B$. Then for $b$ in a neighbourhood of $0$ in $B$, the Hodge numbers of $X_b$ are the same as the Hodge numbers of $X_0$.
Proof. See C.Voisin, Hodge Theory and Complex Algebraic Geometry, I, p.235.
Remark. It actually follows that all nearby fibers are Kähler, but it is not needed in the proof.
From this point I can only speculate: I don't think the Hodge numbers are topological invariants. If they were, then the assumption that one fiber is Kähler is superfluous. So, this suggests that it seems likely that there may be topologically equivalent complex manifolds with different Hodge numbers. I would even expect Kähler ones and perhaps even diffeomorphic ones, but that might be going out on a limb.
Catanese and Manetti gave various examples of orientedly diffeomorphic but not deformation equivalent smooth projective algebraic surfaces of general type. I wonder if those would perhaps have different Hodge numbers. The above theorem definitely does not apply. That in itself, of course, does not prove anything...
EDIT: (Added later)
As pointed out by Greg Kuperberg here, this same question had been debated before in another forum.
A: 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.
A: It depends on how you define Hodge Number, as far as I know you don't even need Kahler structure to define Hodge Number. See Griffiths-Harris pp 105, there the Hodge number is defined as the dimension of qth Cech Cohomology group of the sheaf of Holomorphic p-forms over the complex manifold M. However it is not dependent on the complex structure over M. 
