[Math] Question about Hodge number

complex-geometrysg.symplectic-geometry

Hi. I am studying Hodge theory on Kahler manifolds.
I have several questions.

  1. Is Hodge number a topological invariant? (I mean, is it independent of the choice of
    Kahler structure?)

  2. 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.

  3. 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?

  4. Is there a symplectic version of Hodge-Riemann bilnear relation?

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

Best Answer

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.

Related Question