I have just started reading through the paper of Cattani--Kaplan--Schmid -- Degeneration of Hodge structures (Annals of Mathematics, 123 (1986), 457--535). For the purposes here, take $f : X \to S$ to be a surjective holomorphic map from a compact Kähler manifold onto a complex manifold $S$ of strictly lower dimension. In fact, just take $S$ to be a curve. Let $V_s : = f^{-1}(s)$ denote the fibre over $s \in S$.

On the first page of the CKS paper, the authors write that the Hodge subspaces $$H^{p,q}(V_s)$$ do not vary holomorphically. Is there a nice way of seeing this? I would have thought that the Hodge bundle was a holomorphic vector bundle given the amount of time spent on studying holomorphic vector bundles with integrable connections.

I apologise in advance for my ignorance. Thank you for your time.


Perhaps my original answer was a bit technical. So let add a few comments at the beginning. The first question, is how does one define $H^{p,q}(V_s)$? Initially, it's defined as the space of $(p,q)$ forms which are harmonic with respect to a K"ahler metric. There is no reason why this would vary holomorphically. However, Griffiths realized in the 1960's that the Hodge filtration does vary homomorphically. The $H^{pq}(V_s)$ can be redefined using it. That's what I was describing below:

The Hodge bundles, which are usually taken to be $$F^p = \bigcup_s \bigoplus_{p'\ge p} H^{p',i-p}(V_s)\subset \bigcup_s H^i(V_s) $$ are holomorphic subbundles of the thing on the right (which should really be taken to be $R^if_*\mathbb{C}\otimes \mathcal{O}_S$). The problem is $$\bigcup_s H^{pq}(V_s)= F^p\cap \overline{F}^q$$ needs a complex conjugate, so it's only a $C^\infty$-subbundle.

May I suggest starting with Schmid, Variation of Hodge structure, Inventiones (1973).

  • $\begingroup$ Thank you, I'll take a look at Schmid's paper! $\endgroup$
    – AshyK
    Jan 12 at 22:36
  • $\begingroup$ I should probably ask this as a separate post, but do you have a good reference for why, in defining a polarized Hodge structure, this is to be done on the primitive cohomology, then extended to the whole cohomology? It's not clear to me why every reference mentions primitive cohomology, e.g., top of page 461 CKS. $\endgroup$
    – AshyK
    Jan 12 at 22:41
  • 1
    $\begingroup$ For polarization, one wants bilinear form satisfying the Hodge-Riemann relations. It's possible to do this explicitly on primitive cohomology. In general, you can reduce to that by the hard Lefschetz theorem, but the formulas get complicated. $\endgroup$ Jan 12 at 23:06

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