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.



Is the Hodge bundle a holomorphic vector bundle?
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).


