# Is the Hodge bundle a holomorphic vector bundle?

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.