ag.algebraic geometry – 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.