# ag.algebraic geometry – Rank of Hodge bundles

Does anyone have a reference for the rank of Hodge bundles? More precisely, let $$X to Y$$ be a projective family and let $$(mathbb{H}, nabla)$$ be a variation of Hodge structure. The Hodge filtration produces a graded bundle: $$mathcal{E}^p : = textbf{R}^{n-p} f_{ast} Omega_{X/Y}^p,$$ where $$textbf{R}^{n-p}f_{ast}$$ is the right-derived functor of the direct image sheaf and $$Omega_{X/Y}^p$$ is the sheaf of relative $$p$$–forms.

Question: Is the rank of these vector bundles known?