# Degeneration twisted Hodge to de Rham spectral sequence

Let $$X$$ be a proper and smooth scheme over $$\mathbf{C}$$ and let $$\mathbb{L}$$ be a local system of finite dimensional $$\mathbf{C}$$-vector spaces. By the Riemann Hilbert correspondence, to $$\mathbb{L}$$ one can associate a locally free sheaf $$\mathcal{F}$$ of $$\mathcal{O}_X$$-modules with an integral connection $$\nabla \colon \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X$$ such that $$\mathbb{L} = \mathcal{F}^{\nabla = 0}$$. In particular the de rham complex of $$\mathcal{F}$$: $$0 \to \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X \to \cdots$$ gives a resolution of $$\mathbb{L}$$. This resolution is not injective, but still we can use hypercohomology of the complex to compute the cohomology of $$\mathbb{L}$$, and we get a spectral sequence of hypercohomology.

If $$\mathbb{L} = \mathbf{C}$$ is constant, this is the usual Hodge to de Rham spectral sequence, that degenerates, immediately. What can be said in general? I guess that it not true that the sequence always degenerates. (But for example it should be true if $$\mathbb{L} = f_\ast \mathbf{C}$$ for a proper and smooth morphism $$f \colon Y \to X$$.)

This is true for any local system underlying a polarized $$\mathbb{C}$$-variation of Hodge structure, e.g. it is true for $$R^nf_*\mathbb{C}$$ where $$f:X\to Y$$ is smooth proper and $$n\ge 0$$. Indeed, the theory of harmonic forms and Kähler identities extends to this case. In the case of real VHS this is worked out in Section 2 of S. Zucker "Hodge Theory with Degenerating Coefficients", Annals of Mathematics 109 (1979), pp. 415-476. Even more generally, the $$E_1$$-degeneration holds for perverse sheaves underlying pure Hodge modules, by the theory of Morihiko Saito.
Let me supplement Chris' answer with a few additional remarks. When $$\mathcal{F}$$ underlies a polarizable complex variation of Hodge structure, then it carries a filtration $$F^\bullet\mathcal{F}$$ which induces one on the de Rham complex $$F^p\mathcal{F}\to F^{p-1}\mathcal{F}\otimes \Omega_X^1 \to \ldots$$ The argument of Deligne sketched in Zucker's paper, implies that the spectral sequence $$E_1= H^{p+q}(Gr^p_F(\mathcal{F}\otimes \Omega_X^\bullet))$$ associated to the above filtration degenerates at $$E_1$$. If $$\mathcal{F}$$ is a flat unitary bundle, then this is the same as spectral sequence implicit in your question, but not in general. In fact, I would expect that the more naive spectral sequence would fail to degenerate for a reasonable complicated example.