Degeneration twisted Hodge to de Rham spectral sequence

Question

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 $f \colon Y \to X$.)

Answer 1

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.

Answer 2

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.