Let $M$ be a smooth compact Kahler manifold and let $\mathcal{F}$ be a local system on $M$.

Question 1: I assume that there exists a twisted Hodge to de Rham spectral sequence converging to $H^{p+q}(M;\mathcal{F})$ whose $E_1$-page is of the form

$$E_1^{p,q} = H^p(M;\mathcal{F} \otimes \Omega^q).$$

Is this correct, and if so does anyone know a good reference for it?

Question 2: What conditions can I put on $\mathcal{F}$ to ensure that the above spectral sequence degenerates at the $E_1$-page? For example, what if there exists some finite unbranched cover $\widetilde{M} \rightarrow M$ such that the pullback of $\mathcal{F}$ to $\widetilde{M}$ is just the sheaf of locally constant functions?


(Edit: I answered Q2 initially, ignoring Q1.)

Q1: The spectral sequence is right except that it starts at $E_1$.

Q2: It is true if $\mathcal{F}$ is unitary in the sense that the underlying representation of $\pi_1(M)$ is unitary, then the spectral sequence degenerates at the $E_1$ page. This seems like a folklore fact, so I would need to think to find a good reference. But one that comes to mind is Timmerscheidt's appendix to a paper of Esnault and Viehweg, Inventiones, vol 86, pp 189-194. (He proves something more general, so it might be hard to follow.) By the way, this should be enough to conclude that condition of the last sentence of your Q2 is sufficient.

I don't think degeneration is true for non unitary local systems. The best result in general is due to Simpson that, at least when $M$ is projective, $H^*(M,\mathcal{F})$ is isomorphic to the hypercohomology of the Higgs complex of the associated Higgs bundle. But perhaps this going too far afield.

Added Actually why don't just give you the proof. The key analytic fact that makes the usual degeneration work is the Kähler identity $$d d^*+d^*d= 2(\bar \partial \bar\partial^* +\bar\partial^*\bar\partial)$$ This continues to hold for a unitary local system, where $d$ is now the connection. The point is that the arguments needed to establish the usual identity carry over almost word for word when you work with a unitary local frame.

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  • $\begingroup$ This is a very helpful answer! Also, thanks for pointing out that I had confused myself into thinking that the spectral sequence started at the $E_2$ page rather than the $E_1$ page (I fixed that in the question). $\endgroup$ – Lisa Sep 14 '15 at 22:09

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