Let $X$ be a smooth projective algebraic variety over the complex numbers, and let us name


$$f : X\_{\rm an}\to X\_{\rm Zar}$$


the morphism of sites induced by sending a Zariski open $U\subset X$ to $U^{\rm an}$.


Consider the constant sheaf $\mathbf{Q}(1)$ on $X\_{\rm an}$ whose value group is $2\pi i\cdot\mathbf{Q}$ and let $\mathbf{Q}(n) := \mathbf{Q}(1)^{\otimes n}$.


Does the Leray spectral sequence


$$H^p(X\_{\rm Zar}, R^qf\_\*\mathbf{Q}(n))\Rightarrow H^{p+q}(X\_{\rm an}, \mathbf{Q}(n))$$


degenerate at the $E\_2$-page? That is, are all the differentials in the $E\_2$ page of the spectral sequence, zero?


This would follow if $$Rf\_\*\mathbf{Q}(n) = \bigoplus\_pR^pf\_\*\mathbf{Q}(n)[-p]$$


in the derived category of abelian Zariski sheaves on $X$, so I'm wondering: is some Hard Lefschetz Theorem for the sheaves $R^pf\_\*\mathbf{Q}(n)$ true?


*(This question originated from clever remarks by the user Merlin, on a previous question of mine. I expect the answer to be negative, but if so, I would like to understand why exactly.)*



Spectral sequence in Betti cohomology
No; this spectral sequence usually does not degenerate. It appears that the group in the left hand side are often infinite dimensional. In particular, the $E\_2^{p,p}$-term is the group of codimension $p$ cycles modulo algebraic equivalence (see Corollary 7.4 of the paper Bloch, S., & Ogus, A. (1974). Gersten’s conjecture and the homology of schemes. Ann. Sci. École Norm. Sup.(4), 7(181-201), 12); hence the edge map to $H^{2p}(X,\mathbb{Q})$ is often not injective (starting from $p=2$; see Corollary 7.5 of loc. cit.). 


