Let $X$ be a smooth projective variety over an algebraically closed characteristic $0$ field $\mathbb{K}$, and let $\Omega^\bullet_X$ be the de Rham sheaf of complexes of $\mathcal{O}_X$-modules of algebraic differential forms on $X$. Let $\mathbf{R}\Gamma\Omega^\bullet_X$ be (a complex representing) the derived gloval sections of $\Omega^\bullet_X$, and let $End(\mathbf{R}\Gamma\Omega^\bullet_X)$ be the total complex of hom-bicomplex $Hom(\mathbf{R}\Gamma\Omega^\bullet_X,\mathbf{R}\Gamma\Omega^\bullet_X)$.

I have two questions.

i) Is $End(\mathbf{R}\Gamma\Omega^\bullet_X)$ quasi-isomorphic to $End(H^\bullet(X;\mathbb{K}))$as differential graded Lie algebras? (where the latter has trivial differential). (edit: this should be easy and true in general. thanks Torsten).

ii) Now consider a sheaf endomorphism $f:\Omega^\bullet_X\to \Omega^\bullet_X$ (as sheaves of complexes of vector spaces) such that $f:ker(d)\to Im(d)$, so that $f$ will induce the zero morphism on the cohomology sheaf $\mathcal{H}^\bullet(\Omega_X^\bullet)$. And consider the induced morphism $\mathbf{R}\Gamma f: \mathbf{R}\Gamma \Omega^\bullet_X \to \mathbf{R}\Gamma \Omega^\bullet_X$. Is it true that $\mathbf{R}\Gamma f: ker(d)\to Im(d)$? (I suspect the answer is yes in this case: a morphism of sheaves wich is trivial on local cohomology should be trivial on global cohomology, since there is a spectral sequence with coefficients in local cohomology abutting to global cohomology)

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