In the book "Mixed Hodge structures", Proposition 4.3 reads ($U$ a smooth variety, $X$ a simple normal crossings compactification):
In inclusion of complexes $$\Omega_X^\bullet(\log E)\to j_*\Omega_U^\bullet$$ is a quasi-isomorphism and induces a natural identification $$H^k(U;\mathbb{C})\cong \mathbb{H}^k(X,\Omega_X^\bullet(\log D)).$$ Furthermore, the natural map $j_*\Omega_U^\bullet\to Rj_*\Omega_U^\bullet$ is a quasi-isomorphism.
I do not really understand the logic in this formulation. There are three statements here. The first statement is a local computation (which can be proven in Stein opens), and gives us that for all $k$, $$\mathbb{H}^k(X,\Omega_X^\bullet(\log E)) = \mathbb{H}^k(X,j_*\Omega_U^\bullet).$$ The formulation suggests that this already gives that $\mathbb{H}^k(X,\Omega_X^\bullet(\log E)) \cong H^k(U;\mathbb{C})$, without using the third statement (this is suggested by the word "furthermore"). So it seems that without the third statement it should be clear that $$H^k(U;\mathbb{C})\cong \mathbb{H}^k(X,j_*\Omega_U^\bullet).$$ I do not quite see how to deduce this without using point $3$ though. The right hand side is equal to $\mathbb{H}^k(X,R^0j_*\Omega_U^\bullet)$. By the Poincare lemma, this is equal to $\mathbb{H}^k(X,R^0j_*\underline{\mathbb{C}})$. But these terms form only a single column on the $E_2$ page of the Leray spectral sequence for $j$, which converges to $\mathbb{H}^k(U,\underline{\mathbb{C}})=H^k(U;\mathbb{C})$. So it seems that to get the isomorphism we are looking for we need that the Leray spectral sequence degenerates, e.g. by showing that that $R^ij_*\underline{\mathbb{C}}=0$ for $i>0$, but this is essentially point (3).
So what am I missing here? How do we deduce $H^k(U;\mathbb{C})\cong \mathbb{H}^k(X,\Omega_X^\bullet(\log D))$ from the first claim directly without using the "furthermore" claim?
The inclusion $\Omega_X^\bullet(\log E) \hookrightarrow j_*\Omega_U^\bullet$ induces an isomorphism $$\mathbb H^k(X, j_*\Omega_U^\bullet) \cong \mathbb H^k(X, \Omega_X^\bullet(\log E)).$$ The fact that $H^k(U, \mathbf C) \cong \mathbb H^k(X, j_*\Omega_U^\bullet)$ follows from the (abstract) de Rham theorem (theorem B.18 in "Mixed Hodge Structures"). See also the first paragraph of the proof of Prop. 4.3.