Let $X$ be an $n$-dimensional smooth algebraic variety, and let $Y$ be a compactification with $E=Y\setminus X$ simple normal crossings. There is the natural quotient map $$\Omega_Y^n(\log E)\to \mathcal{H}^n(\Omega_Y^\bullet(\log E))\to 0.$$ I want to understand the image of this map on global sections. If I am not mistaken this map can be identified with the $n$-fold residue map $$\Omega_Y^n(\log E)\to \oplus_{p_i}\mathbb{C}\to 0,$$ where the $p_i$ are all $n$-fold intersections of the components of $E$. The kernel of the residue map is $W_{n-1}\Omega_Y^{n}(\log E)$. So the image of the residue map on global sections is equal to the kernel of $$\oplus_{p_i}\mathbb{C}\to H^1(Y,W_{n-1}\Omega_Y^n(\log E)).$$ Is there a way to understand this map and its kernel more concretely?

Let $X$ be an $n$-dimensional smooth algebraic variety, and let $Y$ be a compactification with $E=Y\setminus X$ simple normal crossings. There is the natural quotient map $$\Omega_Y^n(\log E)\to \mathcal{H}^n(\Omega_Y^\bullet(\log E))\to 0.$$ I want to understand the image of this map on global sections. If I am not mistaken this map can be identified with the $n$-fold residue map $$\Omega_Y^n(\log E)\to \oplus_{p_i}\mathbb{C}\to 0,$$ where the $p_i$ are all $n$-fold intersections of the components of $E$. The kernel of the residue map is $W_{n-1}\Omega_Y^{n}(\log E)$. So the image of the residue map on global sections is equal to the kernel of $$\oplus_{p_i}\mathbb{C}\to H^1(Y,W_{n-1}\Omega_Y^n(\log E)).$$ Is there a way to understand this map and its kernel more concretely?

Edit: is it maybe true, as in the case of Riemann surfaces, that the image is exactly equal to $\{\sum r_i=0\}\subset \oplus_{p_i}\mathbb{C}$?