Is there a way to describe the image of the $n$-fold residue map from $H^0(Y,\Omega_Y^n(\log E))$?

Question

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 $\{\sum r_i=0\}\subset \oplus_{p_i}\mathbb{C}$?

Answer 1

Your guess in the last sentence is not quite right, but there is a natural higher-dimensional generalization of the statement. What happens is that you have a "boundary map" $$ \bigoplus_{p_i} \mathbb C \to \bigoplus_{C_j} \mathbb C$$ where the points $p_i$ are all the $n$-fold intersections of boundary components and the curves $C_j$ are all the $(n-1)$-fold intersections; the summand corresponding to $p_i$ is mapped to those summands $C_j$ such that $p_i \in C_j$. The image is precisely the kernel of this map.

All of these things come from Deligne's Hodge II. You have two filtrations on $\Omega^\bullet(\log E)$ giving you two different spectral sequences, one of which degenerates immediately (giving you the Hodge filtration) and one of which degenerates after the first differential (giving you the weight filtration), the maps you care about are seen in these spectral sequences and you have to think about what things mean. The upshot is first of all that $\Gamma(Y,\Omega^n(\log E)) = F^n H^n(X,\mathbb C)$. This then maps surjectively onto $\mathrm{Gr}^W_{2n} H^n(X,\mathbb C)$ which in turn injects via an edge map in the Leray spectral sequence into $\bigoplus_{p_i} \mathbb C$. Since the Leray spectral sequence degenerates after the first differential, the image of this map will just be the kernel of the differential from this spot of the spectral sequence, and that differential is the thing I wrote above.