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Dan Petersen
Dan Petersen
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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_n H^n(X,\mathbb C)$$\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.

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 $\mathrm{Gr}^W_n H^n(X,\mathbb C)$ 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.

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.

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Dan Petersen
Dan Petersen
  • 40.3k
  • 2
  • 114
  • 201

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 $\mathrm{Gr}^W_n H^n(X,\mathbb C)$ 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.