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Allen Knutson
Allen Knutson
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Dear Zaz,

If you look at Deligne's "Hodge I" article (actually an ICM talk, maybe from 1970 or thereabouts, which should be available at the IMU's electronic database of ICM talks), you will find a table giving at least some info about weights and Hodge numbers in various contexts (smooth but open, proper but possibly singular, etc.); more precisely, he states the ranges that the various numbers can lie in.

Otherwise, the basic principle is that (say in the case of a smooth variety first) you compactify your variety, write down the various long exact sequences that come to mind relating the cohomology of the open variety to that of its compactification, and then use the fact the maps of MHS are strict for weight and Hodge filtrations, so that you can read of the numbers of the MHS you care about (the cohomology of the open variety) from the MHS of other things appearing (which are compact, and so in some sense known: more precisely, the compactification is compact and smooth, so is known --- in principalprinciple --- by usual Hodge theory, while the boundary will be a normal crossings divisor, so is compact but possibly singular --- but in the mildest possible way --- and is also of lower dimension, so you can imagine that you know it by induction on dimension and/or because it's compact and very close to being smooth).

This is not the same as giving you a table, unfortunately; you wanted fish and I am giving you (at best) some kind of fishing implement (or maybe a spoon). Sorry about that.

Dear Zaz,

If you look at Deligne's "Hodge I" article (actually an ICM talk, maybe from 1970 or thereabouts, which should be available at the IMU's electronic database of ICM talks), you will find a table giving at least some info about weights and Hodge numbers in various contexts (smooth but open, proper but possibly singular, etc.); more precisely, he states the ranges that the various numbers can lie in.

Otherwise, the basic principle is that (say in the case of a smooth variety first) you compactify your variety, write down the various long exact sequences that come to mind relating the cohomology of the open variety to that of its compactification, and then use the fact the maps of MHS are strict for weight and Hodge filtrations, so that you can read of the numbers of the MHS you care about (the cohomology of the open variety) from the MHS of other things appearing (which are compact, and so in some sense known: more precisely, the compactification is compact and smooth, so is known --- in principal --- by usual Hodge theory, while the boundary will be a normal crossings divisor, so is compact but possibly singular --- but in the mildest possible way --- and is also of lower dimension, so you can imagine that you know it by induction on dimension and/or because it's compact and very close to being smooth).

This is not the same as giving you a table, unfortunately; you wanted fish and I am giving you (at best) some kind of fishing implement (or maybe a spoon). Sorry about that.

Dear Zaz,

If you look at Deligne's "Hodge I" article (actually an ICM talk, maybe from 1970 or thereabouts, which should be available at the IMU's electronic database of ICM talks), you will find a table giving at least some info about weights and Hodge numbers in various contexts (smooth but open, proper but possibly singular, etc.); more precisely, he states the ranges that the various numbers can lie in.

Otherwise, the basic principle is that (say in the case of a smooth variety first) you compactify your variety, write down the various long exact sequences that come to mind relating the cohomology of the open variety to that of its compactification, and then use the fact the maps of MHS are strict for weight and Hodge filtrations, so that you can read of the numbers of the MHS you care about (the cohomology of the open variety) from the MHS of other things appearing (which are compact, and so in some sense known: more precisely, the compactification is compact and smooth, so is known --- in principle --- by usual Hodge theory, while the boundary will be a normal crossings divisor, so is compact but possibly singular --- but in the mildest possible way --- and is also of lower dimension, so you can imagine that you know it by induction on dimension and/or because it's compact and very close to being smooth).

This is not the same as giving you a table, unfortunately; you wanted fish and I am giving you (at best) some kind of fishing implement (or maybe a spoon). Sorry about that.

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Emerton
Emerton
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Dear Zaz,

If you look at Deligne's "Hodge I" article (actually an ICM talk, maybe from 1970 or thereabouts, which should be available at the IMU's electronic database of ICM talks), you will find a table giving at least some info about weights and Hodge numbers in various contexts (smooth but open, proper but possibly singular, etc.); more precisely, he states the ranges that the various numbers can lie in.

Otherwise, the basic principle is that (say in the case of a smooth variety first) you compactify your variety, write down the various long exact sequences that come to mind relating the cohomology of the open variety to that of its compactification, and then use the fact the maps of MHS are strict for weight and Hodge filtrations, so that you can read of the numbers of the MHS you care about (the cohomology of the open variety) from the MHS of other things appearing (which are compact, and so in some sense known: more precisely, the compactification is compact and smooth, so is known --- in principal --- by usual Hodge theory, while the boundary will be a normal crossings divisor, so is compact but possibly singular --- but in the mildest possible way --- and is also of lower dimension, so you can imagine that you know it by induction on dimension and/or because it's compact and very close to being smooth).

This is not the same as giving you a table, unfortunately; you wanted fish and I am giving you (at best) some kind of fishing implement (or maybe a spoon). Sorry about that.