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When we have a variety and a resolution of singularities, but it is not semi-small (i.e. the dimensions of the fibres do not satisfy the right conditions), then what can we say about the intersection cohomology? Someone was telling me something about this with shifting the IC or something, but I cannot remember the precise statement.

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I'm not sure exactly what question you're asking. I think you may be looking for the following answer. By the decomposition theorem, the intersection cohomology of the variety is a direct summand of the cohomology of the resolution. I'm not sure there's anything more specific you can say than that in general.

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The decomposition theorem also says something about what the other summands look like. – Alexander Woo Dec 16 '09 at 2:29
I'm not sure if I'm aware of a description of the other summands in this case. I'd be happy to read an answer explaining this if you're up for it. – Mike Skirvin Dec 16 '09 at 2:37
thank you very much, The decomposition theorem was what I was looking for (I didn't remember it at the time, and when I was searching on google for things on IC it didn't come up for some reason). I'm sorry for posting a silly question like this, I should have been more patient and asked someone in person instead. – Vinoth Dec 22 '09 at 13:35

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