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.
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.