Sarah Kitchen:
Koszul Categories and Mixed Hodge Modules
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In this talk, I will report on joint work with Pramod Achar. We
considered the following problem: Can we generate a Koszul category
from the category of mixed Hodge modules on a smooth complex variety X
(constructible along an affine stratification S) by a general
procedure, which gives a grading on the category of S-constructible
rational perverse sheaves on X? We were motivated by the fact that in
their paper on Koszul Duality, Beilinson, Ginzburg and Soergel (BGS)
produce their grading from mixed Hodge modules in a way specific to
the Bruhat stratification of a flag variety, whereas their approach to
l-adic perverse sheaves was more general. I will explain how to
"winnow" the category of mixed Hodge modules to come up with the
desired Koszul category, and how to obtain a grading on
S-constructible perverse sheaves from this, plus the relationship to
the grading obtained by BGS.