Sarah Kitchen: Koszul Categories and Mixed Hodge Modules

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.