Summary: KOSZUL DUALITY AND MIXED HODGE MODULES
PRAMOD N. ACHAR AND S. KITCHEN
Abstract. We prove that on a certain class of smooth complex varieties
(those with "affine even stratifications"), the category of mixed Hodge mod-
ules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are
eliminated. For flag varieties, this was proved earlier by BeilinsonGinzburg
Soergel using a rather different construction.
In their seminal paper on Koszul duality in representation theory [BGS], Beilin-
son, Ginzburg, and Soergel established the Koszulity of two important geometric
categories: the category of mixed perverse sheaves on a flag variety over a finite
field, and the category of mixed Hodge modules on a flag variety over C. More
precisely, they are each "almost" Koszul, in that they contain some unwanted ex-
tensions, but once those are removed, what remains is a Koszul category.
A key step in [BGS] is, of course, that of giving a concrete description of the
extensions to be removed. However, the two cases are treated very differently.
For -adic perverse sheaves, the description preceding [BGS, Theorem 4.4.4] is
very general; it applies to any variety satisfying a couple of axioms (cf. [BGS,
Lemma 4.4.1]), and the proof of Koszulity uses only very general results about
´etale cohomology and homological algebra. In contrast, for mixed Hodge modules