Koszul duality and mixed Hodge modules

Details Koszul duality and mixed Hodge modules Koszul duality and mixed Hodge modules

2011-05-11

arxiv.org/abs/1105.2181v2

Mathematics

Representation Theory

24 pages; substantially rewritten due to a mistake in the previous version

Scientific paper

We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. For flag varieties, this was proved earlier by Beilinson-Ginzburg-Soergel using a rather different construction. Note: In version 1 of this paper, Lemma 3.2 is false (except under an additional Hom-vanishing assumption), and this mistake affects many of the subsequent arguments. In particular, the functor \beta{} that appears in the proof of Proposition 3.4 may have good properties only at the level of abelian categories, rather than for triangulated categories. For version 2, the paper has been substantially rewritten to rely only on this weaker statement about \beta. Nevertheless, the main result--the fact that the "winnowing" of MHM(X) is a Koszul category when X has an affine even stratification--is essentially unchanged.

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