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Perverse motives and graded derived category \({\mathcal{O}}\). (English) Zbl 1436.14015
Summary: For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finite-dimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category \({\mathcal{O}}\) and Koszul duality.

MSC:
14C15 (Equivariant) Chow groups and rings; motives
14F42 Motivic cohomology; motivic homotopy theory
14M15 Grassmannians, Schubert varieties, flag manifolds
16S37 Quadratic and Koszul algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
18E10 Abelian categories, Grothendieck categories
18G80 Derived categories, triangulated categories
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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