K-Motives and Koszul Duality

Abstract

We construct an ungraded version of Beilinson-Ginzburg-Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of $K$-theory. For this, we introduce and study categories ${\mathrm{DK}}_{\mathcal{S}}(X)$ of $\mathcal{S}$-constructible $K$-motivic sheaves on varieties $X$ with an affine stratification $\mathcal{S}$. We show that there is a natural and geometric functor, called Beilinson realisation, from $\mathcal{S}$-constructible mixed sheaves $D_{\mathcal{S}}^{mix}(X)$ to ${\mathrm{DK}}_{\mathcal{S}}(X)$. We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible $K$-motivic sheaves on Langlands dual flag varieties.

Improved exposition over last version

References in corpus (3)

• The weight complex functor is symmetric monoidal
• Morita theory for stable derivators
• Equivariant motives and geometric representation theory. (with an appendix by F. Hörmann and M. Wendt)

Cited by in corpus (2)

• Noncommutative resolutions and intersection cohomology for quotient singularities
• Intersection K-theory