Perverse motives and graded derived category $\mathcal{O}$

TL;DR

This paper explores categories of motivic sheaves on stratified varieties, establishing a weight structure and revealing geometric interpretations of graded category O and Koszul duality, especially in flag varieties.

Contribution

It introduces a new framework for stratified mixed Tate motives with a weight structure and links to geometric interpretations of category O and Koszul duality.

Findings

Categories of stratified mixed Tate motives have a natural weight structure.

Tilting provides an equivalence under pointwise purity.

Connections to geometric interpretations of graded category O and Koszul duality.

Abstract

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.