Singular Rouquier Complexes

TL;DR

This paper extends Rouquier complexes to singular Soergel bimodules, demonstrating they preserve key properties and applying them to establish Hodge theory in this broader context.

Contribution

It introduces singular Rouquier complexes, generalizing existing constructions and proving they maintain essential properties, enabling new applications in Hodge theory.

Findings

Singular Rouquier complexes are delta-split.

They satisfy a vanishing formula.

They are perverse when Soergel's conjecture holds.

Abstract

We generalise the construction of Rouquier complexes to the setting of singular Soergel bimodules by taking minimal complexes of the restriction of Rouquier complexes. We show that they retain many of the properties of ordinary Rouquier complexes: they are delta-split, they satisfy a vanishing formula and, when Soergel's conjecture holds they are perverse. As an application, we use singular Rouquier complexes to establish Hodge theory for singular Soergel bimodules.