Local Hodge theory of Soergel bimodules

TL;DR

This paper establishes local Hodge-theoretic properties for Soergel bimodules, providing algebraic proofs of the Jantzen conjectures and revealing dependence of the Jantzen filtration on deformation directions.

Contribution

It introduces local Hodge theory for Soergel bimodules and offers an algebraic proof of the Jantzen conjectures, highlighting new dependencies in the filtration process.

Findings

Proved local hard Lefschetz theorem for Soergel bimodules

Established local Hodge-Riemann bilinear relations

Demonstrated Jantzen filtration dependence on deformation direction

Abstract

We prove the local hard Lefschetz theorem and local Hodge-Riemann bilinear relations for Soergel bimodules. Using results of Soergel and K\"ubel one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.