Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules

TL;DR

This paper constructs co-$t$-structures on derived categories of coherent sheaves related to the nilpotent cone and Springer resolutions, and applies these to study support varieties of tilting modules, proving a version of the Humphreys conjecture in type A.

Contribution

It introduces new co-$t$-structures on derived categories of coherent sheaves on nilpotent cones and Springer resolutions, linking them via translation functors and applying to tilting module support varieties.

Findings

Constructed co-$t$-structures on derived categories of coherent sheaves.

Showed that push-forwards of exotic parity objects are indecomposable in the co-$t$-structure's coheart.

Proved a scheme-theoretic form of the Humphreys conjecture for type A, p > h.

Abstract

We construct a co-$t$-structure on the derived category of coherent sheaves on the nilpotent cone $\mathcal{N}$ of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the "exotic parity objects" along the (classical) Springer resolution give indecomposable objects inside the coheart of the co-$t$-structure on $\mathcal{N}$. We also demonstrate how the various parabolic co-$t$-structures can be related by introducing an analogue to the usual translation functors. As an application, we give a proof of a scheme-theoretic formulation of the relative Humphreys conjecture on support varieties of tilting modules in type $A$ for $p>h$.