Real variations of stability conditions for noncommutative symplectic resolutions

TL;DR

This paper explores how different localizations of cyclotomic rational Cherednik algebras induce varying stability conditions on derived categories, linking algebraic, geometric, and topological aspects.

Contribution

It provides an explicit construction of tilting bundles for $n=2$ and relates stability condition variations to the topology of Hilbert schemes and algebra representations.

Findings

Explicit tilting bundles for $n=2$ constructed.

Variations in $t$-structures are governed by real stability conditions.

Connections established between stability conditions, topology, and algebra representations.

Abstract

A localization theorem for the cyclotomic rational Cherednik algebra $H_c=H_c((\mathbb{Z}/l)^n\rtimes \mathfrak{S}_n)$ over a field of positive characteristic has been proved by Bezrukavnikov, Finkelberg and Ginzburg. Localizations with different parameters give different $t$-structures on the derived category of coherent sheaves on the Hilbert scheme of points on a surface. In this short note, we concentrate on the comparison between different $t$-structures coming from different localizations. When $n=2$, we show an explicit construction of tilting bundles that generates these $t$-structures. These $t$-structures are controlled by a real variation of stability conditions, a notion related to Bridgeland stability conditions. We also show its relation to the topology of Hilbert schemes and irreducible representations of $H_c$.