Stability conditions and crepant small resolutions

TL;DR

This paper explores the structure of stability conditions on categories linked to three-dimensional crepant small resolutions, revealing chamber structures, covering space properties, and Fourier-Mukai transforms.

Contribution

It characterizes the stability condition spaces for these resolutions, detailing their chamber structures and covering space properties, which was previously not well-understood.

Findings

Spaces have chamber structures with each chamber corresponding to a birational model.

The stability condition spaces are covering spaces over open subsets of vector spaces.

Deck transformations of these spaces are explicitly determined.

Abstract

In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special Fourier-Mukai transform. We observe these spaces are covering spaces over certain open subsets of finite dimensional vector spaces, and determine their deck transformations.