Spherical objects in dimensions two and three

TL;DR

This paper classifies spherical objects in dimensions two and three across various geometric settings, revealing their structure via braid group actions and establishing a broad framework for understanding t-structures and autoequivalences.

Contribution

It introduces a general classification of objects with no negative Exts in derived categories, linking them to braid group actions and classifying bounded t-structures in complex geometric contexts.

Findings

Objects with no negative Exts are images of simples under braid group actions.

All bounded t-structures can be classified using the developed techniques.

In silting discrete algebras, objects with no negative Exts lie in the heart of some bounded t-structure.

Abstract

This paper classifies spherical objects in various geometric settings in dimensions two and three, including both minimal and partial crepant resolutions of Kleinian singularities, as well as arbitrary flopping 3-fold contractions with only Gorenstein terminal singularities. The main result is much more general: in each such setting, we prove that all objects in the associated null category with no negative Ext groups are the image, under the action of an appropriate braid or pure braid group, of some object in the heart of a bounded t-structure. The corollary is that all objects which admit no negative Exts, and for which the self-Hom space is one dimensional, are the images of the simples. A variation on this argument goes further, and classifies all bounded t-structures. There are multiple geometric, topological and algebraic consequences, primarily to autoequivalences and stability conditions. Our main new technique also extends into representation theory, and we establish that in the derived category of a finite dimensional algebra which is silting discrete, every object with no negative Ext groups lies in the heart of a bounded t-structure. As a consequence, every semibrick complex can be completed to a simple minded collection.