Spherical objects in dimensions two and three

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 $\mathcal{C}$ 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 $x$ which admit no negative Exts, and for which ${\mathrm{Hom}}_{\mathcal{C}}(x,x)=\mathbb{C}$, are the images of the simples. A variation on this argument goes further and classifies all bounded t-structures on $\mathcal{C}$. 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.