Spherical objects in dimension two and threeThis paper classifies spherical objects in various geometric settings in
dimension 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.
arxiv.org