Spherical objects in dimension two and three

Hara, W. and Wemyss, M. (2023) Spherical objects in dimension two and three. Journal of the European Mathematical Society, (Accepted for Publication)

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Abstract

This 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.

Item Type:Articles
Additional Information:The authors were supported by EPSRC grant EP/R034826/1, and MW additionally by ERC Consolidator Grant 101001227 (MMiMMa).
Status:Accepted for Publication
Refereed:Yes
Glasgow Author(s) Enlighten ID:Wemyss, Professor Michael and Hara, Dr Wahei
Authors: Hara, W., and Wemyss, M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of the European Mathematical Society
Publisher:European Mathematical Society
ISSN:1435-9855
ISSN (Online):1435-9863
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
300490Enhancing Representation Theory, Noncommutative Algebra And Geometry Through Moduli, Stability And DeformationsMichael WemyssEngineering and Physical Sciences Research Council (EPSRC)WT5128463 EP/R034826/1M&S - Mathematics
310007MMiMMAMichael WemyssEuropean Commission (EC)101001227M&S - Mathematics