Noncommutative enhancements of contractions

Donovan, W. and Wemyss, M. (2019) Noncommutative enhancements of contractions. Advances in Mathematics, 344, pp. 99-136. (doi:10.1016/j.aim.2018.11.019)

Donovan, W. and Wemyss, M. (2019) Noncommutative enhancements of contractions. Advances in Mathematics, 344, pp. 99-136. (doi:10.1016/j.aim.2018.11.019)

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Abstract

Given a contraction of a variety X to a base Y, we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one. In all cases, this produces a global invariant. In the crepant setting, we then apply this to study derived autoequivalences of X. We work generally, dropping many of the usual restrictions, and so both extend and unify existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D, and then, under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we also control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Wemyss, Professor Michael
Authors: Donovan, W., and Wemyss, M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Advances in Mathematics
Publisher:Elsevier
ISSN:0001-8708
ISSN (Online):1090-2082
Published Online:04 January 2019
Copyright Holders:Copyright © 2019 The Authors
First Published:First published in Advances in Mathematics 344: 99-136
Publisher Policy:Reproduced under a Creative Commons License
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
756641The Homological Minimal Model ProgramMichael WemyssEngineering and Physical Sciences Research Council (EPSRC)EP/K021400/2M&S - MATHEMATICS
3015810The Homological Minimal Model ProgramMichael WemyssEngineering and Physical Sciences Research Council (EPSRC)EP/R009325/1M&S - Mathematics