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)
|
Text
171847.pdf - Published Version Available under License Creative Commons Attribution. 680kB |
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 |
Related URLs: |
University Staff: Request a correction | Enlighten Editors: Update this record