Twists and braids for general 3-fold flops

Donovan, W. and Wemyss, M. (2019) Twists and braids for general 3-fold flops. Journal of the European Mathematical Society, 21(6), pp. 1641-1701. (doi: 10.4171/JEMS/868)

Abstract

Given a quasi-projective 3-fold X with only Gorenstein terminal singularities, we prove that the flop functors beginning at X satisfy higher degree braid relations, with the combinatorics controlled by a real hyperplane arrangement H. This leads to a general theory, incorporating known special cases with degree 3 braid relations, in which we show that higher degree relations can occur even for two smooth rational curves meeting at a point. This theory yields an action of the fundamental group of the complexified complement π1(Cn\HC) on the derived category of X, for any such 3-fold that admits individually floppable curves. We also construct such an action in the more general case where individual curves may flop analytically, but not algebraically, and furthermore we lift the action to a form of affine pure braid group under the additional assumption that X is Q-factorial. Along the way, we produce two new types of derived autoequivalences. One uses commutative deformations of the scheme-theoretic fibre of a flopping contraction, and the other uses noncommutative deformations of the fibre with reduced scheme structure, generalising constructions of Toda and the authors [T07, DW1] which considered only the case when the flopping locus is irreducible. For type A flops of irreducible curves, we show that the two autoequivalences are related, but that in other cases they are very different, with the noncommutative twist being linked to birational geometry via the Bridgeland–Chen [B02, C02] flop–flop functor.

• Item Type: Articles
• Additional Information: The first author was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan; by EPSRC grant EP/G007632/1; and by the Hausdorff Research Institute for Mathematics, Bonn. The second author was supported by EPSRC grant EP/K021400/1.
• 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: Journal of the European Mathematical Society
• Publisher: European Mathematical Society
• ISSN: 1435-9855
• ISSN (Online): 1435-9863
• Copyright Holders: Copyright © 2016 European Mathematical Society
• First Published: First published in Journal of the European Mathematical Society 21(6):1641-1701
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher