The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

Invent. math. DOI 10.1007/s00222-016-0665-5 The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds Arend Bayer1 · Emanuele Macrì2,3 · Paolo Stellari4 Received: 15 October 2014 / Accepted: 12 April 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We describe a connected component of the space of stability conditions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. Our proof includes the following essential steps: 1. We simultaneously strengthen a conjecture by the first two authors and Toda, and prove that it follows from a more natural and seemingly weaker statement. This conjecture is a Bogomolov-Gieseker type inequality involv- B Arend Bayer http://www.maths.ed.ac.uk/∼abayer/ Emanuele Macrì http://nuweb15.neu.edu/emacri/ Paolo Stellari http://users.unimi.it/stellari/ 1 School of Mathematics and Maxwell Institute, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK 2 Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA 3 Present Address: Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA 4 Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milan, Italy 123 A. Bayer et al. ing the third Chern character of “tilt-stable” two-term complexes on smooth projective threefolds; we extend it from complexes of tilt-slope zero to arbitrary tilt-slope. 2. We show that this stronger conjecture implies the so-called support property of Bridgeland stability conditions, and the existence of an explicit open subset of the space of stability conditions. 3. We prove our conjecture for abelian threefolds, thereby reproving and generalizing a result by Maciocia and Piyaratne. Important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall-crossing, closely related to the support property. Mathematics Subject Classification Primary 14F05; Secondary 14J30 · 18E30 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Review: tilt-stability and the conjectural BG inequality . . . . . . . . . . . . . . . . . . . 3 Classical Bogomolov-Gieseker type inequalities . . . . . . . . . . . . . . . . . . . . . . 4 Generalizing the main conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Reduction to small α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Tilt stability and étale Galois covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Abelian threefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Construction of Bridgeland stability conditions . . . . . . . . . . . . . . . . . . . . . . . 9 The space of stability conditions on abelian threefolds . . . . . . . . . . . . . . . . . . . 10 The space of stability conditions on some Calabi-Yau threefolds . . . . . . . . . . . . . . Appendix 1: Support property via quadratic forms . . . . . . . . . . . . . . . . . . . . . . . Appendix 2: Deforming tilt-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction In this paper, we determine the space of Bridgeland stability conditions on abelian threefolds and on Calabi-Yau threefolds obtained either as a finite quotient of an abelian threefold, or as the crepant resolution of such a quotient. More precisely, we describe a connected component of the space of stability conditions for which the central charge only depends on the degrees H 3−i chi ( ), i = 0, 1, 2, 3, of the Chern character1 with respect to a given polarization H , and that satisfy the support property. 1 In the case of crepant resolutions, we take the Chern character after applying BKR-equivalence [8] between the crepant resolution and the orbifold quotient. 123 The space of stability conditions on abelian threefolds… Stability conditions on threefolds via a conjectural Bogomolov-Gieseker type inequality The existence of stability conditions on three-dimensional varieties in general, and more specifically on Calabi-Yau threefolds, is often considered the biggest open problem in the theory of Bridgeland stability conditions. Until recent work by Maciocia and Piyaratne [29,30], they were only known to exist on threefolds whose derived category admits a full exceptional collection. Possible applications of stability conditions range from modularity properties of generating functions of Donaldson-Thomas invariants [43,45] to Reider-type theorems for adjoint linear series [6]. In [11], the first two authors and Yukinobu Toda, also based on discussions with Aaron Bertram, proposed a general approach towards the construction of stability conditions on a smooth projective threefold X . The construction is based on the auxiliary notion of tilt-stability for twoterm complexes, and a conjectural Bogomolov-Gieseker type inequality for the third Chern character of tilt-stable objects; we review these notions in Sect. 2 and the precise inequality in Conjecture 2.4. It depends on the choice of two divisor classes ω, B ∈ NS(X )R with ω ample. It was shown that this conjecture would imply the existence of Bridgeland stability conditions,2 and, in the companion paper [6], a version of an open case of Fujita’s conjecture, on the very ampleness of adjoint line bundles on threefolds. Our first main result is the following, generalizing the result of [29,30] for the case when X has Picard rank one: Theorem 1.1 The Bogomolov-Gieseker type inequality for tilt-stable objects, Conjecture 2.4, holds when X is an abelian threefold, and ω is a real multiple of an integral ample divisor class. There are Calabi-Yau threefolds that admit an abelian variety as a finite étale cover; we call them Calabi-Yau threefolds of abelian type. Our result applies similarly in these cases: Theorem 1.2 Conjecture 2.4 holds when X is a Calabi-Yau threefold of abelian type, and ω is a real multiple of an integral ample divisor class. Combined with the results of [11], these theorems imply the existence of Bridgeland stability conditions in either case. There is one more type of CalabiYau threefolds whose derived category is closely related to those of abelian threefolds: namely Kummer threefolds, that are obtained as the crepant resolution of the quotient of an abelian threefold X by the action of a finite group 2 Not including the so-called “support property” reviewed further below. 123 A. Bayer et al. G. Using the method of “inducing” stability conditions on the G-equivariant derived category of X and the BKR-equivalence [8], we can also treat this case. Over (...truncated)