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Stability conditions in families
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Bridgeland stability on blow ups and counterexamples

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  • Published: 17 May 2021

Stability conditions in families

  • Arend Bayer1,
  • Martí Lahoz2 nAff3,
  • Emanuele Macrì4 nAff5,
  • Howard Nuer4 nAff6,
  • Alexander Perry7 nAff8 &
  • …
  • Paolo Stellari9 

Publications mathématiques de l'IHÉS volume 133, pages 157–325 (2021)Cite this article

  • 1199 Accesses

  • 13 Citations

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Abstract

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.

Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.

Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

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Author information

Author notes

  1. Martí Lahoz

    Present address: Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007, Barcelona, Spain

  2. Emanuele Macrì

    Present address: Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, Rue Michel Magat, Bât. 307, 91405, Orsay, France

  3. Howard Nuer

    Present address: Department of Mathematics, Technion, Israel Institute of Technology, Amado 914, Haifa, 32000, Israel

  4. Alexander Perry

    Present address: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

Authors and Affiliations

  1. School of Mathematics and Maxwell Institute, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK

    Arend Bayer

  2. Université Paris Diderot – Paris 7, Bâtiment Sophie Germain, Case 7012, 75205, Paris Cedex 13, France

    Martí Lahoz

  3. Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA, 02115, USA

    Emanuele Macrì & Howard Nuer

  4. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Alexander Perry

  5. Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133, Milano, Italy

    Paolo Stellari

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  1. Arend Bayer
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Corresponding author

Correspondence to Arend Bayer.

Additional information

A. B. was supported by the ERC Starting Grant ERC-2013-StG-337039-WallXBirGeom, by the ERC Consolidator Grant ERC-2018-CoG-819864-WallCrossAG, and by the NSF Grant DMS-1440140 while the author was in residence at the MSRI in Berkeley, during the Spring 2019. M. L. was supported by a Ramón y Cajal fellowship and partially by the Spanish MINECO research project PID2019-104047GB-I00. E. M. was partially supported by the NSF grant DMS-1700751, by a Poincaré Chair from the Institut Henri Poincaré and the Clay Mathematics Institute, by the Institut des Hautes Études Scientifiques (IHÉS), by a Poste Rouge CNRS at Université Paris-Sud, by the ERC Synergy Grant ERC-2020-SyG-854361-HyperK, and by the National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA-INdAM). H. N. was partially supported by the NSF postdoctoral fellowship DMS-1606283, by the NSF RTG grant DMS-1246844, and by the NSF FRG grant DMS-1664215. A. P. was partially supported by the NSF postdoctoral fellowship DMS-1606460 and the NSF grant DMS-2002709. P. S. was partially supported by the ERC Consolidator Grant ERC-2017-CoG-771507-StabCondEn, by the research project PRIN 2017 “Moduli and Lie Theory”, and by research project FARE 2018 HighCaSt (grant number R18YA3ESPJ)

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Bayer, A., Lahoz, M., Macrì, E. et al. Stability conditions in families. Publ.math.IHES 133, 157–325 (2021). https://doi.org/10.1007/s10240-021-00124-6

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  • Received: 26 February 2019

  • Revised: 27 April 2021

  • Accepted: 27 April 2021

  • Published: 17 May 2021

  • Issue Date: June 2021

  • DOI: https://doi.org/10.1007/s10240-021-00124-6

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