Selecta Mathematica

, Volume 23, Issue 2, pp 1507–1561 | Cite as

Nef divisors for moduli spaces of complexes with compact support

  • Arend Bayer
  • Alastair CrawEmail author
  • Ziyu Zhang
Open Access


In Bayer and Macrì (J Am Math Soc 27(3):707–752, 2014), the first author and Macrì constructed a family of nef divisors on any moduli space of Bridgeland-stable objects on a smooth projective variety X. In this article, we extend this construction to the setting of any separated scheme Y of finite type over a field, where we consider moduli spaces of Bridgeland-stable objects on Y with compact support. We also show that the nef divisor is compatible with the polarising ample line bundle coming from the GIT construction of the moduli space in the special case when Y admits a tilting bundle and the stability condition arises from a \(\theta \)-stability condition for the endomorphism algebra. Our main tool generalises the work of Abramovich–Polishchuk (J Reine Angew Math 590:89–130, 2006) and Polishchuk (Mosc Math J 7(1):109–134, 2007): given a t-structure on the derived category \(D_c(Y)\) on Y of objects with compact support and a base scheme S, we construct a constant family of t-structures on a category of objects on \(Y \times S\) with compact support relative to S.


Bridgeland stability conditions Derived categories t-structures Moduli spaces of sheaves and complexes Nef divisors 

Mathematics Subject Classification

Primary 14D20 Secondary 14F05 14J28 18E30 


  1. 1.
    Arinkin, D., Bezrukavnikov, R.: Perverse coherent sheaves. Mosc. Math. J. 10(1), 3–29, 271 (2010)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Arcara, D., Bertram, A.: Bridgeland-stable moduli spaces for \(K\)-trivial surfaces. J. Eur. Math. Soc. (JEMS) 15(1), 1–38 (2013). (With an appendix by Max Lieblich)Google Scholar
  3. 3.
    Aspinwall, P. S., Bridgeland, T., Craw, A., Douglas, M.R., Gross, M., Kapustin, A., Moore, G.W., Segal, G., Szendrői, B., Wilson, P.M.H.: Dirichlet branes and mirror symmetry, volume 4 of Clay Mathematics Monographs. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA (2009)Google Scholar
  4. 4.
    Arcara, D., Bertram, A., Coskun, I., Huizenga, J.: The minimal model program for the Hilbert scheme of points on \(\mathbb{P}^2\) and Bridgeland stability. Adv. Math. 235, 580–626 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Anno, R., Bezrukavnikov, R., Mirković, I.: Stability conditions for Slodowy slices and real variations of stability (2011). arXiv:1108.1563
  6. 6.
    Abramovich, D., Polishchuk, A.: Sheaves of \(t\)-structures and valuative criteria for stable complexes. J. Reine Angew. Math. 590, 89–130 (2006)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras. Vol. 1, volume 65 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge. Techniques of representation theory (2006)Google Scholar
  8. 8.
    Bertram, A., Coskun, I.: The birational geometry of the Hilbert scheme of points on surfaces. Birational geometry, rational curves, and arithmetic, pp. 15–55. Springer, New York (2013)Google Scholar
  9. 9.
    Bolognese, B., Huizenga, J., Lin, Y., Riedl, E., Schmidt, B., Woolf, M., Zhao, X.: Nef cones of Hilbert schemes of points on surfaces (2015). arXiv:1509.04722
  10. 10.
    Butler, M., King, A.: Minimal resolutions of algebras. J. Algebra 212(1), 323–362 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Bayer, A., Macrì, E.: The space of stability conditions on the local projective plane. Duke Math. J. 160(2), 263–322 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Bayer, A., Macrì, E.: MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones. Lagrangian Fibrations. Invent. Math. 198(3), 505–590 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Bayer, A., Macrì, E.: Projectivity and birational geometry of Bridgeland moduli spaces. J. Am. Math. Soc. 27(3), 707–752 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Bayer, A., Macrì, E., Stellari, P.: The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds (2014). arXiv:1410.1585
  15. 15.
    Bertram, A., Martinez, C., Wang, J.: The birational geometry of moduli space of sheaves on the projective plane (2013). arXiv:1301.2011
  16. 16.
    Bridgeland, T.: Stability conditions on a non-compact Calabi-Yau threefold. Commun. Math. Phys. 266(3), 715–733 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Bridgeland, T.: Stability conditions on \(K3\) surfaces. Duke Math. J. 141(2), 241–291 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Bridgeland, T.: Stability conditions and Kleinian singularities. Int. Math. Res. Not. (IMRN) 21, 4142–4157 (2009)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Bridgeland, T., Stern, D.: Helices on del Pezzo surfaces and tilting Calabi-Yau algebras. Adv. Math. 224(4), 1672–1716 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Bocklandt, R., Schedler, T., Wemyss, M.: Superpotentials and higher order derivations. J. Pure Appl. Algebra 214(9), 1501–1522 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Brav, C., Thomas, H.: Braid groups and Kleinian singularities. Math. Ann. 351(4), 1005–1017 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Choi, J., Chung, K.: The geometry of the moduli space of one-dimensional sheaves. Sci. China Math. 58(3), 487–500 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Coskun, I., Huizenga, J.: The ample cone of moduli spaces of sheaves on the plane (2014). arXiv:1409.5478
  25. 25.
    Coskun, I., Huizenga, J.: Interpolation, Bridgeland stability and monomial schemes in the plane. J. Math. Pures Appl. (9) 102(5), 930–971 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Coskun, I., Huizenga, J.: The birational geometry of the moduli spaces of sheaves on \(\mathbb{P}^2\). In: Proceedings of the Gökova Geometry-Topology Conference 2014, pp. 114–155. Gökova Geometry/Topology Conference (GGT), Gökova (2015)Google Scholar
  27. 27.
    Coskun, I., Huizenga, J.: The nef cone of the moduli space of sheaves and strong Bogomolov inequalities (2015). arXiv:1512.02661
  28. 28.
    Coskun, I., Huizenga, J., Woolf, M.: The effective cone of the moduli space of sheaves on the plane (2014). arXiv:1401.1613
  29. 29.
    Craw, A., Ishii, A.: Flops of \(G\)-Hilb and equivalences of derived categories by variation of GIT quotient. Duke Math. J. 124(2), 259–307 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Eisenbud, D.: Commutative Algebra, Volume 150 of Graduate Texts in Mathematics. Springer, New York (1995). (With a view toward algebraic geometry)Google Scholar
  31. 31.
    Hassett, B., Tschinkel, Y.: Extremal rays and automorphisms of holomorphic symplectic varieties (2015). arXiv:1506.08153
  32. 32.
    Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  33. 33.
    Hille, L., Van den Bergh, M.: Fourier-Mukai transforms. In: Handbook of tilting theory, volume 332 of London Math. Soc. Lecture Note Ser., pp. 147–177. Cambridge Univ. Press, Cambridge (2007)Google Scholar
  34. 34.
    Ishii, A., Ueda, K.: Dimer models and crepant resolutions. To appear in Hokkaido Mathematical Journal (2013). arXiv:1303.4028
  35. 35.
    Ishii, A., Ueda, K., Uehara, H.: Stability conditions on \(A_n\)-singularities. J. Differ. Geom. 84(1), 87–126 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Karmazyn, J.: Quiver GIT for varieties with tilting bundles (2014). arXiv:1407.5005
  37. 37.
    King, A.: Moduli of representations of finite-dimensional algebras. Q. J. Math. Oxford Ser. (2) 45(180), 515–530 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    King, A.: Tilting bundles on some rational surfaces (1997). Unpublished
  39. 39.
    Knudsen, F.F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand. 39(1), 19–55 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Lieblich, M.: Moduli of complexes on a proper morphism. J. Algebraic Geom. 15(1), 175–206 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Li, C., Zhao, X.: The MMP for deformations of Hilbert schemes of points on the projective plane (2013). arXiv:1312.1748
  42. 42.
    Li, C., Zhao, X.: Birational models of moduli spaces of coherent sheaves on the projective plane (2016). arXiv:1603.05035
  43. 43.
    Manivel, L.: Théorèmes d’annulation sur certaines variétés projectives. Comment. Math. Helv. 71(3), 402–425 (1996)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Miličić, D.: Lectures on Derived Categories (2003). Unpublished lecture notes
  45. 45.
    Minamide, H., Yanagida, S., Yoshioka, K.: Fourier-mukai transforms and the wall-crossing behavior for Bridgeland’s stability conditions (2011). arXiv:1106.5217
  46. 46.
    Minamide, H., Yanagida, S., Yoshioka, K.: Some moduli spaces of Bridgeland’s stability conditions. Int. Math. Res. Not. 19, 5264–5327 (2014)zbMATHMathSciNetGoogle Scholar
  47. 47.
    Meachan, C., Zhang, Z.: Birational geometry of singular moduli spaces of O’Grady type. Adv. Math. 296, 210–267 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Amer. Math. Soc. 9(1), 205–236 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    Neeman, A.: Triangulated Categories, Volume 148 of Annals of Mathematics Studies. Princeton University Press, Princeton (2001)Google Scholar
  50. 50.
    Nuer, H.: Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface (2014). arXiv:1406.0908
  51. 51.
    Orlov, D.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk Ser. Math. 56(4), 852–862 (1992)zbMATHGoogle Scholar
  52. 52.
    Polishchuk, A.: Constant families of \(t\)-structures on derived categories of coherent sheaves. Mosc. Math. J. 7(1), 109–134 (2007). 167zbMATHMathSciNetGoogle Scholar
  53. 53.
    Rouquier, R.: Derived categories and algebraic geometry. In: Triangulated Categories, Volume 375 of London Math. Soc. Lecture Note Ser., pp 351–370. Cambridge Univ. Press, Cambridge (2010)Google Scholar
  54. 54.
    Théorie des intersections et théorème de Riemann-Roch. (French) Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. Lecture Notes in Mathematics, vol. 225, xii\(+\)700, pp. 14–06. Springer, Berlin, New York (1971)Google Scholar
  55. 55.
    The Stacks Project Authors: Stacks project (2016). Accessed 29 Sept 2016
  56. 56.
    Thomas, R. P.: Stability conditions and the braid group. In: Superstring Theory, Volume 1 of Adv. Lect. Math. (ALM), pp 209–233. Int. Press, Somerville, MA (2008)Google Scholar
  57. 57.
    Toda, Y.: Stability conditions and crepant small resolutions. Trans. Am. Math. Soc. 360(11), 6149–6178 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    Toda, Y.: Stability conditions and Calabi-Yau fibrations. J. Algebraic Geom. 18(1), 101–133 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    van den Bergh, M.: Non-commutative crepant resolutions. In: The legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004)Google Scholar
  60. 60.
    Matthew W.: Nef and effective cones on the moduli space of torsion sheaves on the projective plane (2013). arXiv:1305.1465
  61. 61.
    Yoshioka, K.: Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface (2012). arXiv:1206.4838

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Authors and Affiliations

  1. 1.School of Mathematics and Maxwell InstituteThe University of EdinburghEdinburghScotland, UK
  2. 2.Department of Mathematical SciencesUniversity of BathClaverton Down, BathUK
  3. 3.Institute for Algebraic GeometryLeibniz University HannoverHannoverGermany

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