Mathematische Annalen

, Volume 354, Issue 1, pp 1–42 | Cite as

Cusps of the Kähler moduli space and stability conditions on K3 surfaces

  • Heinrich Hartmann


Ma (Int J Math 20(6):727–750, 2009) established a bijection between Fourier–Mukai partners of a K3 surface and cusps of the Kähler moduli space. The Kähler moduli space can be described as a quotient of Bridgeland’s stability manifold. We study the relation between stability conditions σ near to a cusp and the associated Fourier–Mukai partner Y in the following ways. (1) We compare the heart of σ to the heart of coherent sheaves on Y. (2) We construct Y as moduli space of σ-stable objects. An appendix is devoted to the group of auto-equivalences of \({\mathcal{D}^b(X)}\) which respect the component \({Stab^{\dagger}(X)}\) of the stability manifold.

Mathematics Subject Classification (2000)

14F05 14J28 18E30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baily W.L., Borel A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. 84(2), 442–528 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Beauville A., Bourguignon J.-P., Demazure M.: Géométrie des surfaces K3: modules et périodes. Société Mathématique de France, Paris (1985)Google Scholar
  3. 3.
    Borel, A., Ji, L.: Compactifications of symmetric and locally symmetric spaces. In: Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston (2006)Google Scholar
  4. 4.
    Bridgeland T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bridgeland T.: Stability conditions on K3 surfaces. Duke Math. J. 141(2), 241–291 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bridgeland, T.: Spaces of stability conditions. In: Algebraic geometry—Seattle 2005. Proc. Sympos. Pure Math., Part 1, vol. 80, pp. 1–21. Amer Math. Soc., Providence (2009)Google Scholar
  7. 7.
    Dolgachev I.V.: Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81(3), 2599–2630 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Grothendieck, A.: Théorie des intersections et théorème de Riemann-Roch (SGA6). In: Lecture Notes in Mathematics, vol. 225. Springer, Berlin (1971)Google Scholar
  9. 9.
    Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, No. 52. Springer, New York (1977)Google Scholar
  10. 10.
    Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. In: Pure and Applied Mathematics, vol. 80. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1978)Google Scholar
  11. 11.
    Hosono S., Lian B.H., Oguiso K., Yau S.-T.: Auto equivalences of derived category of a K3 surface and monodromy transformations. J. Algebraic Geom. 13(3), 513–545 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Huybrechts, D.: Fourier–Mukai transforms in algebraic geometry. In: Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, Oxford (2006)Google Scholar
  13. 13.
    Huybrechts D.: Derived and abelian equivalence of K3 surfaces. J. Algebraic Geom. 17(2), 375–400 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Huybrechts D., Macrì E., Stellari P.: Derived equivalences of K3 surfaces and orientation. Duke Math. J. 149(3), 461– (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Huybrechts D., Stellari P.: Equivalences of twisted K3 surfaces. Math. Ann. 332(4), 901–936 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Ji L., MacPherson R.: Geometry of compactifications of locally symmetric spaces. Ann. Inst. Fourier (Grenoble) 52(2), 457–559 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lieblich M.: Moduli of complexes on a proper morphism. J. Algebraic Geom. 15(1), 175–206 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Looijenga E.: Compactifications defined by arrangements. II. Locally symmetric varieties of type IV. Duke Math. J. 119(3), 527–588 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ma S.: Fourier-Mukai partners of a K3 surface and the cusps of its Kahler moduli. Int. J. Math. 20(6), 727–750 (2009)zbMATHCrossRefGoogle Scholar
  20. 20.
    Ma S.: On the 0-dimensional cusps of the K ähler moduli of a K3 surface. Math. Ann. 348(1), 57–80 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Mukai, S.: On the moduli space of bundles on K3 surfaces. I. Tata Inst. Fund. Res. Stud. Math., vol. 11. Tata Inst. Fund. Res., Bombay (1987)Google Scholar
  22. 22.
    Mumford, D.: Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics No. 5. Tata Institute of Fundamental Research, Bombay (1970)Google Scholar
  23. 23.
    Ploog, D.: Groups of autoequivalences of derived categories of smooth projective varieties. PhD thesis, FU Berlin (2005)Google Scholar
  24. 24.
    Toda Y.: Moduli stacks and invariants of semistable objects on K3 surfaces. Adv. Math. 217(6), 2736–2781 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institute for MathematicsUniversity of BonnBonnGermany

Personalised recommendations