Geometric and Functional Analysis

, 19:1065

Moving and Ample Cones of Holomorphic Symplectic Fourfolds



We analyze the ample and moving cones of holomorphic symplectic manifolds, in light of recent advances in the minimal model program. As an application, we establish a numerical criterion for ampleness of divisors on fourfolds deformation-equivalent to punctual Hilbert schemes of K3 surfaces.

Keywords and phrases

Birational geometry ample cones punctual Hilbert schemes of K3 surfaces 

2000 Mathematics Subject Classification

14J35 (14C30) 


  1. B.
    Bauer T.: On the cone of curves of an abelian variety. Amer. J. Math. 120(5), 997–1006 (1998)MATHCrossRefMathSciNetGoogle Scholar
  2. Be.
    Beauville A.: Variétés kählériennes compactes avec c 1 =  0, in “Geometry of K3 surfaces: moduli and periods (Palaiseau, 1981/1982)”. Astérisque 126, 181–192 (1985)MathSciNetGoogle Scholar
  3. BiCHM.
    C. Birkar, P. Cascini, Ch.D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type (2006);
  4. Bo.
    Boucksom S.: Le cône kählérien d’une variété hyperkählérienne. C.R.Acad. Sci. Paris Sér. I Math. 333(10), 935–938 (2001)MATHMathSciNetGoogle Scholar
  5. CMS.
    K. Cho, Y. Miyaoka, N.I. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds, in “Higher Dimensional Birational Geometry (Kyoto, 1997)”, Adv. Stud. Pure Math. 35, Math. Soc. Japan, Tokyo (2002), 1–88.Google Scholar
  6. HT.
    Hassett B., Tschinkel Y.: Rational curves on holomorphic symplectic fourfolds. Geom. Funct. Anal. 11(6), 1201–1228 (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. HuK.
    Hu Yi, Keel S.: Mori dream spaces and GIT. Michigan Math. J 48, 331–348 (2000)MATHCrossRefMathSciNetGoogle Scholar
  8. Huy1.
    Huybrechts D.: Compact hyper-Kähler manifolds: basic results. Invent. Math. 135(1), 63–113 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. Huy2.
    D. Huybrechts, Erratum to “Compact hyper-Kähler manifolds: basic results”, (Invent. Math. 135:1 (1999), 63–113), Invent. Math. 152:1 (2003), 209–212.Google Scholar
  10. Huy3.
    Huybrechts D.: The Kähler cone of a compact hyperkähler manifold. Math. Ann. 326(3), 499–513 (2003)MATHMathSciNetGoogle Scholar
  11. K1.
    Kawamata Y.: On the cone of divisors of Calabi-Yau fiber spaces. Internat. J. Math. 8(5), 665–687 (1997)MATHCrossRefMathSciNetGoogle Scholar
  12. K2.
    Y. Kawamata, Flops connect minimal models (2007);
  13. Ko1.
    J. Kollár, Ed., Flips and Abundance for Algebraic Threefolds, Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque 211 (1992).Google Scholar
  14. Ko2.
    J. Kollár, Singularities of pairs, in “Algebraic Geometry – Santa Cruz 1995”, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI (1997), 221–287.Google Scholar
  15. KoM.
    J. Kollár, S. Mori, (with the collaboration of C.H. Clemens and A. Corti, trans. from 1998 Japanese original), Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics 134, Cambridge University Press, Cambridge (1998).Google Scholar
  16. Kov.
    Kovács S.J.: The cone of curves of a K3 surface. Math. Ann. 300(4), 681–691 (1994)MATHCrossRefMathSciNetGoogle Scholar
  17. LP.
    E. Looijenga, C. Peters, Torelli theorems for Kähler K3 surfaces, Compositio Math. 42:2 (1980/81), 145–186.Google Scholar
  18. M.
    Matsushita D.: On fibre space structures of a projective irreducible symplectic manifold. Topology 38(1), 79–83 (1999)MATHCrossRefMathSciNetGoogle Scholar
  19. Mu.
    Mukai S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77(1), 101–116 (1984)MATHCrossRefMathSciNetGoogle Scholar
  20. N.
    Namikawa Y.: Deformation theory of singular symplectic n-folds, Math. Ann. 319(3), 597–623 (2001)MATHMathSciNetGoogle Scholar
  21. O1.
    O’Grady K.G.: The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface. J. Algebraic Geom. 6(4), 599–644 (1997)MATHMathSciNetGoogle Scholar
  22. O2.
    O’Grady K.G.: A new six-dimensional irreducible symplectic variety. J. Algebraic Geom. 12(3), 435–505 (2003)MATHMathSciNetGoogle Scholar
  23. O3.
    K.G. O’Grady, Irreducible symplectic 4-folds numerically equivalent to Hilb2(K3), Communications in Contemporary Mathematics, to appear;
  24. V.
    C. Voisin, Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes, in “Complex projective geometry (Trieste, 1989/Bergen, 1989)”, London Math. Soc. Lecture Note Ser. 179, Cambridge Univ. Press, Cambridge (1992), pages 294–303.Google Scholar
  25. W.
    Wierzba J.: Contractions of symplectic varieties. J. Algebraic Geom. 12(3), 507–534 (2003)MATHMathSciNetGoogle Scholar
  26. WW.
    Wierzba J., Wiśniewski J.A.: Small contractions of symplectic 4-folds. Duke Math. J. 120(1), 65–95 (2003)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsRice UniversityHoustonUSA
  2. 2.Courant Institute of Mathematical Sciences, NYUNew YorkUSA

Personalised recommendations