Inventiones mathematicae

, Volume 198, Issue 3, pp 505–590 | Cite as

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

Article

Abstract

We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space \(M\) of stable sheaves on a K3 surface \(X\): (a) We describe the nef cone, the movable cone, and the effective cone of \(M\) in terms of the Mukai lattice of \(X\). (b) We establish a long-standing conjecture that predicts the existence of a birational Lagrangian fibration on \(M\) whenever \(M\) admits an integral divisor class \(D\) of square zero (with respect to the Beauville–Bogomolov form). These results are proved using a natural map from the space of Bridgeland stability conditions \(\mathop {\mathrm {Stab}}\nolimits (X)\) to the cone \(\mathop {{\mathrm {Mov}}}\nolimits (X)\) of movable divisors on \(M\); this map relates wall-crossing in \(\mathop {\mathrm {Stab}}\nolimits (X)\) to birational transformations of \(M\). In particular, every minimal model of \(M\) appears as a moduli space of Bridgeland-stable objects on \(X\).

Mathematics Subject Classification (2010)

14D20 (Primary) 18E30 14J28 14E30 (Secondary ) 

References

  1. 1.
    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). arXiv:0708.2247Google Scholar
  2. 2.
    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). arXiv:1203.0316MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Amerik, E.: A remark on a question of Beauville about lagrangian fibrations. Mosc. Math. J. 12(4), 701–704 (2012). arXiv:1110.2852MATHMathSciNetGoogle Scholar
  4. 4.
    Abramovich, D., Polishchuk, A.: Sheaves of \(t\)-structures and valuative criteria for stable complexes. J. Reine Angew. Math. 590, 89–130 (2006). arXiv:math/0309435MATHMathSciNetGoogle Scholar
  5. 5.
    Bayer, A.: Polynomial Bridgeland stability conditions and the large volume limit. Geom. Topol. 13(4), 2389–2425 (2009). arXiv:0712.1083MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bertram, A., Coskun, I.: The birational geometry of the Hilbert scheme of points on surfaces. In: Birational Geometry, Rational Curves and Arithmetic, Simons Symposia, pp. 15–55. Springer, Berlin (2013) http://homepages.math.uic.edu/coskun/
  7. 7.
    Birkar, C., Cascini, P., Hacon, C.D., M\(^{{\rm c}}\)Kernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010). arXiv:math/0610203Google Scholar
  8. 8.
    Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782 (1984)Google Scholar
  9. 9.
    Beauville, A.: Systèmes hamiltoniens complètement intégrables associés aux surfaces \(K3\). In: Problems in the Theory of Surfaces and Their Classification (Cortona, 1988), Sympos. Math., vol. XXXII, pp. 25–31. Academic Press, London (1991)Google Scholar
  10. 10.
    Beauville, A.: Counting rational curves on \(K3\) surfaces. Duke Math. J. 97(1), 99–108 (1999). arXiv:alg-geom/9701019MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bayer, A., Hassett, B., Tschinkel, Y.: Mori cones of holomorphic symplectic varieties of K3 type (2013). arXiv:1307.2291Google Scholar
  12. 12.
    Bakker, B., Jorza, A.: Lagrangian hyperplanes in holomorphic symplectic varieties (2011). arXiv:1111.0047.Google Scholar
  13. 13.
    Bayer, A., Macrì, E.: The space of stability conditions on the local projective plane. Duke Math. J. 160(2), 263–322 (2011). arXiv:0912.0043MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Bayer, A., Macrì, E.: Projectivity and birational geometry of Bridgeland moduli spaces (2012). arXiv:1203.4613Google Scholar
  15. 15.
    Bayer, A., Macrì, E., Toda, Y.: Bridgeland stability conditions on threefolds I: Bogomolov–Gieseker type inequalities. J. Geom. Alg. (2011). arXiv:1103.5010Google Scholar
  16. 16.
    Bertram, A., Martinez, C., Wang, J.: The birational geometry of moduli space of sheaves on the projective plane (2013). arXiv:1301.2011Google Scholar
  17. 17.
    Boucksom, S.: Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4) 37(1), 45–76 (2004)MATHMathSciNetGoogle Scholar
  18. 18.
    Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007). arXiv:math/0212237MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Bridgeland, T.: Stability conditions on \(K3\) surfaces. Duke Math. J. 141(2), 241–291 (2008). arXiv:math/0307164MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Căldăraru, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds. ProQuest LLC, Ph.D. Thesis. Cornell University, Ann Arbor (2000)Google Scholar
  21. 21.
    Căldăraru, A.: Nonfine moduli spaces of sheaves on \(K3\) surfaces. Int. Math. Res. Not. 20(20), 1027–1056 (2002). arXiv:math/0108180Google Scholar
  22. 22.
    Ciliberto, C., Knutsen, A.L.: On \(k\)-gonal loci in Severi varieties on general \({K}3\) surfaces and rational curves on hyperkähler manifolds.J. Math. Pures Appl. (2012). arXiv:1204.4838Google Scholar
  23. 23.
    Dolgachev, I.V., Hu, Y.: Variation of geometric invariant theory quotients. Inst. Hautes Études Sci. Publ. Math. 87(87), 5–56 (1998) (with an appendix by Nicolas Ressayre). arXiv:alg-geom/9402008Google Scholar
  24. 24.
    Ellingsrud, G., Göttsche, L.: Variation of moduli spaces and Donaldson invariants under change of polarization. J. Reine Angew. Math. 467, 1–49 (1995). arXiv:alg-geom/9410005MATHMathSciNetGoogle Scholar
  25. 25.
    Friedman, R., Qin, Z.: Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces. Commun. Anal. Geom. 3(1—-2), 11–83 (1995). arXiv:alg-geom/9410007MATHMathSciNetGoogle Scholar
  26. 26.
    Fedorchuk, M., Smyth, D.I.: Alternate compactifications of moduli spaces of curves. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli, vol. I, volume 24 of Adv. Lect. Math., pp. 331–414. Int. Press, Somerville (2013). arXiv:1012.0329Google Scholar
  27. 27.
    Fujino, O.: On Kawamata’s theorem. In: Classification of Algebraic Varieties, EMS Ser. Congr. Rep., pp. 305–315. Eur. Math. Soc., Zürich (2011). arXiv:0910.1156Google Scholar
  28. 28.
    Gross, M., Huybrechts, D., Joyce, D.: Calabi–Yau manifolds and related geometries. Universitext. Springer, Berlin (2003) (Lectures from the Summer School held in Nordfjordeid, June 2001)Google Scholar
  29. 29.
    Greb, D., Lehn, C., Rollenske, S.: Lagrangian fibrations on hyperkähler fourfolds. Izv. Mat. (2011). arXiv:1110.2680Google Scholar
  30. 30.
    Greb, D., Lehn, C., Rollenske, S.: Lagrangian fibrations on hyperkähler manifolds—on a question of Beauville. Ann. École Norm. Sup. Sci. (2011). arXiv:1105.3410Google Scholar
  31. 31.
    Hartmann, H.: Cusps of the Kähler moduli space and stability conditions on K3 surfaces. Math. Ann. 354(1), 1–42 (2012). arXiv:1012.3121MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Harvey, D., Hassett, B., Tschinkel, Y.: Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces. Commun. Pure Appl. Math. 65(2), 264–286 (2012). arXiv:1011.1285MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (2010)Google Scholar
  34. 34.
    Hosono, S., Lian, B.H., Oguiso, K., Yau, S.-T.: Autoequivalences of derived category of a \(K3\) surface and monodromy transformations. J. Algebraic Geom. 13(3), 513–545 (2004). arXiv:math/0201047MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Huybrechts, D., Macrì, E., Stellari, P.: Stability conditions for generic \(K3\) categories. Compos. Math. 144(1), 134–162 (2008). arXiv:math/0608430MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Huybrechts, D., Macrì, E., Stellari, P.: Derived equivalences of \(K3\) surfaces and orientation. Duke Math. J. 149(3), 461–507 (2009). arXiv:0710.1645MATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Huybrechts, D., Stellari, P.: Equivalences of twisted \(K3\) surfaces. Math. Ann. 332(4), 901–936 (2005). arXiv:math/0409030MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Huybrechts, D., Stellari, P.: Proof of Căldăraru’s conjecture. Appendix to: “Moduli spaces of twisted sheaves on a projective variety” by K. Yoshioka. In Moduli spaces and arithmetic geometry, volume 45 of Adv. Stud. Pure Math., pp. 31–42. Math. Soc. Japan, Tokyo (2006). arXiv:math/0411541Google Scholar
  39. 39.
    Hassett, B., Tschinkel, Y.: Rational curves on holomorphic symplectic fourfolds. Geom. Funct. Anal. 11(6), 1201–1228 (2001). arXiv:math/9910021MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Hassett, B., Tschinkel, Y.: Moving and ample cones of holomorphic symplectic fourfolds. Geom. Funct. Anal. 19(4), 1065–1080 (2009). arXiv:0710.0390MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Hassett, B., Tschinkel, Y.: Intersection numbers of extremal rays on holomorphic symplectic varieties. Asian J. Math. 14(3), 303–322 (2010). arXiv:0909.4745MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Huizenga, J.: Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles (2012). arXiv:1210.6576Google Scholar
  43. 43.
    Huybrechts, D.: Compact hyper-Kähler manifolds: basic results. Invent. Math. 135(1), 63–113 (1999). arXiv:alg-geom/9705025MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Huybrechts, D.: The Kähler cone of a compact hyperkähler manifold. Math. Ann. 326(3), 499–513 (2003). arXiv:math/9909109MATHMathSciNetGoogle Scholar
  45. 45.
    Huybrechts, D.: Derived and abelian equivalence of \(K3\) surfaces. J. Algebraic Geom. 17(2), 375–400 (2008). arXiv:math/0604150MATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Huybrechts, D.: A global Torelli theorem for hyperkähler manifolds (after Verbitsky) (2011). arXiv:1106.5573Google Scholar
  47. 47.
    Hwang, J.-M., Weiss, R.M.: Webs of Lagrangian tori in projective symplectic manifolds. Invent. Math. 192(1), 83–109 (2013). arXiv:1201.2369MATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Hwang, J.-M.: Base manifolds for fibrations of projective irreducible symplectic manifolds. Invent. Math. 174(3), 625–644 (2008). arXiv:0711.3224MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Kawamata, Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79(3), 567–588 (1985)MATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Kaledin, D., Lehn, M., Sorger, Ch.: Singular symplectic moduli spaces. Invent. Math. 164(3), 591–614 (2006). arXiv:math/0504202MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Kovács, S.J.: The cone of curves of a \(K3\) surface. Math. Ann. 300(4), 681–691 (1994)MATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson–Thomas invariants and cluster transformations (2008). arXiv:0811.2435Google Scholar
  53. 53.
    Li, J.: Algebraic geometric interpretation of Donaldson’s polynomial invariants. J. Differ. Geom. 37(2), 417–466 (1993)MATHGoogle Scholar
  54. 54.
    Lieblich, M.: Moduli of twisted sheaves. Duke Math. J. 138(1), 23–118 (2007). arXiv:math/0411337MATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Lo, J.: On some moduli of complexes on \({K}3\) surfaces (2012). arXiv:1203.1558.Google Scholar
  56. 56.
    Le Potier, J.: Dualité étrange, sur les surfaces (2005)Google Scholar
  57. 57.
    Lo, J., Qin, Z.: Mini-walls for Bridgeland stability conditions on the derived category of sheaves over surfaces (2011). arXiv:1103.4352Google Scholar
  58. 58.
    Maciocia, A.: Computing the walls associated to Bridgeland stability conditions on projective surfaces. Asian J. Math. (2012). arXiv:1202.4587Google Scholar
  59. 59.
    Markman, E.: Brill-Noether duality for moduli spaces of sheaves on \(K3\) surfaces. J. Algebraic Geom. 10(4), 623–694 (2001). arXiv:math/9901072MATHMathSciNetGoogle Scholar
  60. 60.
    Markushevich, D.: Rational Lagrangian fibrations on punctual Hilbert schemes of \(K3\) surfaces. Manuscr. Math. 120(2), 131–150 (2006). arXiv:math/0509346MATHMathSciNetCrossRefGoogle Scholar
  61. 61.
    Markman, E.: Integral constraints on the monodromy group of the hyperKähler resolution of a symmetric product of a \(K3\) surface. Int. J. Math. 21(2), 169–223 (2010). arXiv:math/0601304MATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    Markman, E.: A survey of Torelli and monodromy results for holomorphic-symplectic varieties. In: Complex and Differential Geometry, volume 8 of Springer Proc. Math., pp. 257–322. Springer, Heidelberg (2011). arXiv:1101.4606Google Scholar
  63. 63.
    Markman, E.: Lagrangian fibrations of holomorphic-symplectic varieties of \({K3}^{[n]}\)-type (2013). arXiv:1301.6584Google Scholar
  64. 64.
    Markman, E.: Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections. Kyoto J. Math. 53(2), 345–403 (2013). arXiv:0912.4981MATHMathSciNetCrossRefGoogle Scholar
  65. 65.
    Matsushita, D.: On fibre space structures of a projective irreducible symplectic manifold. Topology 38(1), 79–83 (1999). arXiv:alg-geom/9709033MATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    Matsushita, D.: Addendum: “On fibre space structures of a projective irreducible symplectic manifold” [Topology 38 (1999), no. 1, 79–83; MR1644091 (99f:14054)]. Topology 40(2), 431–432 (2001). arXiv:math/9903045Google Scholar
  67. 67.
    Matsushita, D.: On almost holomorphic Lagrangian fibrations (2012). arXiv:1209.1194Google Scholar
  68. 68.
    Matsushita, D.: On isotropic divisors on irreducible symplectic manifolds (2013). arXiv:1310.0896Google Scholar
  69. 69.
    Markman, E., Mehrotra, S.: Hilbert schemes of K3 surfaces are dense in moduli (2012). arXiv:1201.0031Google Scholar
  70. 70.
    Maciocia, A., Meachan, C.: Rank one Bridgeland stable moduli spaces on a principally polarized abelian surface. Int. Math. Res. Not. IMRN 9, 2054–2077 (2013). arXiv:1107.5304MathSciNetGoogle Scholar
  71. 71.
    Marian, A., Oprea, D.: A tour of theta dualities on moduli spaces of sheaves. In: Curves and Abelian Varieties, volume 465 of Contemp. Math., pp. 175–201. Amer. Math. Soc., Providence (2008). arXiv:0710.2908Google Scholar
  72. 72.
    Mongardi, G: A note on the Kähler and Mori cones of manifolds of \({K3}^{[n]}\) type (2013). arXiv:1307.0393Google Scholar
  73. 73.
    Morrison, D.R.: On \(K3\) surfaces with large Picard number. Invent. Math. 75(1), 105–121 (1984)MATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    Mukai, S.: Duality between \(D(X)\) and \(D(\hat{X})\) with its application to Picard sheaves. Nagoya Math. J. 81, 153–175 (1981)MATHMathSciNetGoogle Scholar
  75. 75.
    Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or \(K3\) surface. Invent. Math. 77(1), 101–116 (1984)MATHMathSciNetCrossRefGoogle Scholar
  76. 76.
    Mukai, S.: On the moduli space of bundles on \(K3\) surfaces. I. In: Vector Bundles on Algebraic Varieties (Bombay, 1984), volume 11 of Tata Inst. Fund. Res. Stud. Math., pp. 341–413. Tata Inst. Fund. Res., Bombay (1987)Google Scholar
  77. 77.
    Mukai, S.: Fourier functor and its application to the moduli of bundles on an abelian variety. In: Algebraic Geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pp. 515–550. North-Holland, Amsterdam (1987)Google Scholar
  78. 78.
    Matsuki, K., Wentworth, R.: Mumford–Thaddeus principle on the moduli space of vector bundles on an algebraic surface. Int. J. Math. 8(1), 97–148 (1997). arXiv:alg-geom/9410016Google Scholar
  79. 79.
    Minamide, H., Yanagida, S., Yoshioka, K.: Fourier-mukai transforms and the wall-crossing behavior for Bridgeland’s stability conditions (2011). arXiv:1106.5217Google Scholar
  80. 80.
    Minamide, H., Yanagida, S., Yoshioka, K.: Some moduli spaces of Bridgeland’s stability conditions (2011). arXiv:1111.6187Google Scholar
  81. 81.
    Namikawa, Y.: Deformation theory of singular symplectic \(n\)-folds. Math. Ann. 319(3), 597–623 (2001). arXiv:math/0010113MATHMathSciNetCrossRefGoogle Scholar
  82. 82.
    O’Grady, K.G.: Desingularized moduli spaces of sheaves on a \(K3\). J. Reine Angew. Math. 512, 49–117 (1999). arXiv:alg-geom/9708009MATHMathSciNetGoogle Scholar
  83. 83.
    Orlov, D.O.: Equivalences of derived categories and \(K3\) surfaces. J. Math. Sci. (New York) 84(5), 1361–1381 (1997) (Algebraic, geometry, 7). arXiv:alg-geom/9606006Google Scholar
  84. 84.
    Ploog, D.: Groups of autoequivalences of derived categories of smooth projective varieties. PhD-thesis, Berlin (2005)Google Scholar
  85. 85.
    Polishchuk, A.: Constant families of \(t\)-structures on derived categories of coherent sheaves. Mosc. Math. J. 7(1), 109–134, 167 (2007). arXiv:math/0606013Google Scholar
  86. 86.
    Sawon, J.: Abelian fibred holomorphic symplectic manifolds. Turkish J. Math. 27(1), 197–230 (2003). arXiv:math/0404362MATHMathSciNetGoogle Scholar
  87. 87.
    Sawon, J.: Lagrangian fibrations on Hilbert schemes of points on \(K3\) surfaces. J. Algebraic Geom. 16(3), 477–497 (2007). arXiv:math/0509224MATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001). arXiv:math/0001043MATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    Thaddeus, M.: Geometric invariant theory and flips. J. Am. Math. Soc. 9(3), 691–723 (1996). arXiv:alg-geom/9405004MATHMathSciNetCrossRefGoogle Scholar
  90. 90.
    Toda, Y.: Moduli stacks and invariants of semistable objects on \(K3\) surfaces. Adv. Math. 217(6), 2736–2781 (2008). arXiv:math.AG/0703590MATHMathSciNetCrossRefGoogle Scholar
  91. 91.
    Verbitsky, M.: Cohomology of compact hyper-Kähler manifolds and its applications. Geom. Funct. Anal. 6(4), 601–611 (1996). arXiv:alg-geom/9511009MATHMathSciNetCrossRefGoogle Scholar
  92. 92.
    Verbitsky, M.: A global Torelli theorem for hyperkähler manifolds. Duke Math. J. (2009). arXiv:0908.4121Google Scholar
  93. 93.
    Verbitsky, M.: HyperKähler SYZ conjecture and semipositive line bundles. Geom. Funct. Anal. 19(5), 1481–1493 (2010). arXiv:0811.0639MATHMathSciNetCrossRefGoogle Scholar
  94. 94.
    Wierzba, J.: Contractions of symplectic varieties. J. Algebraic Geom. 12(3), 507–534 (2003). arXiv:math/9910130MATHMathSciNetCrossRefGoogle Scholar
  95. 95.
    Yoshioka, K.: Irreducibility of moduli spaces of vector bundles on K3 surfaces (1999). arXiv:math/9907001Google Scholar
  96. 96.
    Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321(4), 817–884 (2001). arXiv:math/0009001MATHMathSciNetCrossRefGoogle Scholar
  97. 97.
    Yoshioka, K.: Moduli spaces of twisted sheaves on a projective variety. In: Moduli Spaces and Arithmetic Geometry, volume 45 of Adv. Stud. Pure Math., pp. 1–30. Math. Soc. Japan, Tokyo (2006). arXiv:math/0411538Google Scholar
  98. 98.
    Yoshioka, K.: Fourier-Mukai transform on abelian surfaces. Math. Ann. 345(3), 493–524 (2009). arXiv:math/0605190MATHMathSciNetCrossRefGoogle Scholar
  99. 99.
    Yoshioka, K.: Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface (2012). arXiv:1206.4838Google Scholar
  100. 100.
    Yanagida, S., Yoshioka, K.: Bridgeland’s stabilities on abelian surfaces (2012). arXiv:1203.0884Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of MathematicsThe University of EdinburghEdinburghScotland, UK
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA

Personalised recommendations