Skip to main content
SpringerLink
Log in

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

  • Published:
Inventiones mathematicae Aims and scope

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 \(X\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. We will prove this and the following results more generally for moduli spaces of Bridgeland-stable complexes.

  2. In [90, Section 3], this Theorem is only stated for families \({\mathcal { E}}\) satisfying \(\mathop {\mathrm {Ext}}\nolimits ^{<0}({\mathcal { E}}_s, {\mathcal { E}}_s) = 0\) for all \(s \in S\). However, Toda’s proof in Lemma 3.13 and Proposition 3.18 never uses that assumption.

  3. In the sense of Definition 5.4.

  4. Note that the restriction of \(\alpha \) to any curve vanishes, hence the structure sheaf \({\mathcal { O}}_D\) is a coherent sheaf on \((Y, \alpha ')\).

References

  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:0708.2247

  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:1203.0316

    Article  MATH  MathSciNet  Google Scholar 

  3. Amerik, E.: A remark on a question of Beauville about lagrangian fibrations. Mosc. Math. J. 12(4), 701–704 (2012). arXiv:1110.2852

    MATH  MathSciNet  Google Scholar 

  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/0309435

    MATH  MathSciNet  Google Scholar 

  5. Bayer, A.: Polynomial Bridgeland stability conditions and the large volume limit. Geom. Topol. 13(4), 2389–2425 (2009). arXiv:0712.1083

    Article  MATH  MathSciNet  Google Scholar 

  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. 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/0610203

  8. Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782 (1984)

    Google Scholar 

  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)

  10. Beauville, A.: Counting rational curves on \(K3\) surfaces. Duke Math. J. 97(1), 99–108 (1999). arXiv:alg-geom/9701019

    Article  MATH  MathSciNet  Google Scholar 

  11. Bayer, A., Hassett, B., Tschinkel, Y.: Mori cones of holomorphic symplectic varieties of K3 type (2013). arXiv:1307.2291

  12. Bakker, B., Jorza, A.: Lagrangian hyperplanes in holomorphic symplectic varieties (2011). arXiv:1111.0047.

  13. Bayer, A., Macrì, E.: The space of stability conditions on the local projective plane. Duke Math. J. 160(2), 263–322 (2011). arXiv:0912.0043

    Article  MATH  MathSciNet  Google Scholar 

  14. Bayer, A., Macrì, E.: Projectivity and birational geometry of Bridgeland moduli spaces (2012). arXiv:1203.4613

  15. Bayer, A., Macrì, E., Toda, Y.: Bridgeland stability conditions on threefolds I: Bogomolov–Gieseker type inequalities. J. Geom. Alg. (2011). arXiv:1103.5010

  16. Bertram, A., Martinez, C., Wang, J.: The birational geometry of moduli space of sheaves on the projective plane (2013). arXiv:1301.2011

  17. Boucksom, S.: Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4) 37(1), 45–76 (2004)

    MATH  MathSciNet  Google Scholar 

  18. Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007). arXiv:math/0212237

    Article  MATH  MathSciNet  Google Scholar 

  19. Bridgeland, T.: Stability conditions on \(K3\) surfaces. Duke Math. J. 141(2), 241–291 (2008). arXiv:math/0307164

    Article  MATH  MathSciNet  Google Scholar 

  20. Căldăraru, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds. ProQuest LLC, Ph.D. Thesis. Cornell University, Ann Arbor (2000)

  21. Căldăraru, A.: Nonfine moduli spaces of sheaves on \(K3\) surfaces. Int. Math. Res. Not. 20(20), 1027–1056 (2002). arXiv:math/0108180

    Google Scholar 

  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:1204.4838

  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/9402008

    Google Scholar 

  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/9410005

    MATH  MathSciNet  Google Scholar 

  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/9410007

    MATH  MathSciNet  Google Scholar 

  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:1012.0329

  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:0910.1156

  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)

  29. Greb, D., Lehn, C., Rollenske, S.: Lagrangian fibrations on hyperkähler fourfolds. Izv. Mat. (2011). arXiv:1110.2680

  30. Greb, D., Lehn, C., Rollenske, S.: Lagrangian fibrations on hyperkähler manifolds—on a question of Beauville. Ann. École Norm. Sup. Sci. (2011). arXiv:1105.3410

  31. Hartmann, H.: Cusps of the Kähler moduli space and stability conditions on K3 surfaces. Math. Ann. 354(1), 1–42 (2012). arXiv:1012.3121

    Article  MATH  MathSciNet  Google Scholar 

  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:1011.1285

    Article  MATH  MathSciNet  Google Scholar 

  33. Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (2010)

  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/0201047

    Article  MATH  MathSciNet  Google Scholar 

  35. Huybrechts, D., Macrì, E., Stellari, P.: Stability conditions for generic \(K3\) categories. Compos. Math. 144(1), 134–162 (2008). arXiv:math/0608430

    Article  MATH  MathSciNet  Google Scholar 

  36. Huybrechts, D., Macrì, E., Stellari, P.: Derived equivalences of \(K3\) surfaces and orientation. Duke Math. J. 149(3), 461–507 (2009). arXiv:0710.1645

    Article  MATH  MathSciNet  Google Scholar 

  37. Huybrechts, D., Stellari, P.: Equivalences of twisted \(K3\) surfaces. Math. Ann. 332(4), 901–936 (2005). arXiv:math/0409030

    Article  MATH  MathSciNet  Google Scholar 

  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/0411541

  39. Hassett, B., Tschinkel, Y.: Rational curves on holomorphic symplectic fourfolds. Geom. Funct. Anal. 11(6), 1201–1228 (2001). arXiv:math/9910021

    Article  MATH  MathSciNet  Google Scholar 

  40. Hassett, B., Tschinkel, Y.: Moving and ample cones of holomorphic symplectic fourfolds. Geom. Funct. Anal. 19(4), 1065–1080 (2009). arXiv:0710.0390

    Article  MATH  MathSciNet  Google Scholar 

  41. Hassett, B., Tschinkel, Y.: Intersection numbers of extremal rays on holomorphic symplectic varieties. Asian J. Math. 14(3), 303–322 (2010). arXiv:0909.4745

    Article  MATH  MathSciNet  Google Scholar 

  42. Huizenga, J.: Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles (2012). arXiv:1210.6576

  43. Huybrechts, D.: Compact hyper-Kähler manifolds: basic results. Invent. Math. 135(1), 63–113 (1999). arXiv:alg-geom/9705025

    Article  MATH  MathSciNet  Google Scholar 

  44. Huybrechts, D.: The Kähler cone of a compact hyperkähler manifold. Math. Ann. 326(3), 499–513 (2003). arXiv:math/9909109

    MATH  MathSciNet  Google Scholar 

  45. Huybrechts, D.: Derived and abelian equivalence of \(K3\) surfaces. J. Algebraic Geom. 17(2), 375–400 (2008). arXiv:math/0604150

    Article  MATH  MathSciNet  Google Scholar 

  46. Huybrechts, D.: A global Torelli theorem for hyperkähler manifolds (after Verbitsky) (2011). arXiv:1106.5573

  47. Hwang, J.-M., Weiss, R.M.: Webs of Lagrangian tori in projective symplectic manifolds. Invent. Math. 192(1), 83–109 (2013). arXiv:1201.2369

    Article  MATH  MathSciNet  Google Scholar 

  48. Hwang, J.-M.: Base manifolds for fibrations of projective irreducible symplectic manifolds. Invent. Math. 174(3), 625–644 (2008). arXiv:0711.3224

    Article  MATH  MathSciNet  Google Scholar 

  49. Kawamata, Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79(3), 567–588 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  50. Kaledin, D., Lehn, M., Sorger, Ch.: Singular symplectic moduli spaces. Invent. Math. 164(3), 591–614 (2006). arXiv:math/0504202

    Article  MATH  MathSciNet  Google Scholar 

  51. Kovács, S.J.: The cone of curves of a \(K3\) surface. Math. Ann. 300(4), 681–691 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  52. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson–Thomas invariants and cluster transformations (2008). arXiv:0811.2435

  53. Li, J.: Algebraic geometric interpretation of Donaldson’s polynomial invariants. J. Differ. Geom. 37(2), 417–466 (1993)

    MATH  Google Scholar 

  54. Lieblich, M.: Moduli of twisted sheaves. Duke Math. J. 138(1), 23–118 (2007). arXiv:math/0411337

    Article  MATH  MathSciNet  Google Scholar 

  55. Lo, J.: On some moduli of complexes on \({K}3\) surfaces (2012). arXiv:1203.1558.

  56. Le Potier, J.: Dualité étrange, sur les surfaces (2005)

  57. Lo, J., Qin, Z.: Mini-walls for Bridgeland stability conditions on the derived category of sheaves over surfaces (2011). arXiv:1103.4352

  58. Maciocia, A.: Computing the walls associated to Bridgeland stability conditions on projective surfaces. Asian J. Math. (2012). arXiv:1202.4587

  59. Markman, E.: Brill-Noether duality for moduli spaces of sheaves on \(K3\) surfaces. J. Algebraic Geom. 10(4), 623–694 (2001). arXiv:math/9901072

    MATH  MathSciNet  Google Scholar 

  60. Markushevich, D.: Rational Lagrangian fibrations on punctual Hilbert schemes of \(K3\) surfaces. Manuscr. Math. 120(2), 131–150 (2006). arXiv:math/0509346

    Article  MATH  MathSciNet  Google Scholar 

  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/0601304

    Article  MATH  MathSciNet  Google Scholar 

  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:1101.4606

  63. Markman, E.: Lagrangian fibrations of holomorphic-symplectic varieties of \({K3}^{[n]}\)-type (2013). arXiv:1301.6584

  64. Markman, E.: Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections. Kyoto J. Math. 53(2), 345–403 (2013). arXiv:0912.4981

    Article  MATH  MathSciNet  Google Scholar 

  65. Matsushita, D.: On fibre space structures of a projective irreducible symplectic manifold. Topology 38(1), 79–83 (1999). arXiv:alg-geom/9709033

    Article  MATH  MathSciNet  Google Scholar 

  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/9903045

  67. Matsushita, D.: On almost holomorphic Lagrangian fibrations (2012). arXiv:1209.1194

  68. Matsushita, D.: On isotropic divisors on irreducible symplectic manifolds (2013). arXiv:1310.0896

  69. Markman, E., Mehrotra, S.: Hilbert schemes of K3 surfaces are dense in moduli (2012). arXiv:1201.0031

  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:1107.5304

    MathSciNet  Google Scholar 

  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:0710.2908

  72. Mongardi, G: A note on the Kähler and Mori cones of manifolds of \({K3}^{[n]}\) type (2013). arXiv:1307.0393

  73. Morrison, D.R.: On \(K3\) surfaces with large Picard number. Invent. Math. 75(1), 105–121 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  74. Mukai, S.: Duality between \(D(X)\) and \(D(\hat{X})\) with its application to Picard sheaves. Nagoya Math. J. 81, 153–175 (1981)

    MATH  MathSciNet  Google Scholar 

  75. Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or \(K3\) surface. Invent. Math. 77(1), 101–116 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

  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/9410016

    Google Scholar 

  79. Minamide, H., Yanagida, S., Yoshioka, K.: Fourier-mukai transforms and the wall-crossing behavior for Bridgeland’s stability conditions (2011). arXiv:1106.5217

  80. Minamide, H., Yanagida, S., Yoshioka, K.: Some moduli spaces of Bridgeland’s stability conditions (2011). arXiv:1111.6187

  81. Namikawa, Y.: Deformation theory of singular symplectic \(n\)-folds. Math. Ann. 319(3), 597–623 (2001). arXiv:math/0010113

    Article  MATH  MathSciNet  Google Scholar 

  82. O’Grady, K.G.: Desingularized moduli spaces of sheaves on a \(K3\). J. Reine Angew. Math. 512, 49–117 (1999). arXiv:alg-geom/9708009

    MATH  MathSciNet  Google Scholar 

  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/9606006

  84. Ploog, D.: Groups of autoequivalences of derived categories of smooth projective varieties. PhD-thesis, Berlin (2005)

  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/0606013

  86. Sawon, J.: Abelian fibred holomorphic symplectic manifolds. Turkish J. Math. 27(1), 197–230 (2003). arXiv:math/0404362

    MATH  MathSciNet  Google Scholar 

  87. Sawon, J.: Lagrangian fibrations on Hilbert schemes of points on \(K3\) surfaces. J. Algebraic Geom. 16(3), 477–497 (2007). arXiv:math/0509224

    Article  MATH  MathSciNet  Google Scholar 

  88. Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001). arXiv:math/0001043

    Article  MATH  MathSciNet  Google Scholar 

  89. Thaddeus, M.: Geometric invariant theory and flips. J. Am. Math. Soc. 9(3), 691–723 (1996). arXiv:alg-geom/9405004

    Article  MATH  MathSciNet  Google Scholar 

  90. Toda, Y.: Moduli stacks and invariants of semistable objects on \(K3\) surfaces. Adv. Math. 217(6), 2736–2781 (2008). arXiv:math.AG/0703590

    Article  MATH  MathSciNet  Google Scholar 

  91. Verbitsky, M.: Cohomology of compact hyper-Kähler manifolds and its applications. Geom. Funct. Anal. 6(4), 601–611 (1996). arXiv:alg-geom/9511009

    Article  MATH  MathSciNet  Google Scholar 

  92. Verbitsky, M.: A global Torelli theorem for hyperkähler manifolds. Duke Math. J. (2009). arXiv:0908.4121

  93. Verbitsky, M.: HyperKähler SYZ conjecture and semipositive line bundles. Geom. Funct. Anal. 19(5), 1481–1493 (2010). arXiv:0811.0639

    Article  MATH  MathSciNet  Google Scholar 

  94. Wierzba, J.: Contractions of symplectic varieties. J. Algebraic Geom. 12(3), 507–534 (2003). arXiv:math/9910130

    Article  MATH  MathSciNet  Google Scholar 

  95. Yoshioka, K.: Irreducibility of moduli spaces of vector bundles on K3 surfaces (1999). arXiv:math/9907001

  96. Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321(4), 817–884 (2001). arXiv:math/0009001

    Article  MATH  MathSciNet  Google Scholar 

  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/0411538

  98. Yoshioka, K.: Fourier-Mukai transform on abelian surfaces. Math. Ann. 345(3), 493–524 (2009). arXiv:math/0605190

    Article  MATH  MathSciNet  Google Scholar 

  99. Yoshioka, K.: Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface (2012). arXiv:1206.4838

  100. Yanagida, S., Yoshioka, K.: Bridgeland’s stabilities on abelian surfaces (2012). arXiv:1203.0884

Download references

Acknowledgments

Conversations with Ralf Schiffler dissuaded us from pursuing a failed approach to the birationality of wall-crossing, and we had extremely useful discussions with Daniel Huybrechts. Tom Bridgeland pointed us towards Corollary 1.3, and Dragos Oprea towards the results in Sect. 15. We also received helpful comments from Daniel Greb, Antony Maciocia, Alina Marian, Eyal Markman, Dimitri Markushevich, Daisuke Matsushita, Ciaran Meachan, Misha Verbitsky, Kōta Yoshioka, Ziyu Zhang, and we would like to thank all of them. We would also like to thank the referee very much for an extremely careful reading of the paper, and for many useful suggestions. The authors were visiting the Max-Planck-Institut Bonn, respectively the Hausdorff Center for Mathematics in Bonn, while working on this paper, and would like to thank both institutes for their hospitality and stimulating atmosphere. A. B.  is partially supported by NSF grant DMS-1101377. E. M.  is partially supported by NSF grant DMS-1001482/DMS-1160466, Hausdorff Center for Mathematics, Bonn, and by SFB/TR 45.

Author information

Authors and Affiliations

  1. School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, UK

    Arend Bayer

  2. Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH, 43210-1174, USA

    Emanuele Macrì

Authors
  1. Arend Bayer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emanuele Macrì
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arend Bayer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bayer, A., Macrì, E. MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations. Invent. math. 198, 505–590 (2014). https://doi.org/10.1007/s00222-014-0501-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-014-0501-8

Mathematics Subject Classification (2010)

Search

Navigation