Skip to main content
SpringerLink
Log in

Lagrangian Fibrations of Holomorphic-Symplectic Varieties of K3[n]-Type

  • Conference paper
  • First Online:
Algebraic and Complex Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 71))

Abstract

Let X be a compact Kähler holomorphic-symplectic manifold, which is deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface. Let \(\mathcal{L}\) be a nef line-bundle on X, such that the top power \(c_{1}(\mathcal{L})^{2n}\) vanishes and \(c_{1}(\mathcal{L})\) is primitive. Assume that the two dimensional subspace H 2, 0(X) ⊕ H 0, 2(X) of \(H^{2}(X, \mathbb{C})\) intersects \(H^{2}(X, \mathbb{Z})\) trivially. We prove that the linear system of \(\mathcal{L}\) is base point free and it induces a Lagrangian fibration on X. In particular, the line-bundle \(\mathcal{L}\) is effective. A determination of the semi-group of effective divisor classes on X follows, when X is projective. For a generic such pair \((X,\mathcal{L})\), not necessarily projective, we show that X is bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion sheaves, each with pure one dimensional support, on a projective K3 surface.

Partially supported by Simons Foundation Collaboration Grant 245840 and by NSA grant H98230-13-1-0239.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Prop. 5.6 and Lemma 6.22 in the last reference [23]. The same convention will be used throughout the paper for all citations with multiple references.

  2. 2.

    The saturation of a sublattice L ′ of Λ is the maximal sublattice L of Λ, of the same rank as L ′, which contains L ′.

  3. 3.

    I thank K. Oguiso and S. Rollenske for pointing out to me that in the non-algebraic case the result should follow from the above via the results of Ref. [8].

  4. 4.

    I thank C. Lehn for Ref. [18, Prop. 2.4], used in an earlier argument, and T. Peternell and Y. Kawamata for suggesting the current more direct argument.

References

  1. A. Altman, A. Iarrobino, S. Kleiman, Irreducibility of the compactified Jacobian, in Real and Complex Singularities (Sijtho and Noordho, Alphen aan den Rijn). Proceedings of Ninth Nordic Summer School/NAVF Symposium in Mathematics, Oslo, 1976 (1997), pp. 1–12

    Google Scholar 

  2. W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 4 (Springer, Berlin, 2004)

    Google Scholar 

  3. A. Bayer, E. Macri, MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations (2013, Electronic preprint). arXiv:1301.6968

    Google Scholar 

  4. A. Beauville, Varietes Kähleriennes dont la premiere classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983)

    MathSciNet  MATH  Google Scholar 

  5. A. Beauville, Holomorphic symplectic geometry: a problem list, in Complex and Differential Geometry, ed. by W. Ebeling et al. Springer Proceedings in Mathematics, vol. 8 (Springer, Dordrecht, 2011), pp.49–63

    Google Scholar 

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

    Google Scholar 

  7. F. Campana, Isotrivialité de certaines familles hyperkählérian de variété non-projective. Math. Z. 252(1), 147–156 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. F. Campana, K. Oguiso, T. Peternell, Non-algebraic hyperkähler manifolds. J. Diff. Geom. 85, 397–424 (2010)

    MathSciNet  MATH  Google Scholar 

  9. J. Carlson, M. Green, P. Griffiths, J. Harris, Infinitesimal variations of Hodge structure I. Compos. Math. 50(2–3), 109–205 (1983)

    MathSciNet  MATH  Google Scholar 

  10. I. Dolgachev, M. Gross, Elliptic three-folds I: Ogg-Shafarevich theory. J. Algebraic Geom. 3(1) 39–80 (1994)

    MathSciNet  MATH  Google Scholar 

  11. O. Fujino, On Kawamata’s theorem, in Classification of Algebraic Varieties, ed. by C. Faber, G. van der Geer, E. Looijenga. EMS Series of Congress Reports (European Mathematical Society, Zurich, 2011), pp. 305–315

    Google Scholar 

  12. R. Howlett, Introduction to Coxeter groups. Lectures delivered at the A.N.U. Geometric Group Theory Workshop, Jan/Feb 1996. http://www.maths.usyd.edu.au/u/bobh/bobhres.html#infCoxpubs

  13. D. Huybrechts, Compact hyperkähler manifolds: basic results. Invent. Math. 135(1), 63–113 (1999); Erratum: Invent. Math. 152(1), 209–212 (2003)

    Google Scholar 

  14. D. Huybrechts, A global Torelli theorem for hyperkähler manifolds (after Verbitsky). Séminaire Bourbaki, 63 année, 2010–2011, no. 1040

    Google Scholar 

  15. D. Huybrechts, M. Lehn, The Geometry of Moduli Spaces of Sheaves, 2nd edn. (Cambridge University Press, Cambridge, 2010)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Kodaira, On compact analytic surfaces, II. Ann. Math. (2) 77, 563–626 (1963); III. Ann. Math. 78, 1–40 (1963)

    Google Scholar 

  18. C. Lai, Varieties fibered by good minimal models (Electronic preprint). Math. Ann. 350(3), 533–547 (2011). arXiv:0912.3012v2

    Google Scholar 

  19. E. Looijenga, C. Peters, Torelli theorems for Kähler K3 surfaces. Compos. Math. 42(2), 145–186 (1980)

    MathSciNet  MATH  Google Scholar 

  20. E. Markman, Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces. Adv. Math. 208, 622–646 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Markman, On the monodromy of moduli spaces of sheaves on K3 surfaces. J. Algebra. Geom. 17, 29–99 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Markman, Integral constraints on the monodromy group of the hyperkähler resolution of a symmetric product of a K3 surface. Internat. J. Math. 21(2), 169–223 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Markman, A survey of Torelli and monodromy results for hyperkahler manifolds, in Complex and Differential Geometry, ed. by W. Ebeling et al. Springer Proceedings in Mathematics, vol. 8 (Springer, Dordrecht, 2011), pp. 257–323

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  26. D. Markushevich, Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces. Manuscr. Math. 120(2), 131–150 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Matsushita, On deformations of Lagrangian fibrations (Preprint). arXiv:0903.2098

    Google Scholar 

  28. D. Matsushita, On isotropic divisors on irreducible symplectic manifolds (Electronic preprint). arXiv:1310.0896

    Google Scholar 

  29. D. Matsushita, Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds. Math. Res. Lett. 7(4), 389–392 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. D. Matsushita, Higher direct images of dualizing sheaves of Lagrangian fibrations. Am. J. Math. 127(2), 243–259 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Matsushita, A talk given at Hyper-Kähler Geometry, Simons center for Geometry and Physics, Oct 2012. A video is available at http://scgp.stonybrook.edu/archives/3434

  32. A.L. Mayer, Families of K3 surfaces. Nagoya Math. J. 48, 1–17 (1972)

    MathSciNet  Google Scholar 

  33. D. Morrison, Some remarks on the moduli of K3 surfaces, in Classification of Algebraic and Analytic Manifolds, ed. by K. Ueno. Progress in Mathematics, vol. 39 (Birkhäuser, Boston, 1983), pp. 303–332

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Mukai, On the moduli space of bundles on K3 surfaces I, in Vector Bundles on Algebraic Varieties. Proceedings of Bombay Conference, 1984, Tata Institute of Fundamental Research Studies, no. 11, Bombay (Oxford University Press, 1987), pp. 341–413

    Google Scholar 

  36. V.V. Nikulin, Integral symmetric bilinear forms and some of their applications. Math. USSR Izvestija 14(1), 103–167 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  37. T. Oda, C.S. Seshadri, Compactifications of the generalized Jacobian variety. Trans. Am. Math. Soc. 253, 1–90 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  38. K. O’Grady, The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface. J. Algebra. Geom. 6(4), 599–644 (1997)

    MathSciNet  MATH  Google Scholar 

  39. J. Sawon, Lagrangian fibrations on Hilbert schemes of points on K3 surfaces. J. Algebra. Geom. 16(3), 477–497 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Verbitsky, A global Torelli theorem for hyperkähler manifolds. Duke Math. J. 162(15), 2929–2986 (2013). arXiv:0908.4121

    Google Scholar 

  41. M. Verbitsky, Cohomology of compact hyper-Kähler manifolds and its applications. Geom. Funct. Anal. 6, 601–611 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. C. Voisin, Hodge Theory and Complex Algebraic Geometry I. Cambridge Studies in Advanced Mathematics, vol. 76 (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

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

    Google Scholar 

  44. K. Yoshioka, Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321(4), 817–884 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like thank Yujiro Kawamata, Christian Lehn, Daisuke Matsushita, Keiji Oguiso, Osamu Fujino, Thomas Peternell, Sönke Rollenske, Justin Sawon, and Kota Yoshioka for helpful communications. I would like to thank the two referees for their careful reading and insightful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of mathematics and statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Eyal Markman

Authors
  1. Eyal Markman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eyal Markman .

Editor information

Editors and Affiliations

  1. Leibniz Universität Hannover, Hannover, Germany

    Anne Frühbis-Krüger

  2. Humboldt-University Berlin, Berlin, Germany

    Remke Nanne Kloosterman

  3. Leibniz Universität Hannover, Hannover, Germany

    Matthias Schütt

Additional information

Dedicated to Klaus Hulek on the occasion of his sixtieth birthday.

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Markman, E. (2014). Lagrangian Fibrations of Holomorphic-Symplectic Varieties of K3[n]-Type. In: Frühbis-Krüger, A., Kloosterman, R., Schütt, M. (eds) Algebraic and Complex Geometry. Springer Proceedings in Mathematics & Statistics, vol 71. Springer, Cham. https://doi.org/10.1007/978-3-319-05404-9_10

Download citation

Publish with us

Policies and ethics

Search

Navigation