Skip to main content
SpringerLink
Log in

An integrable system of K3-Fano flags

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Given a K3 surface S, we show that the relative intermediate Jacobian of the universal family of Fano 3-folds V containing S as an anticanonical divisor is a Lagrangian fibration.

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

Similar content being viewed by others

References

  1. Beauville, A.: Vector bundles on the cubic threefold. In: Bertram, A. (ed.) Symposium in Honor of C. H. Clemens, University of Utah, March 2000, Contemp. Math. vol. 312, pp. 71–86 (2002)

  2. Beauville, A.: Fano threefolds and K3 surfaces. The Fano Conference, pp. 175–184. University Torino (2004)

  3. Deligne P. (1971). Théorie de Hodge, II. Publ. Math. IHES 40: 5–57

    MATH  MathSciNet  Google Scholar 

  4. Donagi, R.: ICMP lecture on Heterotic/F-theory duality. arXiv:hep-th/9802093

  5. Donagi, R., Markman, E.: Spectral covers, algebraically completely integrable Hamiltonian systems, and moduli of bundles. In: Francaviglia M. (ed.) Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, vol. 1620, pp. 1–119 (1996)

  6. Green, M.L.: Infinitesimal methods in Hodge theory. Algebraic cycles and Hodge theory (Torino, 1993), Lecture Notes in Mathematics, vol. 1594, pp. 1–92. Springer, Berlin (1994)

  7. Griffiths P. (1968). Periods of integrals on algebraic manifolds II. Local study of the period mapping. Am. J. Math. 90: 805–865

    Article  MATH  Google Scholar 

  8. Hassett B. and Tschinkel Yu. (2000). Abelian fibrations and rational points on symmetric products. Int. J. Math. 11: 1163–1176

    MATH  MathSciNet  Google Scholar 

  9. Iskovskih V.A. (1977). Fano 3-folds I. Izv. Akad. Nauk SSSR Ser. Mat. 41: 516–562

    MATH  MathSciNet  Google Scholar 

  10. Iskovskih V.A. (1978). Fano 3-folds II. Izv. Akad. Nauk SSSR Ser. Mat. 41: 469–506

    Google Scholar 

  11. Iskovskih, V.A.: Double projection from a line onto Fano 3-folds of the first kind. (Russian) Mat. Sb. 180, 260–278 (1989). Eng. Trans. Math. USSR-Sb. 66 265–284 (1990)

  12. Iliev A. and Manivel L. (2007). Prime Fano threefolds and integrable systems. Math. Ann. 339: 937–955

    Article  MATH  MathSciNet  Google Scholar 

  13. Iskovskih, V.A., Prokhorov, Yu.G.: Fano varieties. Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, pp. 1–247. Springer, Berlin (1999)

  14. Kashiwara M. (1985). The asymptotic behavior of a variation of polarized Hodge structure. Publ. Res. Inst. Math. Sci. 21: 853–875

    Article  MATH  MathSciNet  Google Scholar 

  15. Mori, S., Mukai, S.: Classification of Fano 3-folds with B 2 ≥ 2. Manuscripta Math. 36, 147–162 (1981/82); Erratum, Manuscripta Math. 110, 407 (2003)

    Google Scholar 

  16. Newstead P.E. (1968). Stable bundles of rank 2 and odd degree over a curve of genus 2. Topology 7: 205–215

    Article  MATH  MathSciNet  Google Scholar 

  17. Narasimhan M.S. and Ramanan S. (1975). Deformations of the moduli space of vector bundles over an algebraic curve. Ann. Math. 101(2): 391–417

    Article  MathSciNet  Google Scholar 

  18. Thomas, R.: Gauge theory on Calabi–Yau manifolds. Ph.D. thesis. Oxford (1997)

  19. Tyurin A.N. (1990). The moduli spaces of vector bundles on threefolds, surfaces and curves I. Preprint, Erlangen

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathématiques - bât. M2, Université Lille1, 59655, Villeneuve d’Ascq Cedex, France

    D. Markushevich

Authors
  1. D. Markushevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Markushevich.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Markushevich, D. An integrable system of K3-Fano flags. Math. Ann. 342, 145–156 (2008). https://doi.org/10.1007/s00208-008-0227-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-008-0227-2

Mathematics Subject Classification (2000)

Search

Navigation