Skip to main content
SpringerLink
Log in

On a Relative Fourier–Mukai Transform on Genus One Fibrations

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We study relative Fourier–Mukai transforms on genus one fibrations with section, allowing explicitly the total space of the fibration to be singular and non-projective. Grothendieck duality is used to prove a skew–commutativity relation between this equivalence of categories and certain duality functors. We use our results to explicitly construct examples of semi-stable sheaves on degenerating families of elliptic curves.

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. Altman A., Kleiman S. (1980) Compactifying the Picard scheme. Adv. Math. 35, 50–112

    Article  MATH  MathSciNet  Google Scholar 

  2. Altman A., Kleiman S. (1979) Compactifying the Picard scheme II. Am. J. Math. 101, 10–41

    Article  MATH  MathSciNet  Google Scholar 

  3. Andreas, B., Curio, G., Hernández Ruipérez, D., Yau, S.-T. Fourier–Mukai Transform and Mirror Symmetry for D-Branes on Elliptic Calabi-Yau. arXiv: math.AG/0012196

  4. Bartocci C., Bruzzo U., Hernández Ruipérez D., Muñoz Porras J.M. (1998) Mirror symmetry on K3 surfaces via Fourier–Mukai transform. Comm. Math. Phys. 195, 79–93

    Article  MATH  MathSciNet  Google Scholar 

  5. Ben-Zvi, D., Nevins, T. From solitons to many-body systems. arXiv: math.AG/ 0310490

  6. Bondal, A., Orlov, D. Semiorthogonal decomposition for algebraic varieties. arXiv: alg-geom/9506012

  7. Bridgeland T. (1998) Fourier–Mukai transforms for elliptic surfaces. J. Reine Angew. Math. 498, 115–133

    MATH  MathSciNet  Google Scholar 

  8. Bridgeland T. (1999) Equivalences of triangulated categories and Fourier–Mukai transforms. Bull. Lond. Math. Soc. 31, 25–34

    Article  MATH  MathSciNet  Google Scholar 

  9. Bridgeland T. (2002) Flops and derived categories. Invent. Math. 147(3): 613–632

    Article  MATH  MathSciNet  Google Scholar 

  10. Bridgeland T., Maciocia A. (2002) Fourier–Mukai transforms for K3 and elliptic fibrations. J. Algebraic Geom. 11(4): 629–657

    MATH  MathSciNet  Google Scholar 

  11. Bruns, W., Herzog, J. Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, (1998)

  12. Burban I., Kreußler B. (2005) Fourier–Mukai transforms and semi-stable sheaves on nodal Weierstrass cubics. J. Reine Angew. Math. 584, 45–82

    MATH  MathSciNet  Google Scholar 

  13. Caldararu A. (2002) Derived categories of twisted sheaves on elliptic threefolds. J. Reine Angew. Math. 544, 161–179

    MATH  MathSciNet  Google Scholar 

  14. Chen J.-C. (2002) Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities. J. Differ. Geom. 61(2): 227–261

    MATH  Google Scholar 

  15. Conrad B. (2000) Grothendieck Duality and Base Change. Lecture Notes in Mathematics vol. 1750. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  16. Donagi, R., Pantev, T. Torus fibrations, gerbes, and duality. arXiv: math.AG/0306213

  17. Friedman R., Morgan J., Witten E. (1999) Vector bundles over elliptic fibrations. J. Algebr. Geom. 8(2): 279–401

    MATH  MathSciNet  Google Scholar 

  18. Grothendieck, A. Éléments de géométrie algébrique. Publ. Math., Inst. Hautes Étud. Sci. 4, 17, 20, 24, 28, 32 (1960–1967)

  19. Hartshorne R. (1966) Residues and duality. Lecture Notes in Mathematics 20, Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  20. Hernández Ruipérez, D., López Martín, A.C., Sancho de Salas, F. Fourier Mukai Transforms for Gorenstein Schemes. arXiv: math.AG/0506593

  21. Illusie, L. Existance de résolutions globales, Exposé II. Théorie des intersections et théorème de Riemann-Roch, Séminaire de Géométrie Algèbrique du Bois-Marie 1966–1967 (SGA 6), Lecturer Notes in Mathematics 225, 160–221 (1971)

    Google Scholar 

  22. Kawamata Y. (2002) D-equivalence and K-equivalence. J. Differ. Geom. 61(1): 147–171

    MATH  MathSciNet  Google Scholar 

  23. Lipman, J. Notes on Derived Categories and Derived Functors. Available on http://www.math.purdue.edu/~lipman/

  24. Mumford D. (1970) Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, Vol. 5. Oxford University Press, Oxford

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  26. Orlov, D.O. Derived categories of coherent sheaves and equivalences between them. Uspekhi Mat. Nauk 58(3), 89–172 (2003); translation in Russian Math. Surveys 58(3), 511–591 (2003)

  27. Spaltenstein N. (1988) Resolutions of unbounded complexes. Compos. Math. 65(2): 121–154

    MATH  MathSciNet  Google Scholar 

  28. Van den Bergh M. (2004) Three-dimensional flops and noncommutative rings. Duke Math. J. 122(3): 423–455

    Article  MATH  MathSciNet  Google Scholar 

  29. Weibel C. (1994) An introduction to homological Algebra. Cambridge Studies in Advanced Mathematics Vol. 38. Cambridge University Press, Cambridge

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Physik, Mathematik und Informatik, Institut für Mathematik, Johannes-Gutenberg Universität Mainz, 55099, Mainz, Germany

    Igor Burban

  2. Mary Immaculate College, South Circular Road, Limerick, Ireland

    Bernd Kreußler

Authors
  1. Igor Burban
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernd Kreußler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernd Kreußler.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burban, I., Kreußler, B. On a Relative Fourier–Mukai Transform on Genus One Fibrations. manuscripta math. 120, 283–306 (2006). https://doi.org/10.1007/s00229-006-0013-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-006-0013-y

Mathematics Subject Classification (2000)

Search

Navigation