Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Stability conditions and Calabi-Yau fibrations


Author: Yukinobu Toda
Journal: J. Algebraic Geom. 18 (2009), 101-133
DOI: https://doi.org/10.1090/S1056-3911-08-00511-0
Published electronically: April 23, 2008
MathSciNet review: 2448280
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Abstract | References | Additional Information

Abstract: In this paper, we describe the spaces of stability conditions for the triangulated categories associated to three-dimensional Calabi-Yau fibrations. We deal with two cases, the flat elliptic fibrations and smooth $ K3$ (Abelian) fibrations. In the first case, we will see that there exist chamber structures similar to those of the movable cone used in birational geometry. In the second case, we will compare the space with the space of stability conditions for the closed fiber of the fibration.


References [Enhancements On Off] (What's this?)

  • 1. A. Bondal and D. Orlov.
    Semiorthgonal decomposition for algebraic varieties.
    preprint.
    math. AG/9506012.
  • 2. T. Bridgeland.
    Stability conditions and Kleinian singularities.
    preprint.
    math. AG/ 0508257.
  • 3. T. Bridgeland.
    Stability conditions on $ {K}$3 surfaces.
    preprint.
    math. AG/0307164.
  • 4. T. Bridgeland.
    Stability conditions on triangulated categories.
    Ann. of Math (to appear).
    math. AG/0212237.
  • 5. T. Bridgeland.
    Flops and derived categories.
    Invent. Math, Vol. 147, pp. 613-632, 2002. MR 1893007 (2003h:14027)
  • 6. T. Bridgeland.
    Stability conditions on a non-compact Calabi-Yau threefold.
    Comm. Math. Phys., Vol. 266, pp. 715-733, 2006. MR 2238896 (2007d:14075)
  • 7. J-C. Chen.
    Flops and equivalences of derived categories for three-folds with only Gorenstein singularities.
    J. Differential Geom., Vol. 61, pp. 227-261, 2002. MR 1972146 (2004d:14012)
  • 8. M. Van den Bergh.
    Three dimensional flops and noncommutative rings.
    Duke Math. J., Vol. 122, pp. 423-455, 2004. MR 2057015 (2005e:14023)
  • 9. P. Horja.
    Derived Category Automorphisms from Mirror Symmetry.
    Duke Math. J., Vol. 127, pp. 1-34, 2005. MR 2126495 (2006a:14023)
  • 10. A. Ishii and H. Uehara.
    Autoequivalences of derived categories on the minimal resolutions of $ {A}_n$-singularities on surfaces.
    J. Differential Geom., Vol. 71, pp. 385-435, 2005. MR 2198807 (2006k:14024)
  • 11. A. Ishii, K. Ueda, and H. Uehara.
    Stability Conditions on $ {A}_n$-Singularities.
    math. AG/ 0609551.
  • 12. Y. Kawamata.
    On the cone of divisors of Calabi-Yau fiber spaces.
    Internat. J. Math, Vol. 5, pp. 665-687, 1997. MR 1468356 (98g:14043)
  • 13. Y. Kawamata, K. Matsuda, and K. Matsuki.
    Introduction to the Minimal Model Problem.
    Adv. Stud. Pure Math, Vol. 10, pp. 283-360, 1987. MR 946243 (89e:14015)
  • 14. A. Polishchuk.
    Constant families of t-structures on derived categories of coherent sheaves.
    preprint.
    math. AG/0606013. MR 2324559
  • 15. Y. Toda.
    On a certain generalization of spherical twists.
    Bulletin de la SMF (to appear).
    math. AG/0603050.
  • 16. Y. Toda.
    Stability conditions and crepant small resolutions.
    Trans. Amer. Math. Soc. (to appear).
    math. AG/0512648.


Additional Information

Yukinobu Toda
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo, 153-8914, Japan
Address at time of publication: Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa-City, Chiba, 277-8568, Japan
Email: toda@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S1056-3911-08-00511-0
Received by editor(s): October 11, 2006
Received by editor(s) in revised form: September 30, 2007
Published electronically: April 23, 2008
Additional Notes: The author is supported by the Japan Society for the Promotion of Sciences Research Fellowships for Young Scientists, No. 1611452.

American Mathematical Society