Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on Bogomolov-Gieseker type inequality for Calabi-Yau 3-folds


Author: Yukinobu Toda
Journal: Proc. Amer. Math. Soc. 142 (2014), 3387-3394
MSC (2010): Primary 14F05
DOI: https://doi.org/10.1090/S0002-9939-2014-12096-X
Published electronically: June 25, 2014
MathSciNet review: 3238415
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The conjectural Bogomolov-Gieseker (BG) type inequality for tilt semistable objects on projective 3-folds was proposed by Bayer, Macri and the author. In this note, we prove our conjecture for slope stable sheaves with the smallest first Chern class on certain Calabi-Yau 3-folds, e.g. quintic 3-folds.


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

  • [1] D. Arcara and A. Bertram,
    Bridgeland-stable moduli spaces for K-trivial surfaces, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 1-38. MR 2998828
  • [2] A. Bayer, A. Bertram E. Macri, and Y. Toda,
    Bridgeland stability conditions on 3-folds II: An application to Fujita's conjecture.
    Preprint, arXiv:1106.3430.
  • [3] A. Bayer, E. Macri, and Y. Toda,
    Bridgeland stability conditions on 3-folds I: Bogomolov-Gieseker type inequalities, J. Algebraic Geom. 23 (2014), no. 1, 117-163. MR 3121850
  • [4] F. A. Bogomolov, Holomorphic tensors and vector bundles on projective manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 6, 1227-1287, 1439 (Russian). MR 522939 (80j:14014)
  • [5] Tom Bridgeland, Stability conditions on $ K3$ surfaces, Duke Math. J. 141 (2008), no. 2, 241-291. MR 2376815 (2009b:14030), https://doi.org/10.1215/S0012-7094-08-14122-5
  • [6] D. Gieseker, On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math. 101 (1979), no. 1, 77-85. MR 527826 (80j:14015), https://doi.org/10.2307/2373939
  • [7] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870 (98g:14012)
  • [8] Adrian Langer, Moduli spaces of sheaves and principal $ G$-bundles, Algebraic geometry--Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 273-308. MR 2483939 (2010d:14010)
  • [9] Y. Toda,
    Bogomolov-Gieseker type inequality and counting invariants, J. Topol. 6 (2013), no. 1, 217-250. MR 3029426
  • [10] Kōta Yoshioka, An action of a Lie algebra on the homology groups of moduli spaces of stable sheaves, Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), Adv. Stud. Pure Math., vol. 58, Math. Soc. Japan, Tokyo, 2010, pp. 403-459. MR 2676164 (2011j:14019)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14F05

Retrieve articles in all journals with MSC (2010): 14F05


Additional Information

Yukinobu Toda
Affiliation: Kavli Institute for the Physics and Mathematics of the Universe, Todai Institute for Advanced Studies (TODIAS), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
Email: yukinobu.toda@ipmu.jp

DOI: https://doi.org/10.1090/S0002-9939-2014-12096-X
Received by editor(s): January 29, 2012
Received by editor(s) in revised form: October 30, 2012
Published electronically: June 25, 2014
Additional Notes: This work was supported by a World Premier International Research Center Initiative (WPI initiative), MEXT, Japan. This work was also supported by Grant-in-Aid for Scientific Research grant (22684002), and partly (S-19104002), from the Ministry of Education, Culture, Sports, Science and Technology, Japan
Communicated by: Lev Borisov
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society