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Large families of stable bundles on abelian varieties

Author: Tohru Nakashima
Journal: Proc. Amer. Math. Soc. 141 (2013), 2225-2231
MSC (2010): Primary 14J60; Secondary 14K12
Published electronically: February 20, 2013
MathSciNet review: 3043004
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Abstract: A sequence of $ \mu $-stable bundles $ \{E_m\}$ on a polarized variety $ (X,H)$ is said to be a large family if their ranks and the discriminants become arbitrarily large as $ m$ goes to infinity. We prove the existence of large families on a principally polarized abelian variety $ (X,\Theta )$ such that the Neron-Severi group is generated by $ \Theta $.

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

  • 1. M.R. Douglas, R. Reinbacher and S.-T. Yau, Branes, bundles and attractors: Bogomolov and beyond, arXiv:math/0604597
  • 2. M. Gulbrandsen, Vector bundles and monads on abelian threefolds, arXiv:math/09073597
  • 3. Christina Birkenhake and Herbert Lange, Complex abelian varieties, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR 2062673
  • 4. Masaki Maruyama, Moduli of stable sheaves. II, J. Math. Kyoto Univ. 18 (1978), no. 3, 557–614. MR 509499
  • 5. Shigeru Mukai, Duality between 𝐷(𝑋) and 𝐷(𝑋) with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 607081
  • 6. Tohru Nakashima, Reflection of sheaves on a Calabi-Yau variety, Asian J. Math. 6 (2002), no. 3, 567–577. MR 1946347, 10.4310/AJM.2002.v6.n3.a8
  • 7. Tohru Nakashima, Strong Bogomolov inequality for stable vector bundles, J. Geom. Phys. 57 (2007), no. 10, 1977–1983. MR 2348273, 10.1016/j.geomphys.2007.04.002
  • 8. Tohru Nakashima, Existence of stable bundles on Calabi-Yau manifolds, Higher dimensional algebraic varieties and vector bundles, RIMS Kôkyûroku Bessatsu, B9, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 153–161. MR 2509698
  • 9. Raffaella Paoletti, Stability of a class of homogeneous vector bundles on 𝑃ⁿ, Boll. Un. Mat. Ital. A (7) 9 (1995), no. 2, 329–343 (English, with Italian summary). MR 1336240

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Additional Information

Tohru Nakashima
Affiliation: Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women’s University, Mejirodai, Bunkyoku, Tokyo 112-8681, Japan

Keywords: Stable vector bundles, abelian varieties
Received by editor(s): May 11, 2011
Received by editor(s) in revised form: October 9, 2011
Published electronically: February 20, 2013
Additional Notes: The author was supported in part by Grant-in-Aid for Scientific Research (C)(21540049)
The author is grateful to the referee for pointing out several mistakes in the original manuscript and for giving valuable comments
Communicated by: Lev Borisov
Article copyright: © Copyright 2013 American Mathematical Society