A cohomological interpretation of Bogomolov’s instability
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- by Gabriele Di Cerbo PDF
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Abstract:
We give a new proof of Bogomolov’s instability theorem. Furthermore, we prove that it is equivalent to a statement which characterizes when the first cohomology group of a suitable divisor does not vanish.References
- 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
- Lawrence Ein and Robert Lazarsfeld, Global generation of pluricanonical and adjoint linear series on smooth projective threefolds, J. Amer. Math. Soc. 6 (1993), no. 4, 875–903. MR 1207013, DOI 10.1090/S0894-0347-1993-1207013-5
- Guillermo Fernández del Busto, Bogomolov instability and Kawamata-Viehweg vanishing, J. Algebraic Geom. 4 (1995), no. 4, 693–700. MR 1339844
- Robert Friedman, Algebraic surfaces and holomorphic vector bundles, Universitext, Springer-Verlag, New York, 1998. MR 1600388, DOI 10.1007/978-1-4612-1688-9
- D. Gieseker, On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math. 101 (1979), no. 1, 77–85. MR 527826, DOI 10.2307/2373939
- Robert Lazarsfeld, Lectures on linear series, Complex algebraic geometry (Park City, UT, 1993) IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence, RI, 1997, pp. 161–219. With the assistance of Guillermo Fernández del Busto. MR 1442523, DOI 10.1090/pcms/003/03
- Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
- Yoichi Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 449–476. MR 946247, DOI 10.2969/aspm/01010449
- Miles Reid, Bogomolov’s theorem $c_{1}^{2}\leq 4c_{2}$, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 623–642. MR 578877
- Igor Reider, Vector bundles of rank $2$ and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), no. 2, 309–316. MR 932299, DOI 10.2307/2007055
- Fumio Sakai, Reider-Serrano’s method on normal surfaces, Algebraic geometry (L’Aquila, 1988) Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 301–319. MR 1040564, DOI 10.1007/BFb0083346
- Fernando Serrano, Extension of morphisms defined on a divisor, Math. Ann. 277 (1987), no. 3, 395–413. MR 891582, DOI 10.1007/BF01458322
Additional Information
- Gabriele Di Cerbo
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
- Email: gdi@math.princeton.edu
- Received by editor(s): September 28, 2011
- Received by editor(s) in revised form: December 4, 2011
- Published electronically: June 5, 2013
- Communicated by: Lev Borisov
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3049-3053
- MSC (2010): Primary 14F17, 14J60
- DOI: https://doi.org/10.1090/S0002-9939-2013-11621-7
- MathSciNet review: 3068958