Geodesic
Stable vector bundles on projective surfaces
Sbornik. Mathematics, Tome 81 (1995) no. 2, pp. 397-419 
F. A. Bogomolov. Stable vector bundles on projective surfaces. Sbornik. Mathematics, Tome 81 (1995) no. 2, pp. 397-419. http://geodesic.mathdoc.fr/item/SM_1995_81_2_a6/
@article{SM_1995_81_2_a6,
     author = {F. A. Bogomolov},
     title = {Stable vector bundles on projective surfaces},
     journal = {Sbornik. Mathematics},
     pages = {397--419},
     year = {1995},
     volume = {81},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_81_2_a6/}
}
TY  - JOUR
AU  - F. A. Bogomolov
TI  - Stable vector bundles on projective surfaces
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 397
EP  - 419
VL  - 81
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1995_81_2_a6/
LA  - en
ID  - SM_1995_81_2_a6
ER  - 
%0 Journal Article
%A F. A. Bogomolov
%T Stable vector bundles on projective surfaces
%J Sbornik. Mathematics
%D 1995
%P 397-419
%V 81
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1995_81_2_a6/
%G en
%F SM_1995_81_2_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

An effective variant of an arithmetic criterion for instability of vector bundles on a surface is considered. Namely, a lower bound is established for the degree of a destabilizing subsheaf in a vector bundle with positive discriminant. This bound, which depends on the rank and discriminant of the bundle, is used to prove that the restrictions of stable bundles on a surface to curves are stable, and to prove a number of other results.

[1] Bogomolov Fedor A., “Stability of vector bundles on surfaces and curves”, Einstein metrics and Yang–Mills connections, eds. T. Mabuchi, S. Mukai, Marcel Dekker, Inc., New York–Basel–Hong Kong, 1992, 35–49 | MR

[2] Flenner H., “Restrictions of semistable bundles on projective varieties”, Commentarii Math. Helvetici, 59 (1984), 635–650 | DOI | MR | Zbl

[3] Maruyama M., “Openness of a family of torsion free sheaves”, J. Math. Kyoto Univ., 16 (1976), 627–637 | MR | Zbl

[4] Maruyama M., “Boundness of semistable sheaves of small ranks”, Nagoya Math. J., 78 (1980), 65–94 | MR | Zbl

[5] Maruyama M., “Elementary transformations in the theory of algebraic vector bundles”, Proc. of the conference on Algebraic Geometry, LN, 961, Springer-Verlag, 1981, 241–266 | MR

[6] Mehta V. B., Ramanathan A., “Semistable sheaves on projective varieties and their reduction to curves”, Math. Annalen, 258 (1982), 213–224 | DOI | MR | Zbl

[7] Qui Z., “Birational property of moduli spaces of stable locally free rank-$2$ sheaves on algebraic surface”, Manusc. Math., 72 (1991), 163–180 | DOI | MR

[8] Sakai F., “Semistable curves on algebraic surfaces and logarithmic pluricanonical maps”, Math. Annalen, 254 (1980), 84–120 | DOI | MR