Restriction of stable bundles in characteristic $p$
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Abstract:
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a nonsingular projective variety defined over $k$ and $H$ an ample line bundle on $X$. We shall prove that there exists an explicit number $m_{0}$ such that if $E$ is a $\mu$-stable vector bundle of rank at most three, then the restriction $E_{\vert D}$ is $\mu$-stable for all $m\geq m_{0}$ and all smooth irreducible divisors $D\in \vert mH\vert$. This result has implications to the geometry of the moduli space of $\mu$-stable bundles on a surface or a projective space.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
- Fedor A. Bogomolov, Stability of vector bundles on surfaces and curves, Einstein metrics and Yang-Mills connections (Sanda, 1990) Lecture Notes in Pure and Appl. Math., vol. 145, Dekker, New York, 1993, pp. 35–49. MR 1215277
- Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
- J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), no. 1, 53–94 (French). MR 999313, DOI 10.1007/BF01850655
- T. Matsusaka, On polarized varieties of dimension $3$. I, Amer. J. Math. 101 (1979), no. 1, 212–232. MR 527833, DOI 10.2307/2373946
- Hubert Flenner, Restrictions of semistable bundles on projective varieties, Comment. Math. Helv. 59 (1984), no. 4, 635–650. MR 780080, DOI 10.1007/BF02566370
- Rachid Fahlaoui, Stabilité du fibré tangent des surfaces de del Pezzo, Math. Ann. 283 (1989), no. 1, 171–176 (French). MR 973810, DOI 10.1007/BF01457509
- D. Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2) 106 (1977), no. 1, 45–60. MR 466475, DOI 10.2307/1971157
- Masaki Maruyama, Moduli of stable sheaves. I, J. Math. Kyoto Univ. 17 (1977), no. 1, 91–126. MR 450271, DOI 10.1215/kjm/1250522815
- Masaki Maruyama, On boundedness of families of torsion free sheaves, J. Math. Kyoto Univ. 21 (1981), no. 4, 673–701. MR 637512, DOI 10.1215/kjm/1250521908
- Atsushi Moriwaki, A note on Bogomolov-Gieseker’s inequality in positive characteristic, Duke Math. J. 64 (1991), no. 2, 361–375. MR 1136381, DOI 10.1215/S0012-7094-91-06418-5
- Atsushi Moriwaki, Frobenius pull-back of vector bundles of rank $2$ over nonuniruled varieties, Math. Ann. 296 (1993), no. 3, 441–451. MR 1225985, DOI 10.1007/BF01445114
- Atsushi Moriwaki, Arithmetic Bogomolov-Gieseker’s inequality, Amer. J. Math. 117 (1995), no. 5, 1325–1347. MR 1350599, DOI 10.2307/2374978
- V. B. Mehta and A. Ramanathan, Semistable sheaves on projective varieties and their restriction to curves, Math. Ann. 258 (1981/82), no. 3, 213–224. MR 649194, DOI 10.1007/BF01450677
- V. B. Mehta and A. Ramanathan, Restriction of stable sheaves and representations of the fundamental group, Invent. Math. 77 (1984), no. 1, 163–172. MR 751136, DOI 10.1007/BF01389140
- Tohru Nakashima, Bogomolov-Gieseker inequality and cohomology vanishing in characteristic $p$, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3609–3613. MR 1291789, DOI 10.1090/S0002-9939-1995-1291789-9
- Tohru Nakashima, Singularity of the moduli space of stable bundles on surfaces, Compositio Math. 100 (1996), no. 2, 125–130. MR 1383462
- Roberto Paoletti, Seshadri constants, gonality of space curves, and restriction of stable bundles, J. Differential Geom. 40 (1994), no. 3, 475–504. MR 1305979
- N. I. Shepherd-Barron, Unstable vector bundles and linear systems on surfaces in characteristic $p$, Invent. Math. 106 (1991), no. 2, 243–262. MR 1128214, DOI 10.1007/BF01243912
Additional Information
- Tohru Nakashima
- Affiliation: Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo, 192-03 Japan
- Email: nakasima@math.metro-u.ac.jp
- Received by editor(s): April 30, 1996
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 4775-4786
- MSC (1991): Primary 14D20, 14F05
- DOI: https://doi.org/10.1090/S0002-9947-97-02072-2
- MathSciNet review: 1451612