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Restriction of stable bundles in characteristic $\protect {p}$


Author: Tohru Nakashima
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
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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.


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  • [B1] F.Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math. of the USSR, Izvestija 13 (1979), 499-555. MR 80j:14014
  • [B2] F.Bogomolov, Stability of vector bundles on surfaces and curves, Einstein metrics and Yang-Mills connections, Lecture Notes Pure Appl. Math., vol. 145 Marcel Dekker, New York, 1993, 35-49. MR 94i:14021
  • [D] P.Deligne, Cohomologie des intersections complètes, SGA7, exp.XI., Lecture Notes in Math., 340, Springer, 1973, 39-61. MR 50:7135
  • [D-N] J.-M.Drezet, M.S.Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbe algébriques, Invent.Math. 97 (1989), 53-94. MR 90d:14008
  • [E] L.Ein, Stable vector bundles on projective spaces in char $p>0$, Math.Ann. 254 (1980), 53-72. MR 81d:14010
  • [F] H.Flenner, Restriction of semistable bundles on projective varieties, Comment.Math.Helvetici 59 (1984), 635-650. MR 86m:14014
  • [Fa] R.Fahlaoui, Stabilité du fibré tangent des surfaces de del Pezzo, Math.Ann. 283 (1989), 171-176. MR 89k:14063
  • [G] D.Gieseker, On the moduli of vector bundles on an algebraic surface, Ann.Math. 106 (1977), 45-60. MR 81h:14014
  • [Ma1] M.Maruyama, Moduli of stable sheaves I, J.Math. Kyoto Univ. 17 (1977), 91-126. MR 56:8567
  • [Ma2] M.Maruyama, On boundedness of families of torsion free sheaves, J.Math. Kyoto Univ. 21 (1981), 673-701. MR 83a:14019
  • [Mo1] A.Moriwaki, A note on Bogomolov-Gieseker's inequality in positive characteristic, Duke Math.J 64 (1991), 361-375. MR 92m:14054
  • [Mo2] A.Moriwaki, Frobenius pull-back of vector bundles of rank 2 over non-uniruled varieties, Math.Ann. 296 (1993), 441-451. MR 94j:14039
  • [Mo3] A.Moriwaki, Arithmetic Bogomolov-Gieseker's inequality, Amer.J.Math. 117 (1995), 1325-1347. MR 96i:14022
  • [M-R1] V.B.Mehta, A.Ramanathan, Semistable sheaves on projective varieties and their restriction to curves, Math.Ann. 258 (1982), 213-224. MR 83f:14013
  • [M-R2] V.B.Mehta, A.Ramanathan, Restriction of stable sheaves and representations of the fundamental group, Invent.Math. 77 (1984), 163-172. MR 85m:14026
  • [N1] T.Nakashima, Bogomolov-Gieseker inequality and cohomology vanishing in characteristic $p$, Proc.Amer.Math.Soc. 123 (1995), 3609-3613. MR 96b:14058
  • [N2] T.Nakashima, Singularity of the moduli space of stable bundles on surfaces, Compositio Math. 100 (1996), 125-130. MR 97d:14018
  • [P] R.Paoletti, Seshadri constants, gonality of space curves, and restriction of stable bundles, J.Diff.Geom. 40 (1994), 475-504. MR 95k:14046
  • [S-B] N.I.Shepherd-Barron, Unstable vector bundles and linear systems on surfaces in positive characteristic, Invent.Math. 106 (1991), 243-262. MR 92h:14027

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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

DOI: https://doi.org/10.1090/S0002-9947-97-02072-2
Received by editor(s): April 30, 1996
Article copyright: © Copyright 1997 American Mathematical Society

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