Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T05:13:06.045Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable vector bundles on quadric hypersurfaces

Published online by Cambridge University Press:  22 January 2016

L. Ein  and
I. Sols
L. Ein
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass. 02189, USA
I. Sols
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass. 02189, USA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Barth, Hulek and Maruyama have showed that the moduli of stable rank 2 vector bundles on P2 are nonsingular rational varieties. There are also many examples of stable rank 2 vector bundles on P3. On the other hand, there is essentially only one example of rank 2 bundles on P4, which is constructed by Horrocks and Mumford. We hope the study of rank 2 bundles on hypersurfaces in P4 may give more insight to the study of vector bundles on P4. In this paper, we establish some general properties of stable rank 2 bundles on quadric hypersurfaces. We show the restriction theorem (1.4), (1.6), the existence of the spectrum (2.2), and the vanishing theorem (2.4), are also true for the stable rank 2 reflexive sheaves on quadric hypersurfaces just as in the case when the base variety is Pn. Though the methods to prove such results are similar to those we use for projective spaces, there are some technical difficulties. We should also mention that we shall always assume the base field is characteristic 0 and algebraically closed, and we shall use the definition of stability introduced by Mumford and Takemoto.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 96 , December 1984 , pp. 11 - 22
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1984

References

[ 1 ] Barth, W., Moduli of vector bundles on the projective plane, Invent. Math., 42 (1977), 63–91.CrossRefGoogle Scholar
[ 2 ] Barth, W., Some properties of stable vector bundles on Pn , Math. Ann., 226 (1977), 125–150.CrossRefGoogle Scholar
[ 3 ] Barth, W., Elencwajg, G., Concernant la cohomologie des fibres algebriques stable sur Pn(C), In: Varietes analytiques compacts (Nice, 1977). Lecture Notes in Math., 683 (1978), 1–24.Google Scholar
[ 4 ] Ein, L., Stable vector bundles in char p>0, Math. Ann., 254 (1980), 53–72.CrossRefGoogle Scholar
[ 5 ] Ein, L., Hartshorne, R., Vogelaur, H., Restriction theorems for stable rank 3 vector bundles Pn , Math. Ann., 259 (1982), 541–569.CrossRefGoogle Scholar
[ 6 ] Elencwajg, G., Hirschowitz, A. and Schneider, M., Type de scindage generalise pour fibres stables, (Preprint).Google Scholar
[ 7 ] Hartshorne, R., Stable reflexive sheaves, Math. Ann., 254 (1980), 121–176.CrossRefGoogle Scholar
[ 8 ] Hartshorne, R., Stable reflexive sheaves II, Invent. Math., 66 (1982), 165–190.CrossRefGoogle Scholar
[ 9 ] Hartshorne, R., Sols, I., Stable rank 2 vector bundle on P3 with c± = — 1, c2 = 2, Creile J., 325 (1981), 145–152.Google Scholar