×

zbMATH — the first resource for mathematics

The Chern classes of the stable rank 3 vector bundles on \({\mathbb{P}}^ 3\). (English) Zbl 0588.14010
In an earlier work the author states certain technical conditions which the Chern classes \(c_ 1\), \(c_ 2\) and \(c_ 3\) of a stable rank 3 vector bundle on \({\mathbb{P}}^ 3_ k\) must satisfy after normalizing \(c_ 1\) to \(c_ 1=0, -1\) or -2. In the present work it is shown that if these conditions are satisfied and \(c_ 1c_ 2\equiv c_ 3 mod 2\) for a given set \(c_ 1, c_ 2, c_ 3\) of Chern classes, then there is a stable rank 3 vector bundle on \({\mathbb{P}}^ 3_ k\) with the Chern classes \(c_ 1, c_ 2, c_ 3\). Results of R. Miro on stable rank 2 reflexive sheaves are extended. The author ”adds in proof” that R. Miro has determined the Chern classes of the stable rank 3 reflexive sheaves on \({\mathbb{P}}^ 3\) [R. M. Miro-Roig, Math. Ann. (to appear; see the following review)].
Reviewer: P.Cherenack

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
57R20 Characteristic classes and numbers in differential topology
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Bănică, C., Coandă, I.: Existence of rank 3 vector bundles with given Chern classes on homogeneous rational 3-folds. Manuscripta Math.51, 121–143 (1985) · Zbl 0571.14008
[2] Ein, L., Hartshorne, R., Vogelaar, H.: Restriction theorems for stable rank 3 vector bundles on \(\mathbb{P}\) n . Math. Ann.259, 541–569 (1982) · Zbl 0511.14008
[3] Gruson, L., Peskine, C.: Genre des courbes de l’espace projectif (II). Ann. Sci. Ec. Norm. Super. 4e série,15, 401–418 (1982) · Zbl 0517.14007
[4] Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, Vol. 52. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0367.14001
[5] Hartshorne, R.: Algebraic vector bundles on projective spaces: a problem list. Topology18, 117–128 (1979) · Zbl 0417.14011
[6] Miró, R.: Gaps in the Chern classes of rank 2 stable reflexive sheaves. Math. Ann.270, 317–323 (1985) · Zbl 0548.14006
[7] Okonek, C., Spindler, H.: Reflexive Garben vom Rangr>2 auf \(\mathbb{P}\) n . Crelles J.344, 38–64 (1983) · Zbl 0511.14009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.