Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T06:26:41.058Z Has data issue: false hasContentIssue false
  • English
  • Français

A theorem of Matsushima

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura
Hiroshi Umemura*
Affiliation:
Nagoya University
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.

In [7], Matsushima studied the vector bundles over a complex torus. One of his main theorems is: A vector bundle over a complex torus has a connection if and only if it is homogeneous (Theorem (2.3)). The aim of this paper is to prove the characteristic p > 0 version of this theorem. However in the characteristic p > 0 case, for any vector bundle E over a scheme defined over a field k with char, k = p, the pull back F*E of E by the Frobenius endomorphism F has a connection. Hence we have to replace the connection by the stratification (cf. (2.1.1)). Our theorem states: Let A be an abelian variety whose p-rank is equal to the dimension of A. Then a vector bundle over A has a stratification if and only if it is homogeneous (Theorem (2.5)).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 54 , July 1974 , pp. 123 - 134
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1974

References

[1] Atiyah, M. F., On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France, 84 (1956), 307–317.Google Scholar
[2] Atiyah, M. F., Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. (3), 7 (1957), 414–452.Google Scholar
[3] Demazure, M. et Grothendieck, A., Schemas en Groupes (S. G. A. D.), I, II, III, Vol. 151, 152, 153, Lecture Notes in Math., Springer.Google Scholar
[4] Grothendieck, A., Technique de construction et théorèmes d’existence en géométrie algébrique I. Généralités. Descente par morphismes fidèlement plats, Séminaire Bourbaki, 1.12 (1959/60), n° 90.Google Scholar
[5] Grothendieck, A., Technique de construction et théorèmes d’existence en géométrie algébrique IV. Les schémas de Hilbert, Séminaire Bourbaki, 1.13 (1960/61), n°221.Google Scholar
[6] Grothendieck, A., Crystals and the De Rham cohomology of schemes, Dix exposés sur la cohomologie des schémas, 1968 North-Holland Pub. company.Google Scholar
[7] Matsushima, Y., Fibres holomorphes sur un tore complexe, Nagoya Math. J., vol. 14 (1959), 1–24.Google Scholar
[8] Miyanishi, M., Some remarks on algebraic homogeneous vector bundles, Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Akizuki, Y., Kinokuniya, Tokyo (1973), 71–93.Google Scholar
[9] Morimoto, A., Sur le groupe d’automorphismes d’un espace fibre principal analytique complexe, Nogaya Math. J., vol. 13 (1958), 157–168.Google Scholar
[10] Umemura, H., Some results in the theory of vector bundles, Nagoya Math. J., vol. 52 (1973), 97–128.Google Scholar