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A theorem of Matsushima

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura
Hiroshi Umemura*
Affiliation:
Nagoya University
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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

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