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Torus-equivariant vector bundles on projective spaces

Published online by Cambridge University Press:  22 January 2016

Tamafumi Kaneyama*
Affiliation:
Department of Mathematics, Gifu College of Education, Yanaizu-cho Hashima-gun, Gifu 501-61, Japan
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For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Atiyah, M. F., On the Krull-Schmidt Theorem with applications to sheaves, Bull. Soc. Math. France, 84 (1956), 307317.CrossRefGoogle Scholar
[2] Bertin, J. etElencwajg, G., Symétries des fibrés vectorieles sur Pn et nombre d’Euler, Duke Math. J., 49 (1982), 807831.CrossRefGoogle Scholar
[3] Hartshorne, R., Stable vector bundles of rank 2 on P3 , Math. Ann., 238 (1978), 229280.Google Scholar
[4] Kaneyama, T., On equivariant vector bundles on an almost homogeneous variety, Nagoya Math. J., 57 (1975), 6586.Google Scholar
[5] Kempf, and Mumford, , Toroidal embeddings, Springer Lecture notes 339.Google Scholar
[6] Oda, T., Torus embeddings and applications, Tata Institute of Fundamental Research 58, (1978).Google Scholar
[7] Oda, T. and Miyake, K., Almost homogeneous algebraic varieties under torus action, Manifolds Tokyo 1973, 373381, Proceedings of International Conference on Manifolds and Related Topics in Topology.Google Scholar