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Stable Vector Bundles on Algebraic Surfaces

Published online by Cambridge University Press:  22 January 2016

Fumio Takemoto
Fumio Takemoto*
Affiliation:
Department of Mathematics, Nagoya Institute for Technology
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Abstract

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Let k be an algebraically closed field, and X a nonsingular irreducible protective algebraic variety over k. These assumptions will remain fixed throughout this paper. We will consider a family of vector bundles on X of fixed rank r and fixed Chern classes (modulo numerical equivalence). Under what condition is this family a bounded family? When X is a curve, Atiyah [1] showed that it is so if all elements of this family are indecomposable. But when I is a surface, he showed also that this condition is not sufficient. We give the definition of an H-stable vector bundle on a variety X. This definition is a generalization of Mumford’s definition on a curve. Under the condition that all elements of a family are H-stable of rank two on a surface X, we prove that the family is bounded. And we study H-stable bundles, when X is an abelian surface, the protective plane or a geometrically ruled surface.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 47 , September 1972, pp. 29 - 48
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1972

References

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