Stable Vector Bundles on Algebraic Surfaces

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.