Abstract
We prove that a vector bundle on a smooth projective variety is (semi)stable if the restriction on a fixed ample smooth subvariety is (semi)stable.
Highlights
The purpose of this work is to show a property of slope-stability of vector bundles with respect to restriction to a given ample subvariety
Given a slope-stable vector bundle E on a projective variety X, it is rather difficult to prove that the restriction of E to an ample subvariety is stable
This can be done for general subvarieties of sufficiently high degree
Summary
The purpose of this work is to show a property of slope-stability of vector bundles with respect to restriction to a given ample subvariety. Given a slope-stable vector bundle E on a projective variety X, it is rather difficult to prove that the restriction of E to an ample subvariety is stable This can be done for general subvarieties of sufficiently high degree (cf [4, 6]). The purpose of this work is to show that if the restriction of EjY to one given smooth ample hypersurface Y & X (of any degree) is (semi)stable, the vector bundle E is (semi)stable on X This result only appears in the literature as its generic version, which is almost elementary to prove, so we think it is useful to write it here, as we need it as a reference for future use, and for the interest of the result in itself. We notice that, contrarily to its generic version, the result is not granted on slightly broader hypotheses, for example if we consider stability of a
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