A remark on stability and restrictions of vector bundles to hypersurfaces

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.

Hermitian–Yang–Mills connections on some complete non-compact Kähler manifolds

“…Proof Suppose there is a reflexive subsheaf S of E with 0 < rank(S) < rank(E) such that μ(S, c 1 (D)) ≥ c 1 (E, c 1 (D)), we need to show that μ(S, [ω 0 ]) < μ(E, [ω 0 ]). By [18] for any coherent refelxive sheaf E on X , we have…”

“…On the stability condition. Note that global semistability is known [18], if we assume the restriction to D is semistable. There do exist irreducible holomorphic vector bundles which are polystable when restricted to D but not globally stable, even under more restrictive assumptions that X is Fano and D ∈ |K −1 X |.…”