Hermitian-Einstein metrics on parabolic stable bundles

Abstract

Let M̅ be a compact complex manifold of complex dimension two with a smooth Kähler metric and D a smooth divisor on M̅. If E is a rank 2 holomorphic vector bundle on M̅ with a stable parabolic structure along D, we prove the existence of a metric on E'=E|_M̅\D (compatible with the parabolic structure) which is Hermitian-Einstein with respect to the restriction of the Kähler metric to M̅D. A converse is also proved.