STABILITY AND HERMITIAN-EINSTEIN METRICS FOR VECTOR BUNDLES ON FRAMED MANIFOLDS

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that K-X circle times [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization K-X circle times [D] coincides with the degree with respect to the complete Kahler-Einstein metric g(X\D) on X\D. For stable holomorphic vector bundles, we prove the existence of a Hermitian-Einstein metric with respect to g(X\D) and also the uniqueness in an adapted sense.