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

Abstract

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 KX⊗ [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization KX⊗ [D] coincides with the degree with respect to the complete Kähler–Einstein metric gX\Don X\D. For stable holomorphic vector bundles, we prove the existence of a Hermitian–Einstein metric with respect to gX\Dand also the uniqueness in an adapted sense.