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 MumfordTakemoto 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 ⊗ [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization K X ⊗ [D] coincides with the degree with respect to the complete Kähler-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.

Generalized Deligne-Hitchin Twistor Spaces: Construction and Properties

“…Since Stokes-type theorem is still valid for complete Riemannian manifolds [17], by Theorem 5.3, for a holomorphic vector bundle E over TW(X, X ′ ; r), one defines the degree (cf. [31] and [45,Lemma 3.3…”