# Colloquium abstracts

**Matthias Stemmler**

TIFR

February 10, 2011

**Stability and Hermitian-Einstein metrics for vector bundles on framed manifolds**: We adapt the notions of stability in the sense of Mumford-Takemoto and Hermitian-Einstein metrics for holomorphic vector bundles on canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that the canonical divisor of X plus D is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to this polarization coincides with the degree with respect to the complete Kaehler-Einstein metric g on the complement of D in X. For stable holomorphic vector bundles, we prove the existence of a Hermitian-Einstein metric with respect to g and also the uniqueness in an adapted sense.