Stability of line bundles and vector bundles on some surfaces

Donaldson and Uhlenbeck-Yau established the classical result that on a compact Kähler manifold, an irreducible holomorphic vector bundle admits a Hermitian metric solving the Hermitian-Yang-Mills equation if and only if the vector bundle is Mumford-Takemoto stable. A modern analog of this question was posted by Collins-Yau. In this talk, we will discuss partial answers to this modern analog for a set of line bundles and tangent/cotangent bundles on some surfaces. This is based on joint work/work in progress with Tristan Collins, Jason Lo, and Shing-Tung Yau.