Kobayashi-Hitchin correspondence for twisted vector bundles

We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact Kahler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g-polystable if and only if it is g-Hermite-Einstein, and if X is a compact Kahler manifold and g is a Kahler metric on X, then a twisted holomorphic vector bundle on X is g-semistable if and only if it is approximate g-Hermite-Einstein.