On the Kobayashi-Hitchin correspondence for K\"{a}hler currents

In this paper, we show that if a holomorphic vector bundle is slope polystable with respect to a K\"{a}hler class, then it admits a Hermitian-Yang-Mills metric with respect to a suitable K\"{a}hler current with singularities in higher codimension which represents the K\"{a}hler class. Most parts of the proof remains valid for closed positive $(1,1)$-currents representing a nef and big class.