Gauduchon metrics and Hermite-Einstein metrics on non-K\"ahler varieties

We show the existence of Gauduchon metrics on arbitrary compact hermitian varieties, generalizing our previous work on smoothable singularities. These metrics allow us to define the notion of slope stability for torsion-free coherent sheaves on compact normal varieties that are not necessarily K"ahler. Then we prove the existence and uniqueness of singular Hermite-Einstein metrics for slope-stable reflexive sheaves on non-K"ahler normal varieties.