Motivation: If we extend Hitchin-Kobayashi correspondence along holomorphic fibre space such that each vector bundle $E_s$ of fibers $X_s$ be stable then for finding canonical twisted Hermitian-Einstein metric (or relative Hermitian-Einstein metric) we need to work on singular Hermitian metric instead smooth hermitian metric on vector bundles. To run the relative version of Donaldson flow we need to choose a correct initial singular hermitian metric and existence of such singular hermitian initial metric on some vector bundle correspond to my question and so the following question is natural. So finding such positive inital hermitan metric correspond to positivity theory and also note that we can define a singular hermitan metric on positive line bunde due to Demailly, but note that, we can not define singular hermitan metric on vector bundle in general and for some cases it is well defined. There are some examples like Parabolic stable vector bunlde on orbifolds with singular Hermitian-Yang Mills connection
Let $\mathcal E\to X$ be a stable vector bundle over a polarized Kahler manifold $(X,\omega)$. From Donaldson-Uhlenbeck-Yau theorem, $\mathcal E$ admits Hermitian-Einstein metric, i.e., a smooth hrmitian metric $h$, such that
$$ {\rm tr}_\omega F(\mathcal E, h)=\lambda \rm Id, $$ where $F(\mathcal E, h)\in \Lambda^{1,1}(T^*M)\otimes \rm End(\mathcal E)$ is the curvature of Chern connection on $(\mathcal E, h)$
So my question is
Is there a reference for Existence of singular Hermitian-Einstein metric on non-smooth projective variety(which admit mild singularites in the sense of MMP), ?
If a vector bunlde admit a singular Hermitian-Einstein metric on projective variety $X$, then $X$ which type of singularity can have?