Let $V$ be a general type projective algebraic variety with only canonical singularities such that $K_{V}$ is ample. Let $h^{N}$ a smooth hermitian metric on $\omega^{N}_{V}$ such that $\Omega=c_{1}(K_{V},h)$ is a smooth Kähler form on $V$.

There is a unique $\varphi\in{\mathcal{C}}^{0}(V,\mathbb{R})$ such that:

1. (1)

   $\varphi$ is $\Omega$-psh.

2. (2)

   $\Omega+dd^{c}\varphi$ semi Kähler current with ${\mathcal{C}}^{0}$ potential.

3. (3)

   $(\Omega+dd^{c}\varphi)^{n}=e^{\varphi}v(h)$.

Consequently $\Omega+dd^{c}\varphi$ is the unique singular KE metric on $V$ of negative curvature in the canonical class of $V$. The current $\Omega+dd^{c}\varphi$ has continuous potentials and is smooth on $V^{reg}$ where it defines a bona fide KE metric.