Let $X$ be a complex projective manifold such that $K_{X}$ is nef and big. Let $\Omega$ be a smooth Kähler metric on $X$ and consider the Kähler-Ricci flow

$$\frac{\partial\Omega_{t}}{\partial t}=-Ric(\Omega_{t})-\Omega_{t},\ \ \Omega_{0}=\Omega.$$

In [Ts], it was proved that this flow has a global solution for all time $t\in[0,\infty[$, and an argument was given, recently fully completed in [TZ], to the effect that $\Omega_{t}$ converges to a closed positive current $T_{KE}$, independent of $\Omega$, which defines a smooth Kähler-Einstein metric outside the exceptional divisor $E$ of the holomorphic bimeromorphic map $X\to X_{can}$. Its potential satisfies the Monge Ampère équation considered in Theorem 7.8 outside $E$. It follows from proposition 4.4 that the current $T_{KE}$ coincides with the solution produced by Theorem 7.8.

The notes [ST], [TZ] announce a proof of the following properties, already conjectured by [Ts], that $T_{KE}$ has locally bounded potential and satisfies the degenerate Monge-Ampère equation considered in Theorem 7.8. Our Theorem 7.8 in this case gives the precision that $T_{KE}$ has continuous potentials.