Kähler--Einstein metrics on quasi-projective manifolds

Let $X$ be a compact Kähler manifold and $D$ be a simple normal crossing divisor on $X$ such that $K_X+D$ is big and nef. We first prove that the singular Kähler–Einstein metric constructed by Berman–Guenancia is almost-complete on $X\mathrm{\backslash }D$ in the sense of Tian–Yau. In our second main result, we establish the weak convergence of conic Kähler–Einstein metrics of negative curvature to the above-mentioned metric when $K_X+D$ is merely big, answering partly a recent question posed by Biquard–Guenancia. Potentials of low energy play an important role in our approach.