On the uniqueness of Poincaré type cscK metrics

Abstract

Let $X$ be a compact Kähler manifold with a divisor $D$ which may have simple normal crossings. We prove that the $K$-energy is convex on the space of Poincaré type Kähler metrics. Then, we prove that the Poincaré type constant scalar curvature Kähler metric is unique up to an automorphism of $X$ preserving $D$.

We remove the additional assumptions in the previous version about the automorphism group of D and the smoothness of D