The uniqueness of Poincar\'e type constant scalar curvature K\"ahler metric

TL;DR

This paper establishes the uniqueness of Poincaré type constant scalar curvature Kähler metrics with certain singularities on a Kähler manifold, linking it to asymptotic behavior and fixed point problems.

Contribution

It proves uniqueness under specific conditions and proposes a conjecture relating asymptotic behavior to uniqueness, reducing the problem to a fixed point formulation.

Findings

Uniqueness of Poincaré type cscK metrics with no holomorphic vector fields on D.

Affirmative answer to the conjecture for asymptotic behaviors invariant under automorphisms.

Reduction of the conjecture to a fixed point problem.

Abstract

Let $D$ be a smooth divisor on a closed K\"ahler manifold $X$. First, we prove that Poincar\'e type constant scalar curvature K\"ahler (cscK) metric with a singularity at $D$ is unique up to a holomorphic transformation on $X$ that preserves $D$, if there are no nontrivial holomorphic vector fields on $D$. For the general case, we propose a conjecture relating the uniqueness of Poincar\'e type cscK metric to its asymptotic behavior near $D$. We give an affirmative answer to this conjecture for those Poincar\'e type cscK metrics whose asymptotic behavior is invariant under any holomorphic transformation of $X$ that preserve $D$. We also show that this conjecture can be reduced to a fixed point problem.