A Product Model for Generalizing Poincar\'e-Type K\"ahler Metrics

We begin by defining a type of K\"ahler metric near the zero section of a trivial holomorphic open disk bundle $N$ over a compact K\"ahler manifold $X$ by incorporating flows generated by holomorphic vector fields on $X$. These metrics are then shown to deviate exponentially from Poincar\'e-type metrics on $N\setminus X$ in terms of the log-polar distance from $X$ in $N$. Lastly we see that they arise naturally when perturbing classes containing Poincar\'e-type K\"ahler metrics of constant scalar curvature to obtain nearby cscK metrics even when the perturbed class on $X$ does not admit a cscK metric.