A product model for generalizing Poincaré-type Kähler metrics

Abstract

We begin by defining a type of Kähler metric near the zero section of a trivial holomorphic open disk bundleNNover a compact Kähler manifoldXXby incorporating flows generated by holomorphic vector fields onXX. These metrics are then shown to deviate exponentially from Poincaré-type metrics onN∖XN\setminus Xin terms of the log-polar distance fromXXinNN. Lastly we see that they arise naturally when perturbing classes containing Poincaré-type Kähler metrics of constant scalar curvature to obtain nearby cscK metrics even when the perturbed class onXXdoes not admit a cscK metric.