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 bundle $N$ over a compact Kähler manifold $X$ by incorporating flows generated by holomorphic vector fields on $X$. These metrics are then shown to deviate exponentially from Poincaré-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 Poincaré-type Kähler metrics of constant scalar curvature to obtain nearby cscK metrics even when the perturbed class on $X$ does not admit a cscK metric.

14 pages LaTeX; typos corrected; improvement of statement and proof of Lemma 3.3; version accepted for publication in Proc AMS