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

TL;DR

This paper introduces a new class of K"ahler metrics near the zero section of a disk bundle over a compact K"ahler manifold, showing their exponential deviation from Poincaré-type metrics and their natural emergence in perturbation scenarios for constant scalar curvature K"ahler metrics.

Contribution

It defines a novel product model for generalizing Poincaré-type K"ahler metrics, connecting flows generated by holomorphic vector fields with metric perturbations.

Findings

Metrics deviate exponentially from Poincaré-type metrics

Metrics arise naturally in perturbations of cscK classes

Provides a new framework for understanding K"ahler metric deformations

Abstract

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.