On complete constant scalar curvature K\"ahler metrics with Poincar\'e-Mok-Yau asymptotic property

TL;DR

This paper constructs special examples of complete constant scalar curvature K"ahler metrics on non-compact manifolds with Poincaré-Mok-Yau asymptotic behavior, using Calabi's ansatz and moment construction methods.

Contribution

It introduces new explicit examples of complete CSC K"ahler metrics with Poincaré-Mok-Yau asymptotics on non-compact manifolds.

Findings

Constructed explicit examples of complete CSC K"ahler metrics.

Demonstrated the use of Calabi's ansatz and moment construction in this context.

Provided insights into the geometry near subvarieties with higher codimension.

Abstract

Let $X$ be a compact K\"ahler manifold and $S$ a subvariety of $X$ with higher co-dimension. The aim is to study complete constant scalar curvature K\"ahler metrics on non-compact K\"ahler manifold $X-S$ with Poincar\'e--Mok--Yau asymptotic property (see Definition \ref{def}). In this paper, the methods of Calabi's ansatz and the moment construction are used to provide some special examples of such metrics.