Asymptotics of complete Kahler metrics of finite volume on quasiprojective manifolds

TL;DR

This paper analyzes the asymptotic behavior of complete Kahler metrics of finite volume on quasiprojective manifolds, especially under Ricci flow and in relation to Kahler-Einstein metrics, revealing detailed boundary behaviors.

Contribution

It provides a detailed description of the asymptotics of Kahler metrics on quasiprojective manifolds and their evolution under Ricci flow, including boundary regularity and polyhomogeneity of limits.

Findings

Asymptotic behavior described at infinity for Kahler metrics on quasiprojective manifolds.

Persistence of asymptotics under Ricci flow with smooth potential functions.

Different boundary behavior for Kahler-Einstein metrics with logarithmic terms.

Abstract

Let X be a quasiprojective manifold given by the complement of a divisor $\bD$ with normal crossings in a smooth projective manifold $\bX$. Using a natural compactification of $X$ by a manifold with corners $\tX$, we describe the full asymptotic behavior at infinity of certain complete Kahler metrics of finite volume on X. When these metrics evolve according to the Ricci flow, we prove that such asymptotic behaviors persist at later time by showing the associated potential function is smooth up to the boundary on the compactification $\tX$. However, when the divisor $\bD$ is smooth with $K_{\bX}+[\bD]>0$ and the Ricci flow converges to a Kahler-Einstein metric, we show that this Kahler-Einstein metric has a rather different asymptotic behavior at infinity, since its associated potential function is polyhomogeneous with in general some logarithmic terms occurring in its expansion at the boundary.