K\"ahler Immersions of ALE K\"ahler Manifolds into Complex Space Forms

TL;DR

This paper explores conditions under which ALE K"ahler manifolds can be immersed into complex space forms, revealing that positive mass prevents immersion into Euclidean space and any mass sign prevents immersion into hyperbolic space.

Contribution

It establishes a link between the mass of ALE K"ahler manifolds and their immersibility into complex space forms, providing new obstructions based on mass sign.

Findings

Positive mass ALE K"ahler manifolds cannot be immersed into complex Euclidean space.

ALE K"ahler manifolds with any mass sign cannot be immersed into complex hyperbolic space.

The paper relates K"ahler immersion problems to the mass of ALE metrics.

Abstract

In this article, we investigate the K\"ahler immersions of special Asymptotically Locally Euclidean (ALE) K\"ahler metrics into complex space forms. We provide a relation between K\"ahler immersions problem of these metrics and the sign of their mass. Our result shows that these metrics with positive mass do not admit a K\"ahler immersion into complex Euclidean space. Moreover, we also obtain that these metrics with any sign of mass cannot be K\"ahler immersed into complex hyperbolic space.