K\"ahler immersions of K\"ahler manifolds into complex space forms

TL;DR

This paper reviews Calabi's foundational work on Kähler immersions into complex space forms, discussing necessary conditions, classifications, and open problems in the field of Kähler geometry.

Contribution

It provides a comprehensive account of Calabi's original criteria, recent developments, and highlights open problems in classifying Kähler manifolds admitting such immersions.

Findings

Calabi's diastasis function as a key tool

Classification results for certain complex space forms

Open problems in full classification of Kähler immersions

Abstract

The study of K\"ahler immersions of a given real analytic K\"ahler manifold into a finite or infinite dimensional complex space form originates from the pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines a powerful tool, a special (local) potential called diastasis function, which allows him to obtain necessary and sufficient conditions for a neighbourhood of a point to be locally K\"ahler immersed into a finite or infinite dimensional complex space form. As application of its criterion, he also provides a classification of (finite dimensional) complex space forms admitting a K\"ahler immersion into another. Although, a complete classification of K\"ahler manifolds admitting a K\"ahler immersion into complex space forms is not known, not even when the K\"ahler manifolds involved are of great interest, e.g. when they are K\"ahler-Einstein or homogeneous spaces. In fact, the diastasis function is not always explicitely given and Calabi's criterion, although theoretically impeccable, most of the time is of difficult application. Nevertheless, throughout the last 60 years many mathematicians have worked on the subject and many interesting results have been obtained. The aim of this book is to describe Calabi's original work, to provide a detailed account of what is known today on the subject and to point out some open problems.