Lower dimensional $S^1$-invariant K\"ahler-Einstein metrics via integrable structures

TL;DR

This paper investigates the classification of K"ahler-Einstein manifolds with rotational symmetries that can be immersed into complex projective space, using integrable structures to achieve a classification in this special case.

Contribution

It introduces a classification approach for K"ahler-Einstein metrics with rotational symmetries based on integrable distributions, addressing a classical open problem.

Findings

Classification of $S^1$-invariant K"ahler-Einstein metrics

Use of integrable structures for classification

Partial progress on the immersion problem

Abstract

We focus on the classical open problem of the classification of K\"ahler-Einstein manifolds that can be K\"ahler immersed into a complex projective space endowed with the Fubini-Study metric. In particular, we will deal with such problem in the special case of K\"ahler- Einstein metrics admitting symmetries of rotational type. This leads to certain integrable distributions allowing a classification of such metrics.