$\mathbb{T}^n$-invariant Kaehler-Einstein manifolds immersed in complex projective spaces

TL;DR

This paper classifies all non-isometric $ abla$-invariant Kaehler-Einstein manifolds for dimensions up to 6 that can be immersed in complex projective spaces with the Fubini-Study metric, solving a longstanding classical problem.

Contribution

It provides a complete classification for $n \,\leq\, 6$, addressing a classical problem by Calabi and Chern regarding such manifolds.

Findings

Complete list of $ abla$-invariant Kaehler-Einstein manifolds for $n \leq 6$

Identification of all such manifolds that can be immersed in complex projective spaces

Resolution of a classical problem in differential geometry

Abstract

We give a complete list, for $n \leq 6$, of non-isometric $\mathbb{T}^n$-invariant Kaehler-Einstein manifolds immersed in a finite dimensional complex projective space endowed with the Fubini-Study metric. This solves, in the aforementioned case, a classical and long-staying problem addressed among others by Calabi and Chern.