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

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.