On coupled K\"ahler-Einstein metrics and weighted solitons on Fano manifolds

TL;DR

This survey discusses recent progress on coupled K"ahler-Einstein metrics and weighted solitons on Fano manifolds, highlighting their existence criteria linked to algebraic stability conditions.

Contribution

It summarizes recent developments in the theory of coupled K"ahler-Einstein metrics and weighted solitons, emphasizing their algebraic stability characterizations.

Findings

Existence of these metrics is equivalent to generalized K-polystability.

Recent advances connect geometric existence results with algebraic stability conditions.

The survey outlines key recent results and open problems in the field.

Abstract

We consider coupled K\"ahler-Einstein metrics and weighted solitons on Fano manifolds. These are natural generalizations of K\"ahler-Einstein metrics. As in the case of K\"ahler-Einstein metrics, the existence is known to be equivalent to algebraic conditions which generalize the K-polystability. In this survey, we outline recent developments for these two cases.