We consider coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds. These are natural generalizations of Kähler-Einstein metrics. As in the case of Kähler-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.
Akito Futaki (Thu,) studied this question.
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