Existence Problem

Abstract

AbstractIn this chapter, we give some remarks on the existence of extremal Kähler metrics on polarized algebraic manifolds: For the existence of extremal Kähler metrics, as mentioned in Sect. 8.1, a result of He states the following: a polarized algebraic manifold (X, L) admits an extremal Kähler metric in the class c 1(L) if the modified K-energy is proper modulo the action of the centralizer G 0 of the extremal vector field in \(\operatorname {Aut}^0(X)\). Hence the existence of an extremal Kähler metric is reduced to showing the properness of the modified K-energy modulo the action of G 0. In Sect. 8.2, we give some observations on the existence of extremal metrics, which gives an idea of how strong K-stability is useful for the existence. KeywordsThe theorem of Chen, Donaldson and Sun and of TianThe results of Chen, Cheng and He