Abstract
AbstractIn this chapter, we study canonical Kähler metrics in the class c 1(X) for Fano manifolds with no Kähler–Einstein metrics. Typical examples are: Kähler–Ricci solitons, Extremal Kähler metrics, Generalized Kähler–Einstein metrics. Since Kähler-Ricci solitons and extremal Kähler metrics are well-known, we focus on the recent developments of the studies of generalized Kähler–Einstein metrics. In Sects. 9.1 and 9.2, we discuss basic facts concerning holomorphic vector fields (including the extremal vector field) on Fano manifolds. In Sects. 9.3 and 9.4, we discuss obstructions to the existence of generalized Kähler–Einstein metrics. In Sects. 9.5, 9.7, 9.8, and 9.9, we discuss the existence problem of generalized Kähler–Einstein metrics including both Yao’s result and Hisamoto’s result. In Sect. 9.6, we shall show that a generalized Kähler–Einstein manifold always admits an extremal Kähler metric in the class c 1(X). KeywordsGeneralized Kähler–Einstein metricsExtremal vector fieldsYao’s resultHisamoto’s result
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