Abstract
AbstractIn this chapter, we shall introduce various special metrics for compact complex manifolds such as Kähler–Einstein metrics, CSC Kähler metrics, extremal Kähler metrics, Kähler–Ricci solitons and generalized Kähler–Einstein metrics. In Sect. 3.1, we give definitions of these special metrics. Here CSC Kähler metrics and extremal Kähler metrics are defined by using the scalar curvature S ω, while Kähler–Einstein metrics, Kähler–Ricci solitons and generalized Kähler–Einstein metrics are defined by using the Ricci potential f ω. In Sect. 3.2, we shall show that Kähler–Ricci solitons and generalized Kähler–Einstein metrics are Kähler–Einstein analogues in Bakry–Emery geometry by conformal changes via Hamiltonian functions of holomorphic vector fields. KeywordsKähler–Einstein metricsCSC Kähler metricsExtremal Kähler metricsKähler–Ricci solitonsGeneralized Kähler–Einstein metrics
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