Kahler geometry on toric manifolds, and some other manifolds with large symmetry

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This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold
Section 3.3 has been changed to correct a mistake in the original version

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