This chapter is a second trip into the realm of Kähler geometry, focusing on Kähler-Einstein metrics, in particular positive scalar curvature Kähler-Einstein metrics on compact Fano orbifolds, which gives rise to the famous Monge-Amfipere equation. Some basic techniques such as the continuity method, Tian's invariant, and multipliers ideal sheaves are introduced. These provide for proving various existence results concerning orbifold Kähler-Einstein metrics. The Matsushima-Lichnerowicz theorem and Futaki invariant are briefly discussed in the section on obstructions.
Keywords: orbifold canonical bundle, Fano orbifolds, Kähler-Einstein metrics, Futaki invariant, Matsushima-Lichnerowicz theorem, multiplier ideal sheaf, continuity method, Monge-Ampère equations, Tian's invariant
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