These notes present very recent results on compact K?hler-Einstein manifolds of positive scalar curvature. A central role is played here by a Lie algebra character of the complex Lie algebra consisting of all holomorphic vector fields, which can be intrinsically defined on any compact complex manifold and becomes an obstruction to the existence of a K?hler-Einstein metric. Recent results concerning this character are collected here, dealing with its origin, generalizations, sufficiency for the existence of a K?hler-Einstein metric and lifting to a group character. Other related topics such as extremal K?hler metrics studied by Calabi and others and the existence results of Tian and Yau are also reviewed. As the rudiments of K?hlerian geometry and Chern-Simons theory are presented in full detail, these notes are accessible to graduate students as well as to specialists of the subject.

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