Kähler-Einstein Metrics and the Calabi Conjecture

Abstract

Among the known examples of Einstein manifolds, a good many are Kähler. In fact, all compact examples with zero or negative scalar curvature are either Kähler, or locally homogeneous. On a complex manifold, one often gets Kähler-Einstein metrics by specific techniques. One reason is perhaps, in the Kähler case, the relative autonomy of the Ricci tensor with regard to the metric, once the complex structure is given. The Ricci tensor—or, to be precise, the Ricci form—only depends on the volume form. On the other hand, on a compact manifold, the cohomology class of the Ricci form is determined by the complex structure (so that, in particular, the sign of an eventual Kähler-Einstein metric is itself determined by the complex structure). Due to these circumstances, it has been possible to exhibit some existence theorems of Einstein metrics in the Kähler framework (Calabi-Yau and Aubin-Calabi-Yau theorems) which have no counterpart in general Riemannian geometry.