Kähler-Einstein manifolds of positive scalar curvature (Q2771291)

A Riemannian metric on a manifold \(M\) is called Kähler-Einstein if it is Kähler and if its Ricci curvature form is proportional by a real constant to the Kähler form. In particular, such metrics are Einstein. As a further consequence, the first Chern class \(c_1(M)\), if non-zero, is necessarily definite. Conversely, a vanishing or negative definite first Chern class implies the existence of Kähler-Einstein metrics on \(M\). For a positive definite \(c_1(M)\), however, the existence of such a metric is not guaranteed. In this survey paper, the author discusses some basic results in the theory of compact Kähler-Einstein manifolds with positive \(c_1(M)\), centered around this existence problem. NEWLINENEWLINENEWLINEA first holomorphic obstruction to the existence is given by the fact that the Lie algebra \(\eta(M)\) of holomorphic vector fields on \(M\) must be reductive. For complex surfaces with \(c_1(M)>0\), this requirement is already sufficient for a Kähler-Einstein metric to exist; not so, however, for higher-dimensional spaces. A second obstruction is given by the vanishing of the Futaki invariant, which is a complex-valued function on \(\eta(M)\). NEWLINENEWLINENEWLINENext, a complex Monge-Ampere equation is derived, the solutions of which correspond to Kähler-Einstein metrics on the manifold. The solvability of this equation then leads to an analytic criterion for the existence of Kähler-Einstein metrics in terms of a certain functional on a particular set of smooth functions. In the final section, the author relates the existence of Kähler-Einstein metrics to the CM-stability of the underlying manifold.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00021].