Manifolds with Suitable Finite Symmetry

Abstract

In Chapter 4 we discussed two obstructions to the existence of Kähler-Einstein metrics on compact Kähler manifolds with positive anticanonical line bundle. These two obstructions are related to the presence of nonzero holomorphic vector fields. There is a conjecture that any compact Kähler manifold of positive anticanonical line bundle without any nonzero holomorphic vector field admits a Kähler-Einstein metric. The conjecture is still open. The only known examples of Kähler-Einstein metrics of Kähler manifolds of positive anticanonical line bundle are those of Hermitian symmetric manifolds or homogeneous manifolds or certain noncompact manifolds [C5]. So far there is no known way of proving the existence of Kähler-Einstein metrics of compact Kahler manifolds of positive anticanonical line bundle by using the continuity method with reasonable additional assumptions such as the nonexistence of nonzero holomorphic vector fields. In this Chapter we discuss a method [Siul,Siu2] to prove the existence of Kähler-Einstein metrics for compact Kähler manifolds of positive anticanonical line bundle under the additional assumption of the existence of a suitable finite or compact group of symmetry. The method is not very satisfactory, because its applicability is exceedingly limited.