Multiplier Ideal Sheaves and Kahler-Einstein Metrics of Positive Scalar Curvature

Abstract

We present a method for proving the existence of Kiihler-Einstein metrics of positive scalar curvature on certain compact complex manifolds, and use the method to produce a large class of examples of compact Kiihler-Einstein manifolds of positive scalar curvature. Suppose that M is a compact complex manifold of positive first Chern class. As is well-known, the existence of a Kiihler-Einstein metric on M is equivalent to the existence of a solution to a certain complex Monge-Ampere equation on M. To solve this complex MongeAmpere equation by the method of continuity, one needs only to establish the appropriate zeroth order a priori estimate. Suppose now that M does not admit a Kiihler-Einstein metric, so that the zeroth order a priori estimate fails to hold. From this lack of an estimate we extract various global algebro-geometric properties of M by introducing a coherent sheaf of ideals >J on M, called the multiplier sheaf, which carefully measures the extent to which the estimate fails. The sheaf >Y is analogous to the subelliptic multiplier ideal sheaf that J. J. Kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d-Neumann problem. Now >J is a global algebro-geometric object on M, and it so happens that >J satisfies a number of highly nontrivial global algebro-geometric conditions, including a cohomology vanishing theorem. In particular, the complex analytic subspace V c M cut out by >J is nonempty, connected, and has arithmetic genus zero. If V is zero-dimensional then it is a single reduced point, while if V is one-dimensional then its support is a tree of smooth rational curves. The logarithmic-geometric genus of M - V always vanishes. These considerations place nontrivial global algebro-geometric restric