Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature


We present a method for proving the existence of Kähler-Einstein metrics of positive scalar curvature on certain compact complex manifolds, and use the method to produce a large class of examples of compact Kähler-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 Kähler-Einstein metric on $M$ is equivalent to the existence of a solution to a certain complex Monge-Ampère equation on $M$. To solve this complex Monge-Ampère 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 Kähler-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 $\mathscr{J}$ on $M$, called the multiplier ideal sheaf, which carefully measures the extent to which the estimate fails. The sheaf $\mathscr{J}$ 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 $\overline{\partial}$-Neumann problem. Now $\mathscr{J}$ is a global algebro-geometric object on $M$, and it so happens that $\mathcal{J}$ satisfies a number of highly nontrivial global algebro-geometric conditions, including a cohomology vanishing theorem. In particular, the complex analytic subspace $V \subset M$ cut out by $\mathscr{J}$ 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 restrictions on $M$.


Alan Michael Nadel