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

Alan Michael Nadel
Annals of Mathematics
Second Series, Vol. 132, No. 3 (Nov., 1990), pp. 549-596
Published by: Annals of Mathematics
DOI: 10.2307/1971429
Stable URL: http://www.jstor.org/stable/1971429
Page Count: 48
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Multiplier Ideal Sheaves and Kahler-Einstein Metrics of Positive Scalar Curvature
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Abstract

We present a method for proving the existence of Kahler-Einstein metrics of positive scalar curvature on certain compact complex manifolds, and use the method to produce a large class of examples of compact Kahler-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 Kahler-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 Monge-Ampere 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 Kahler-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 \mathcal{J} on M, called the multiplier ideal sheaf, which carefully measures the extent to which the estimate fails. The sheaf 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 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 \subset M cut out by \mathcal{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.

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