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Multiplier Ideal Sheaves and KahlerEinstein Metrics of Positive Scalar Curvature
Alan Michael Nadel
Annals of Mathematics
Second Series, Vol. 132, No. 3 (Nov., 1990), pp. 549596
Published by: Annals of Mathematics
DOI: 10.2307/1971429
Stable URL: http://www.jstor.org/stable/1971429
Page Count: 48
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Abstract
We present a method for proving the existence of KahlerEinstein metrics of positive scalar curvature on certain compact complex manifolds, and use the method to produce a large class of examples of compact KahlerEinstein manifolds of positive scalar curvature. Suppose that M is a compact complex manifold of positive first Chern class. As is wellknown, the existence of a KahlerEinstein metric on M is equivalent to the existence of a solution to a certain complex MongeAmpere 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 KahlerEinstein metric, so that the zeroth order a priori estimate fails to hold. From this lack of an estimate we extract various global algebrogeometric 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 algebrogeometric object on M, and it so happens that J satisfies a number of highly nontrivial global algebrogeometric 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 zerodimensional then it is a single reduced point, while if V is onedimensional then its support is a tree of smooth rational curves. The logarithmicgeometric genus of M  V always vanishes. These considerations place nontrivial global algebrogeometric restrictions on M.
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Annals of Mathematics © 1990 Annals of Mathematics