Topic
Ideal sheaf
About: Ideal sheaf is a research topic. Over the lifetime, 390 publications have been published within this topic receiving 6671 citations.
Papers published on a yearly basis
Papers
More filters
TL;DR: In this article, the existence of Kiihler-Einstein metrics of positive scalar curvature on certain compact complex manifolds was shown to be possible by using the multiplier ideal sheaf.
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 ideal 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
429 citations
367 citations
01 Jan 1989
TL;DR: In this article, the existence of Kiihler-Einstein metrics of positive scalar curvature on certain compact complex manifolds was shown to be possible by using the multiplier ideal sheaf.
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 ideal 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
344 citations
[...]
TL;DR: In this article, the authors define the notion of complete topological algebras as a topological algebra which is complete if and only if, the spectrum of the algebra considered is complete.
Abstract: The“Gel’fand sheaf” of a topological algebra is endowed with auniform structure, this being complete if and only if, the spectrum of the algebra considered is complete. Examples are also provided.
330 citations
TL;DR: In this paper, the moduli problem of pairs consisting of a rank 2 vector bundle and a nonzero section over a fixed smooth curve was studied, and it was shown that moduli space undergoes a sequence of flips in the sense of Mori.
Abstract: We study the moduli problem of pairs consisting of a rank 2 vector bundle and a nonzero section over a fixed smooth curve. The stability condition involves a parameter; as it varies, we show that the moduli space undergoes a sequence of flips in the sense of Mori. As applications, we prove several results about moduli spaces of rank 2 bundles, including the Harder-Narasimhan formula and the SU(2) Verlinde formula. Indeed, we prove a general result on the space of sections of powers of the ideal sheaf of a curve in projective space, which includes the Verlinde formula.
295 citations