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5 - Multiplier ideal sheaves and geometric problems

Published online by Cambridge University Press:  05 November 2011

By
Akito Futaki  and
Yuji Sano
Edited by
Roger Bielawski ,
Kevin Houston  and
Martin Speight
Akito Futaki
Affiliation:
Department of Mathematics, Tokyo Institute of Technology
Yuji Sano
Affiliation:
Department of Mathematics, Kyushu University
Roger Bielawski
Affiliation:
University of Leeds
Kevin Houston
Affiliation:
University of Leeds
Martin Speight
Affiliation:
University of Leeds
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Summary

Abstract

In this expository article we first give an overview on multiplier ideal sheaves and geometric problems in Kählerian and Sasakian geometries. Then we review our recent results on the relationship between the support of the subschemes cut out by multiplier ideal sheaves and the invariant whose non-vanishing obstructs the existence of Kähler-Einstein metrics on Fano manifolds.

Introduction

One of the main problems in Kählerian and Sasakian geometries is the existence problem of Einstein metrics. An obvious necessary condition for the existence of a Kähler-Einstein metric on a compact Kähler manifold M is that the first Chern class c1(M) is negative, zero or positive since the Ricci form represents the first Chern class. This existence problem in Kählerian geometry was settled by Aubin [1] and Yau [58] in the negative case and by Yau [58] in the zero case. In the remaining case when the manifold has positive first Chern class, in which case the manifold is called a Fano manifold in algebraic geometry, there are two known obstructions. One is due to Matsushima [29] which says that the Lie algebra ɧ(M) of all holomorphic vector fields on a compact Kähler-Einstein manifold M is reductive. The other one is due to the first author [16] which is given by a Lie algebra character F : ɧ(M) → ℂ with the property that if M admits a Kähler-Einstein metric then F vanishes identically. Besides, it has been conjectured by Yau [59] that a more subtle condition related to geometric invariant theory (GIT) should be equivalent to the existence of Kähler-Einstein metrics.

Type
Chapter
Information
Variational Problems in Differential Geometry , pp. 68 - 93
Publisher: Cambridge University Press
Print publication year: 2011

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