An Obstruction to the Existence of Einstein Kähler Metrics.

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

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

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

/pdf/uniqueness-of-einstein-kahler-metrics-modulo-connected-group-2hbr3xzf5h.pdf

Uniqueness of Einstein Kähler Metrics Modulo Connected Group Actions

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On donne des exemples de varietes de Kahler-Einstein compactes non homogenes. En particulier on considere des varietes de Kahler-Einstein compactes presque homogenes a ensemble exceptionnel disconnexe

https://projecteuclid.org/download/pdf_1/euclid.ojm/1200781175

Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds. II

This paper deals with compact complex solvmanifolds. Our main purpose is to generalize the theory on the divisor group of a complex torus to these manifolds. By a solvmanifold we mean a homogeneous space of solvable Lie group. Let G be a simply connected complex solvable Lie group and Γ be a lattice of G, that is, a discrete subgroup of G such that G/Γ is compact. The de Rham cohomology group and the Dolbeault cohomology group of a compact complex manifold G/Γ play an important role in studying the divisor group of a complex manifold G/Γ. The de Rham cohomology group of a compact solvmanifold G/Γ has been discussed by Matsushima [7], Nomizu [10] and Mostow [8]. Let M be a compact connected complex manifold and H% ,?(M) denote the Dolbeault cohomology group of M of type (p, q). Let g be a complex Lie algebra and / be the canonical complex structure of g. Then g=gθg~, where g±= \\X<ΞQ\\IX= itV^ΪX}. In section 2, we prove:

/pdf/on-compact-complex-parallelisable-solvmanifolds-2l9gourc2k.pdf

On compact complex parallelisable solvmanifolds

On donne des exemples de varietes d'Einstein-Kahler compactes avec premiere classe de Chein positive qui ne sont pas homogenes. On donne une condition necessaire et suffisante pour l'existence de metriques d'Einstein-Kahler sur des P 1 (C)-fibres sur des espaces symetriques hermitiens de type compact

/pdf/examples-of-compact-einstein-kahler-manifolds-with-positive-150hmfp13m.pdf

Examples of compact Einstein Kähler manifolds with positive Ricci tensor

Homogeneous Einstein metrics on flag manifolds.

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by D’Atri and Ziller (Mem Am Math Soc 18, (215) 1979). In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for n ≥ 6, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) (n ≥ 11), Sp(n) (n ≥ 3), E6, E7, and E8.

Yusuke Sakane

Papers

Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds. II

On compact complex parallelisable solvmanifolds

Examples of compact Einstein Kähler manifolds with positive Ricci tensor

Homogeneous Einstein metrics on flag manifolds.

Einstein metrics on compact Lie groups which are not naturally reductive