scholarly journals Remarks on Kähler-Einstein Manifolds

1972 ◽  
Vol 46 ◽  
pp. 161-173 ◽  
Author(s):  
Yozo Matsushima
Keyword(s):  
Einstein Metrics ◽  
Integral Formula ◽  
Curvature Form ◽  
Einstein Metric ◽  
Kähler Metric ◽  
Kähler Einstein Metrics ◽  
Simply Connected ◽  
Kahler Metric ◽  
Complex Projective ◽  
Ricci Form

The main purpose of this note is to characterize a compact Káhler-Einstein manifold in terms of curvature form. The curvature form Q is an EndT valued differential form of type (1,1) which represents the curvature class of the manifold. We shall prove that the curvature form of a Káhler metric is the harmonic representative of the curvature class if and only if the Káhler metric is an Einstein metric in the generalized sense (g.s.), that is, if the Ricci form of the metric is parallel. It is well known that a Káhler metric is an Einstein metric in the g. s. if and only if it is locally product (globally, if the manifold is simply connected and complete) of Kàhler-Einstein metrics. We obtain an integral formula, involving the integral of the trace of some operators defined by the curvature tensor, which measures the deviation of a Káhler-Einstein metric from a Hermitian symmetric metric. In the final section we shall prove the uniqueness up to equivalence of Kãhler-Einstein metrics in a simply connected compact complex homogeneous space. This result was proved by Berger in the case of a complex projective space and our proof is completely different from Berger’s.

scholarly journals Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Junyan Cao ◽  
Henri Guenancia ◽  
Mihai Păun
Keyword(s):  
General Type ◽  
Einstein Metrics ◽  
Kodaira Dimension ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
Fiber Space ◽  
Generic Fiber ◽  
Lelong Numbers ◽  
Kähler Einstein Metrics

Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

Hermitian–Einstein metrics and jumping lines forSp(n+1)-homogeneous bundles over ℙ2n+1

1993 ◽  
Vol 114 (3) ◽  
pp. 443-451
Author(s):  
Al Vitter
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Vector Bundles ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Holomorphic Vector Bundles ◽  
Kähler Form ◽  
Points Of View ◽  
The Stability ◽  
Complex Projective

Stable holomorphic vector bundles over complex projective space ℙnhave been studied from both the differential-geometric and the algebraic-geometric points of view.On the differential-geometric side, the stability ofE-→ ℙncan be characterized by the existence of a unique hermitian–Einstein metric onE, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn(see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙnand the nullcorrelation bundle (see [9] and [10]).

Conformally Kähler manifolds

1954 ◽  
Vol 50 (1) ◽  
pp. 16-19 ◽  
Author(s):  
W. J. Westlake
Keyword(s):  
Conformal Geometry ◽  
Kähler Manifold ◽  
Hermitian Manifold ◽  
Necessary And Sufficient Condition ◽  
Kähler Metric ◽  
Kahler Manifold ◽  
Hermitian Space ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Kahler Metric

Introduction. The present paper is concerned with the conformal geometry of Hermitian spaces. In the first part we find a necessary and sufficient condition for a Hermitian space to be conformally Kähler, that is, conformal to some Kähler space. The condition is that a certain conformal tensor, , vanishes identically. Then, defining a Hermitian manifold as in Hodge (3), we consider such a manifold where the restriction is made that at every point the tensor is zero. This will be called a conformally Kähler manifold, and conditions under which it may be given a Kähler metric are obtained. It is found that any conformally Kähler manifold may be given a Kähler metric provided it is simply-connected or that its fundamental group is of finite order.

scholarly journals On the Kähler–Ricci flow near a Kähler–Einstein metric

2015 ◽  
Vol 2015 (699) ◽  
Author(s):  
Song Sun ◽  
Yuanqi Wang
Keyword(s):  
Critical Point ◽  
Ricci Flow ◽  
Complex Structure ◽  
Type Inequality ◽  
Gradient Flow ◽  
Einstein Metric ◽  
Fano Manifold ◽  
Kähler Metric ◽  
Canonical Class ◽  
Kahler Metric

AbstractOn a Fano manifold, we prove that the Kähler–Ricci flow starting from a Kähler metric in the anti-canonical class which is sufficiently close to a Kähler–Einstein metric must converge in a polynomial rate to a Kähler–Einstein metric. The convergence cannot happen in general if we study the flow on the level of Kähler potentials. Instead we exploit the interpretation of the Ricci flow as the gradient flow of Perelman's μ functional. This involves modifying the Ricci flow by a canonical family of gauges. In particular, the complex structure of the limit could be different in general. The main technical ingredient is a Lojasiewicz type inequality for Perelman's μ functional near a critical point.

scholarly journals Dynamical construction of Kähler-Einstein metrics

2010 ◽  
Vol 199 ◽  
pp. 107-122
Author(s):  
Hajime Tsuji
Keyword(s):  
Einstein Metrics ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
General Fiber ◽  
New Construction ◽  
Kähler Einstein Metrics ◽  
Singular Hermitian Metric ◽  
Projective Morphism

AbstractIn this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphismf:X→Swith connected fibers such that a general fiber has an ample canonical bundle, and for a positive integerm, we construct a canonical singular Hermitian metrichE,monwith semipositive curvature in the sense of Nakano.

scholarly journals Remarks on extremal Kähler metrics on ruled manifolds

1992 ◽  
Vol 126 ◽  
pp. 89-101 ◽  
Author(s):  
Akira Fujiki
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Einstein Metric ◽  
Kähler Metrics ◽  
Algebraic Subgroup ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kahler Metric ◽  
Compact Kähler Manifold

Let X be a compact Kähler manifold and γ Kähler class. For a Kàhler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kähler metrics g whose Kähler forms belong to γ, where dvg is the volume form associated to g. Such a Kähler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of AutoX (cf. [5]). Note also that any Kähler-Einstein metric is of constant scalar curvature.

scholarly journals KÄHLER–EINSTEIN METRICS WITH CONE SINGULARITIES ON KLT PAIRS

2013 ◽  
Vol 24 (05) ◽  
pp. 1350035 ◽  
Author(s):  
HENRI GUENANCIA
Keyword(s):  
Einstein Metrics ◽  
Einstein Metric ◽  
Technical Assumption ◽  
Kähler Einstein Metrics ◽  
Smooth Locus

We prove that any Kähler–Einstein metric attached to a klt pair (X, D) has cone singularities along D on the log-smooth locus of the pair, under some technical assumption on the cone angles.

Ricci curvature in Kähler geometry

2017 ◽  
Vol 28 (09) ◽  
pp. 1740009
Author(s):  
Song Sun
Keyword(s):  
Ricci Curvature ◽  
Einstein Metrics ◽  
Complex Analysis ◽  
Einstein Metric ◽  
Tangent Cones ◽  
Kahler Geometry ◽  
Fano Manifolds ◽  
Winter School ◽  
Kähler Einstein Metrics ◽  
Hausdorff Limits

These are the notes for lectures given at the Sanya winter school in complex analysis and geometry in January 2016. In Sec. 1, we review the meaning of Ricci curvature of Kähler metrics and introduce the problem of finding Kähler–Einstein metrics. In Sec. 2, we describe the formal picture that leads to the notion of K-stability of Fano manifolds, which is an algebro-geometric criterion for the existence of a Kähler–Einstein metric, by the recent result of Chen–Donaldson–Sun. In Sec. 3, we discuss algebraic structure on Gromov–Hausdorff limits, which is a key ingredient in the proof of the Kähler–Einstein result. In Sec. 4, we give a brief survey of the more recent work on tangent cones of singular Kähler–Einstein metrics arising from Gromov–Hausdorff limits, and the connections with algebraic geometry.

scholarly journals Quaternionic Geometry in Dimension 8

2018 ◽  
pp. 91-114
Author(s):  
Diego Conti ◽  
Thomas Bruun Madsen ◽  
Simon Salamon
Keyword(s):  
Cohomogeneity One ◽  
Kähler Metric ◽  
Quaternionic Geometry ◽  
Simply Connected ◽  
Kahler Metric

This chapter describes the 8-dimensional Wolf spaces as cohomogeneity one SU(3)-manifolds, and discover perturbations of the quaternion-kähler metric on the simply connected 8-manifold G2/SO(4) that carry a closed fundamental 4-form but are not Einstein.

scholarly journals DEGENERATION OF KÄHLER–EINSTEIN MANIFOLDS I: THE NORMAL CROSSING CASE

2004 ◽  
Vol 06 (02) ◽  
pp. 301-313
Author(s):  
WEI-DONG RUAN
Keyword(s):  
Einstein Metrics ◽  
Total Space ◽  
Kähler Manifolds ◽  
Einstein Metric ◽  
Kahler Manifolds ◽  
Einstein Manifolds ◽  
Normal Crossing ◽  
Smooth Part ◽  
Kähler Einstein Metrics

In this paper we prove that the Kähler–Einstein metrics for a degeneration family of Kähler manifolds with ample canonical bundles converge in the sense of Cheeger–Gromov to the complete Kähler–Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil–Peterson metric in this case.

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