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 Greatest lower bounds on the Ricci curvature of Fano manifolds

2010 ◽  
Vol 147 (1) ◽  
pp. 319-331 ◽  
Author(s):  
Gábor Székelyhidi
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Einstein Metrics ◽  
Upper Bounds ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Existence Time ◽  
Kähler Einstein Metrics ◽  
Bounded Below ◽  
Kahler Metric

AbstractOn a Fano manifoldMwe study the supremum of the possibletsuch that there is a Kähler metricω∈c1(M) with Ricci curvature bounded below byt. This is shown to be the same as the maximum existence time of Aubin’s continuity path for finding Kähler–Einstein metrics. We show that onP2blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.

scholarly journals Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Projective Line ◽  
Tangent Cones ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Complex Curves ◽  
Complex Dimensions ◽  
Irregular Case ◽  
Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

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).

Homogeneous Einstein Finsler Metrics on -dimensional Spheres

2018 ◽  
Vol 62 (3) ◽  
pp. 509-523
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Constant Sectional Curvature ◽  
Einstein Metric ◽  
Finsler Metrics ◽  
Dimensional Sphere ◽  
Order Ordinary Differential Equation ◽  
Canonical Metric ◽  
Homogeneous Einstein Metrics

AbstractIn this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.

scholarly journals Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties

2019 ◽  
Vol 2019 (751) ◽  
pp. 27-89 ◽  
Author(s):  
Robert J. Berman ◽  
Sebastien Boucksom ◽  
Philippe Eyssidieux ◽  
Vincent Guedj ◽  
Ahmed Zeriahi
Keyword(s):  
Ricci Flow ◽  
Existence And Uniqueness ◽  
Additional Condition ◽  
Einstein Metrics ◽  
Discrete Version ◽  
Fano Varieties ◽  
Fano Manifolds ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics ◽  
Special Case

AbstractWe prove the existence and uniqueness of Kähler–Einstein metrics on {{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

Affine compact almost-homogeneous manifolds of cohomogeneity one

2009 ◽  
Vol 7 (1) ◽  
Author(s):  
Daniel Guan
Keyword(s):  
Einstein Metrics ◽  
Compact Manifolds ◽  
Complex Manifolds ◽  
Homogeneous Manifolds ◽  
Einstein Manifolds ◽  
Extremal Metrics ◽  
Cohomogeneity One ◽  
Fano Manifolds ◽  
Kähler Einstein Metrics

AbstractThis paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.

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 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.

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