scholarly journals On the boundary behavior of Kähler–Einstein metrics on log canonical pairs

2015 ◽  
Vol 366 (1-2) ◽  
pp. 101-120 ◽  
Author(s):  
Henri Guenancia ◽  
Damin Wu
Keyword(s):  
Einstein Metrics ◽  
Boundary Behavior ◽  
Log Canonical ◽  
Kähler Einstein Metrics

Log Canonical Thresholds of Complete Intersection Log Del Pezzo Surfaces

2015 ◽  
Vol 58 (2) ◽  
pp. 445-483 ◽  
Author(s):  
In-Kyun Kim ◽  
Jihun Park
Keyword(s):  
Complete Intersection ◽  
Einstein Metrics ◽  
Projective Spaces ◽  
Del Pezzo Surfaces ◽  
Log Canonical ◽  
Del Pezzo ◽  
Kähler Einstein Metrics ◽  
Log Del Pezzo Surfaces

AbstractWe compute the global log canonical thresholds of quasi-smooth well-formed complete intersection log del Pezzo surfaces of amplitude 1 in weighted projective spaces. As a corollary we show the existence of orbifold Kähler—Einstein metrics on many of them.

scholarly journals Kähler–Einstein metrics: From cones to cusps

2020 ◽  
Vol 2020 (759) ◽  
pp. 1-27
Author(s):  
Henri Guenancia
Keyword(s):  
Einstein Metrics ◽  
Boundary Behavior ◽  
Cone Angle ◽  
Kähler Manifold ◽  
Strong Sense ◽  
Kahler Manifold ◽  
Kähler Einstein Metrics ◽  
Compact Kähler Manifold

AbstractIn this note, we prove that on a compact Kähler manifold \hskip-0.569055pt{X}\hskip-0.569055pt carrying a smooth divisor D such that {K_{X}+D} is ample, the Kähler–Einstein cusp metric is the limit (in a strong sense) of the Kähler–Einstein conic metrics when the cone angle goes to 0. We further investigate the boundary behavior of those and prove that the rescaled metrics converge to a cylindrical metric on {\mathbb{C}^{*}\times\mathbb{C}^{n-1}}.

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).

scholarly journals Geometry and topology of the space of Kähler metrics on singular varieties

2018 ◽  
Vol 154 (8) ◽  
pp. 1593-1632 ◽  
Author(s):  
Eleonora Di Nezza ◽  
Vincent Guedj
Keyword(s):  
Einstein Metrics ◽  
Normal Space ◽  
Fano Varieties ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Metric Properties ◽  
Singular Varieties ◽  
Underlying Space ◽  
And Topology ◽  
Kähler Einstein Metrics

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

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