scholarly journals Ricci iterations on Kähler classes

2009 ◽  
Vol 8 (4) ◽  
pp. 743-768 ◽  
Author(s):  
Julien Keller
Keyword(s):  
Iterative Procedure ◽  
Einstein Metrics ◽  
Einstein Manifold ◽  
Periodic Points ◽  
Dynamical Behaviour ◽  
Iteration Scheme ◽  
Fano Manifold ◽  
Ricci Operator ◽  
Finite Dimensional ◽  
Kähler Einstein Metrics

AbstractIn this paper we consider the dynamical system involved by the Ricci operator on the space of Kähler metrics of a Fano manifold. Nadel has defined an iteration scheme given by the Ricci operator and asked whether it has some non-trivial periodic points. First, we prove that no such periodic points can exist. We define the inverse of the Ricci operator and consider the dynamical behaviour of its iterates for a Fano Kähler–Einstein manifold. Then we define a finite-dimensional procedure to give an approximation of Kähler–Einstein metrics using this iterative procedure and apply it on ℂℙ2 blown up in three points.

scholarly journals Ricci Iteration for Coupled Kähler–Einstein Metrics

2019 ◽  
Author(s):  
Ryosuke Takahashi
Keyword(s):  
Dynamical System ◽  
Automorphism Group ◽  
Chern Class ◽  
Einstein Metrics ◽  
Fano Manifolds ◽  
Ricci Operator ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics

Abstract In this paper, we introduce the “coupled Ricci iteration”, a dynamical system related to the Ricci operator and twisted Kähler–Einstein metrics as an approach to the study of coupled Kähler–Einstein (CKE) metrics. For negative 1st Chern class, we prove the smooth convergence of the iteration. For positive 1st Chern class, we also provide a notion of coercivity of the Ding functional and show its equivalence to the existence of CKE metrics. As an application, we prove the smooth convergence of the iteration on CKE Fano manifolds assuming that the automorphism group is discrete.

scholarly journals Pluricomplex Green's functions and Fano manifolds

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Nicholas McCleerey ◽  
Valentino Tosatti
Keyword(s):  
Green's Functions ◽  
Einstein Metrics ◽  
Green’S Functions ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Continuity Method ◽  
Homogeneous Complex ◽  
Analytic Singularities ◽  
Kähler Einstein Metrics ◽  
Monge Ampere

We show that if a Fano manifold does not admit Kahler-Einstein metrics then the Kahler potentials along the continuity method subconverge to a function with analytic singularities along a subvariety which solves the homogeneous complex Monge-Ampere equation on its complement, confirming an expectation of Tian-Yau. Comment: EpiGA Volume 3 (2019), Article Nr. 9

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 Multiplier Hermitian structures on Kähler manifolds

2003 ◽  
Vol 170 ◽  
pp. 73-115 ◽  
Author(s):  
Toshiki Mabuchi
Keyword(s):  
Systematic Study ◽  
Einstein Metrics ◽  
Group Actions ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Fano Manifold ◽  
Hermitian Manifolds ◽  
Kähler Einstein Metrics ◽  
Hermitian Structures

AbstractThe main purpose of this paper is to make a systematic study of a special type of conformally Kähler manifolds, called multiplier Hermitian manifolds, which we often encounter in the study of Hamiltonian holomorphic group actions on Kähler manifolds. In particular, we obtain a multiplier Hermitian analogue of Myers’ Theorem on diameter bounds with an application (see [M5]) to the uniquness up to biholomorphisms of the “Kähler-Einstein metrics” in the sense of [M1] on a given Fano manifold with nonvanishing Futaki character.

scholarly journals K-STABILITY OF FANO MANIFOLDS WITH NOT SMALL ALPHA INVARIANTS

2017 ◽  
Vol 18 (3) ◽  
pp. 519-530 ◽  
Author(s):  
Kento Fujita
Keyword(s):  
Automorphism Group ◽  
Einstein Metrics ◽  
Holomorphic Automorphism ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Kähler Einstein Metrics ◽  
Holomorphic Automorphism Group

We show that any $n$-dimensional Fano manifold $X$ with $\unicode[STIX]{x1D6FC}(X)=n/(n+1)$ and $n\geqslant 2$ is K-stable, where $\unicode[STIX]{x1D6FC}(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits Kähler–Einstein metrics and the holomorphic automorphism group $\operatorname{Aut}(X)$ of $X$ is finite.

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).

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