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 On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold

2012 ◽  
Vol 148 (6) ◽  
pp. 1985-2003 ◽  
Author(s):  
Chi Li
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Limit Behavior ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Continuity Method ◽  
Type Singularity ◽  
Moment Polytope ◽  
The Moment ◽  
Monge Ampere

AbstractThis work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.

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 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 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 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 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 Almost Euclidean Isoperimetric Inequalities in Spaces Satisfying Local Ricci Curvature Lower Bounds

2018 ◽  
Vol 2020 (5) ◽  
pp. 1481-1510 ◽  
Author(s):  
Fabio Cavalletti ◽  
Andrea Mondino
Keyword(s):  
Riemannian Manifold ◽  
Isoperimetric Inequality ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
General Framework ◽  
Isoperimetric Inequalities ◽  
Optimal Transportation ◽  
Maximal Volume ◽  
Bounded Below

Abstract Motivated by Perelman’s Pseudo-Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.

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