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Ricci iterations on Kähler classes

Part of: Global differential geometry Surfaces and higher-dimensional varieties Functional-differential and differential-difference equations Complex manifolds Computational number theory

Published online by Cambridge University Press:  30 January 2009

Julien Keller
Julien Keller
Affiliation:
Centre de Mathématiques et Informatique, Université de Provence, 39 rue Frédéric Joliot-Curie, 13453 Marseille cedex 13, France (jkeller@cmi.univ-mrs.fr)
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Abstract

In 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.

Keywords

Kähler–Einstein metricsdynamics of the Ricci operatorKähler–Ricci flownumerical approximationsBergman kernelFano manifold

MSC classification

Secondary: 32Q20: Kähler-Einstein manifolds 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 14J45: Fano varieties 34K28: Numerical approximation of solutions 11Y16: Algorithms; complexity
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 4 , October 2009 , pp. 743 - 768
Copyright
Copyright © Cambridge University Press 2009

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