K\"ahler-Ricci flow on a toric manifold with positive first Chern class

Source: arXiv


In this note, we prove that on an n-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial (S 1) n-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manifold with positive first Chern class by using the Kähler-Ricci flow. 0. Introduction. Let M be a compact toric manifold with positive first Chern class. Let T ∼ = (S 1) n ×R n be a maximal torus which acts on M and K0 ∼ = (S 1) n be its maximal compact subgroup. In this note we discuss a Kähler-Ricci flow with a K0-invariant initial metric on M and we shall prove

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    • "Then, Tian-Zhu [30] (also [33]) proved Proposition 2.6 ([30], [33]). The family {ω˜φt } t converges to a Kähler-Ricci soliton associated to v KRS and˜v t converges to v KRS as t goes to the infinity. "
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    ABSTRACT: The purpose of this paper is to calculate the support of the multiplier ideal sheaves derived from the K\"ahler-Ricci flow on certain toric Fano manifolds with large symmetry. The early idea of this paper has already been in Appendix of \cite{futaki-sano0711}.
    Preview · Article · Dec 2008 · Communications in Analysis and Geometry
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    • "Let σ t = exp(tv). Since g 0 and v are Z 2 -invariant, the arguments in [24] and [28] imply that there is a sequence {t i } i such that ω ′ ti := (σ ti ) * ω ti converges to ω KS , where ω ti and ω KS are Kähler forms of g(t i , ·) and g KS respectively. Let ϕ ′ ti and ψ ti be Kähler potential functions satisfying "
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    ABSTRACT: We extend Nadel's results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal sheaves on toric del Pezzo surfaces which do not admit K\"ahler-Einstein metrics. We also show that one can define multiplier ideal sheaves for K\"ahler-Ricci solitons and extend the result of Nadel using the holomorphic invariant defined by Tian and Zhu.
    Preview · Article · Dec 2007 · Mathematische Annalen
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    ABSTRACT: In this paper we consider the dynamical system involved by the Ricci operator on the space of Kähler metrics. A. Nadel has defined an iter-ation scheme given by the Ricci operator for Fano manifold and asked whether it has some nontrivial 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. In particular we show that the iterates do converge to the Kähler-Ricci soliton for toric manifolds. Finally, we define a finite di-mensional procedure to give an approximation of Kähler-Einstein metrics using this iterative procedure and apply it for P 2 blown up in 3 points.
    Full-text · Article · Oct 2007 · Journal of the Institute of Mathematics of Jussieu
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