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

Source: arXiv

ABSTRACT In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the K\"ahler-Ricci flow with any initial $(S^1)^n$-invariant K\"ahler metric converges to a K\"ahler-Ricci soliton. In particular, we give another proof for the existence of K\"ahler-Ricci solitons on a compact toric manifold with positive first Chern class by using the K\"ahler-Ricci flow.

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    ABSTRACT: These lecture notes give an introduction to the Kahler-Ricci flow. They are based on lectures given by the authors at the conference "Analytic Aspects of Complex Algebraic Geometry", held at the Centre International de Rencontres Mathematiques in Luminy, in February 2011.
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    ABSTRACT: We study the convergence of the K\"ahler-Ricci flow on a compact K\"ahler manifold $(M,J)$ with positive first Chern class $c_1(M;J)$ and vanished Futaki invariant on $\pi c_1(M;J)$. As the application we establish a criterion for the stability of the K\"ahler-Ricci flow (with perturbed complex structure) around a K\"ahler-Einstein metric with positive scalar curvature, under certain local stable condition on the dimension of holomorphic vector fields. In particular this gives a stability theorem for the existence of K\"ahler-Einstein metrics on a K\"ahler manifold with possibly nontrivial holomorphic vector fields. Comment: 23 pages; Remark 4.1 changed due to the comments of Valentino Tosatti
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    ABSTRACT: We prove that the Ricci flow on CP^n blown-up at one point starting with any rotationally symmetric Kahler metric must develop Type I singularities. In particular, if the total volume does not go to zero at the singular time, the parabolic blow-up limit of the Type I Ricci flow along the exceptional divisor is a complete non-flat shrinking gradient Kahler-Ricci soliton on a complete Kahler manifold homeomorphic to C^n blown-up at one point.

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