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 notes are based on a lecture series given at the Park City Math Institute in the summer of 2013. The notes are intended as a leisurely introduction to the K\"ahler-Ricci flow on compact K\"ahler manifolds, aimed at graduate students with some background in differential geometry.
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    ABSTRACT: Let $(M,J)$ be a Fano manifold which admits a Kähler-Einstein metric $g_{KE}$ (or a Kähler-Ricci soliton $g_{KS}$ ). Then we prove that Kähler-Ricci flow on $(M,J)$ converges to $g_{KE}$ (or $g_{KS}$ ) in $C^\infty $ in the sense of Kähler potentials modulo holomorphisms transformation as long as an initial Kähler metric of flow is very closed to $g_{KE}$ (or $g_{KS}$ ). The result improves Main Theorem in [14] in the sense of stability of Kähler-Ricci flow.
    Mathematische Annalen 01/2013; 356(4). DOI:10.1007/s00208-012-0889-7 · 1.20 Impact Factor
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    ABSTRACT: In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified $K$-energy is proper for toric invariant K\"ahler potentials on a toric Fano manifold.

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