## Abstract

On a polarized manifold (X, L), the Bergman iteration φ^{(m)}
_{k} is defined as a sequence of Bergman metrics on L with two integer parameters k, m. We study the relation between the Kähler-Ricci flow φ_{t} at any time t ≥ 0 and the limiting behavior of metrics φ^{(m)}
_{k} when m = m(k) and the ratio m/k approaches to t as k → ∞. Mainly, three settings are investigated: the case when L is a general polarization on a Calabi-Yau manifold X and the case when L = ±K_{X} is the (anti-) canonical bundle. Recently, Berman showed that the convergence φ^{(m)}
_{k} → φ_{t} holds in the C^{0}-topology, in particular, the convergence of curvatures holds in terms of currents. In this paper, we extend Berman’s result and show that this convergence actually holds in the smooth topology.

Original language | English |
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Pages (from-to) | 713-729 |

Number of pages | 17 |

Journal | Osaka Journal of Mathematics |

Volume | 55 |

Issue number | 4 |

Publication status | Published - 2018 Oct 1 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)