Bergman iteration and C-convergence towards KÄhler-Ricci flow

Takahashi Ryosuke

Research output: Contribution to journalArticlepeer-review

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 φt 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 = ±KX is the (anti-) canonical bundle. Recently, Berman showed that the convergence φ(m) k → φt holds in the C0-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 languageEnglish
Pages (from-to)713-729
Number of pages17
JournalOsaka Journal of Mathematics
Volume55
Issue number4
Publication statusPublished - 2018 Oct 1
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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