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.
|Number of pages||17|
|Journal||Osaka Journal of Mathematics|
|Publication status||Published - Oct 2018|
All Science Journal Classification (ASJC) codes