Bergman iteration and C^∞-convergence towards KÄhler-Ricci flow

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 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.