Bergman iteration and $C^{\infty }$-convergence towards Kähler-Ricci flow

Abstract

On a polarized manifold $(X,L)$, the Bergman iteration ${\varphi }_k^{(m)}$ 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 ${\varphi }_t$ at any time $t\ge 0$ and the limiting behavior of metrics ${\varphi }_k^{(m)}$ when $m=m(k)$ and the ratio $m/k$ approaches to $t$ as $k\to \infty$. Mainly, three settings are investigated: the case when $L$ is a general polarization on a Calabi-Yau manifold $X$ and the case when $L=\pm K_X$ is the (anti-) canonical bundle. Recently, Berman showed that the convergence ${\varphi }_k^{(m)}\to {\varphi }_t$ 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.

arXiv comment17 pages, final version