Canonical measures and the dynamical systems of Bergman kernels

Hajime Tsuji

Published 2008-05-13, updated 2011-03-07Version 4

In this article, we construct the canonical semipositive current or the canonical measure ($=$ the potential of the canonical semipositive current) on a smooth projective variety of nonnegative Kodaira dimension in terms of a dynamical system of Bergman kernels. This current is considered to be a generalization of a K\"{a}hler-Einstein metric and coincides the one constructed independently by J. Song and G. Tian. The major difference between their work and the present article is that they found the canonical measure in terms of K\"{a}her-Ricci flows, while I found the canonical measure in terms of the dynamical system of Bergman kernels. Hence the present approach can be viewed as the discrete version of the K\"{a}hler-Ricci flow.