Canonical measures and dynamical systems of Bergman kernels

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 with nonnegative Kodaira dimension in terms of a dynamical system of Bergman kernels. This current is considered to be a generalization of a Kähler-Einstein metric and coincides the one considered independently by J. Song and G. Tian ([S-T]). The major difference between [S-T] and the present article is that they found the canonical measure in terms of Käher-Ricci flows, while I found the canonical measure in terms of dynamical systems of Bergman kernels. Hence the present approach can be viewed as a discrete version of a Kähler-Ricci flow. The advantage of the dynamical construction is two folds. First, it enables us to deduce the logarithmic plurisubharmonic variation propery of the canonical measures on a projective family. Second, we can overcome the difficulty arising from the singularities of the solution of a Kähler-Ricci flow. MSC: 53C25(32G07 53C55 58E11)