Canonical measures and the dynamical system 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 of 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 the Käher-Ricci flow, 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 Kähler-Ricci flow. The advantage of the dynamical construction is two folds. First, it enables us to deduce the 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 Kähler-Ricci flow. Next generalizing the above construction, we define a more natural measure: the supercanonical measure on a smooth projective variety with pseudoeffective canonical line bundle in terms of the dynamical system of Bergman kernels. By the construction, the supercanonical measure also satisfies the plurisubharmonic variation property as the canonical measure. And it is expected that the canonical measure is the limit of the Kähler-Ricci flow on a smooth minimal projective variety without assuming the semiampleness of the canonical bundle. MSC: 53C25(32G07 53C55 58E11)