Ricci iterations and canonical K\"ahler-Einstein currents on log canonical pairs

Abstract

In this article we construct a canonical K\"{a}hler-Einstein current on a LC (log canonical) pairs of log general type as the limit of a sequence of canonical K\"{a}hler-Einstein currents on KLT(Kawamata log terminal) pairs of log general type. We call the volume form associated with the canonical K\"{a}hler-Einstein current the canonical measure of the LC pair. We prove that the relative canonical measure on a projective family of LC pairs of log general type defines a singular hermitian metric on the relative log canonical bundle and the metric has semipositive curvature in the sense of current. This is the first semipositivity result for relative log canonical bundles of a family of LC pairs in general dimension. Our proof depends on certain Ricci iterations and dynamical systems of Bergman kernels.