  We study Riemannian geometry of canonical Kahler-Einstein currents on
projective Calabi-Yau varieties and canonical models of general type with
crepant singularities. We prove that the metric completion of the regular part
of such a canonical current is a compact metric length space homeomorphic to
the original projective variety, with well-defined tangent cones. We also prove
a special degeneration for Kahler-Einstein manifolds of general type as an
approach to establish the compactification of the moduli space of
Kahler-Einstein manifolds of general type. A number of applications are given
for degeneration of Calabi-Yau manifolds and the Kahler-Ricci flow on smooth
minimal models of general type.
