  In this paper, I prove a very general extension theorem for log
pluricanonical systems. The main application of this extension theorem is
(together with Kawamata's subadjunction theorem) to give an optimal
subadjunction theorem which relates the positivities of canonical bundle of the
ambient projective manifold and that of the (maximal) center of log canonical
singularities. This is an extension of the corresponding result in my previous
work where I dealt with log pluricanonical systems of general type. This
subadjunction theorem indicates an approach to solve the abundance conjecture
for canonical divisors (or log canonical divisors) in terms of the induction in
dimension.
