On finiteness of relative log pluricanonical representations

We prove the finiteness of relative log pluricanonical representations in the complex analytic setting.

As an application, we discuss the abundance conjecture for semi-log canonical pairs within this framework.

Furthermore, we establish the existence of log canonical flips for complex analytic spaces.

Roughly speaking, we reduce the abundance conjecture for semi-log canonical pairs to the case of log canonical pairs in the complex analytic setting.

Moreover, we show that the abundance conjecture for projective morphisms of complex analytic spaces can be reduced to the classical abundance conjecture for projective varieties.