On finiteness of relative log pluricanonical representations

TL;DR

This paper proves the finiteness of certain complex analytic representations and applies this to the abundance conjecture and existence of flips for complex spaces, reducing complex problems to classical cases.

Contribution

It establishes the finiteness of relative log pluricanonical representations in the complex analytic setting and connects the abundance conjecture for semi-log canonical pairs to the classical case.

Findings

Finiteness of relative log pluricanonical representations proven.

Reduction of the abundance conjecture for semi-log canonical pairs to the log canonical case.

Existence of log canonical flips for complex analytic spaces established.

Abstract

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.