Remarks on the minimal model theory for log surfaces in the analytic setting

TL;DR

This paper extends the log minimal model theory to complex surface pairs in the analytic setting, confirming key theorems like the minimal model program, abundance, and finite generation.

Contribution

It demonstrates that fundamental results of the log minimal model program hold for log surfaces in the analytic context, broadening their applicability.

Findings

Minimal model program holds for log surfaces in the analytic setting

Abundance theorem is valid for these log pairs

Finite generation of log canonical rings is established

Abstract

We discuss the relative log minimal model theory for log surfaces in the analytic setting. More precisely, we show that the minimal model program, the abundance theorem, and the finite generation of log canonical rings hold for log pairs of complex surfaces which are projective over complex analytic varieties.