On the finiteness of log surfaces

TL;DR

This work proves that for a fixed log surface $(X,B)$, the class of projective weakly log canonical (wlc) klt models crepant birationally equivalent to $(X,B)$ contains only finitely many log surfaces up to log isomorphism, provided the underlying polarizations are bounded. It reduces the finiteness problem to establishing bounded polarization for the underlying varieties and develops a framework using elementary families, isotriviality results, and MMP arguments to propagate boundedness to the space of models. The key technical contribution is a general finiteness theorem: if the corresponding underlying varieties $\mathfrak{D}$ have bounded polarization, then the class of wlc klt models $\mathfrak{C}$ is finite up to log isomorphism; the paper also delineates a precise boundary between Calabi–Yau type and non-Calabi–Yau cases, with a counterexample showing the necessity of klt-type singularities. This provides a concrete strategy to translate boundedness of polarizations into birational finiteness and outlines paths toward higher-dimensional generalizations.

Abstract

We prove that a log surface has only finitely many weakly log canonical projective models with klt singularities up to log isomorphism, by reducing the problem to the boundedness of their polarization.

Theorems & Definitions (52)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Example 2.5