Kulikov-Persson-Pinkham theorem via smoothing of dlt models

TL;DR

This paper provides an alternative, MMP-based proof of the Kulikov-Persson-Pinkham theorem for projective degenerations of $K$-trivial smooth surfaces. By running the Minimal Model Program, the authors produce a minimal dlt model and then resolve its mild singularities through Brieskorn simultaneous resolutions and toric techniques, after a finite base change. The method yields a Kulikov model with $K_{ X} sim_f O_{ X}$ and proves semistability by controlling singularities via adjunction and the different, with a three-step outline that first stabilizes the model, then analyzes singularities, and finally resolves Du Val pieces. The approach clarifies the role of $dlt$-structures in the degenerations, connects with monodromy data on $H^2$, and aligns with related higher-dimensional results, offering a conceptually accessible path to the Kulikov classification in this setting.

Abstract

We give an alternative proof of the Kulikov-Persson-Pinkham Theorem for a projective degeneration of K-trivial smooth surfaces. After running the Minimal Model Program, the obtained minimal dlt model has mild singularities which we resolve via Brieskorn's simultaneous resolutions and toric resolutions.

Theorems & Definitions (5)

• proof : Demostració
• proof : Demostració
• proof : Demostració
• proof : Demostració del Teorema \ref{['complex']}
• proof : Demostració