Combinatorial degenerations of surfaces and Calabi--Yau threefolds

TL;DR

This paper investigates combinatorial degenerations of minimal surfaces and Calabi--Yau threefolds over local fields, linking degeneration types to monodromy and describing the topology of dual intersection graphs.

Contribution

It extends the understanding of degenerations of surfaces and Calabi--Yau threefolds by relating monodromy actions to degeneration types and characterizing the intersection graph topology.

Findings

Degeneration type can be read from monodromy operator.

Dual intersection graph of maximally unipotent Calabi--Yau degenerations is a 3-sphere.

Extends results from complex surface degenerations to arithmetic setting.

Abstract

In this article we study combinatorial degenerations of minimal surfaces of Kodaira dimension 0 over local fields, and in particular show that the `type' of the degeneration can be read off from the monodromy operator acting on a suitable cohomology group. This can be viewed as an arithmetic analogue of results of Persson and Kulikov on degenerations of complex surfaces, and extends various particular cases studied by Matsumoto, Liedtke/Matsumoto and Hern\'andez-Mada. We also study `maximally unipotent' degenerations of Calabi--Yau threefolds, following Koll\'ar/Xu, showing in this case that the dual intersection graph is a 3-sphere.