Combinatorial degenerations of surfaces and Calabi--Yau threefolds

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.