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 Hernández-Mada. We also study `maximally unipotent' degenerations of Calabi--Yau threefolds, following Kollár/Xu, showing in this case that the dual intersection graph is a 3-sphere.

27 pages. Final version, published in Algebra & Number Theory

References in corpus (2)

• The dual complex of Calabi--Yau pairs
• Monodromy Criterion for the Good Reduction of K3 Surfaces

Cited by in corpus (6)

• Minimal model program for semi-stable threefolds in mixed characteristic
• BCOV invariants of Calabi--Yau manifolds and degenerations of Hodge structures
• Degeneration of Kummer surfaces
• On the Shafarevich conjecture for Enriques surfaces
• Congruences of the cardinalities of rational points of log Fano varieties and log Calabi-Yau varieties over the log points of finite fields
• A Kulikov-Type Classification Theorem for a One Parameter Family of K3-Surfaces Over a p-ADIC Field and a Good Reduction Criterion