Vol. 10, No. 10, 2016

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Combinatorial degenerations of surfaces and Calabi–Yau threefolds

Bruno Chiarellotto and Christopher Lazda

Vol. 10 (2016), No. 10, 2235–2266
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 and Matsumoto, and Hernández Mada. We also study “maximally unipotent” degenerations of Calabi–Yau threefolds, following Kollár and Xu, showing in this case that the dual intersection graph is a 3-sphere.

Keywords
monodromy, surfaces, good reduction
Mathematical Subject Classification 2010
Primary: 14J28
Secondary: 14G20, 11G25
Milestones
Received: 26 April 2016
Revised: 28 July 2016
Accepted: 5 September 2016
Published: 9 December 2016
Authors
Bruno Chiarellotto
Dipartimento di Matematica “Tullio Levi-Civita”
Università degli Studi di Padova
Via Trieste, 63
I-35121 Padova
Italy
Christopher Lazda
Dipartimento di Matematica “Tullio Levi-Civita”
Università degli Studi di Padova
Via Trieste, 63
I-35121 Padova
Italy