Combinatorial degenerations of surfaces and Calabi–Yau threefolds

Chiarellotto, Bruno; Lazda, Christopher

doi:10.2140/ant.2016.10.2235

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Combinatorial degenerations of surfaces and Calabi–Yau threefolds

Bruno Chiarellotto and Christopher Lazda

Vol. 10 (2016), No. 10, 2235–2266

DOI: 10.2140/ant.2016.10.2235

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

Vol. 10, No. 10, 2016