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The non-archimedean SYZ fibration

Part of: Birational geometry Surfaces and higher-dimensional varieties Non-Archimedean analysis

Published online by Cambridge University Press:  30 April 2019

Johannes Nicaise ,
Chenyang Xu  and
Tony Yue Yu
Johannes Nicaise
Affiliation:
Department of Mathematics, Imperial College, South Kensington Campus, London SW72AZ, UK email j.nicaise@imperial.ac.uk Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium
Chenyang Xu
Affiliation:
Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, USA email cyxu@math.mit.edu
Tony Yue Yu
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay, France email yuyuetony@gmail.com
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Abstract

We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away from a codimension-two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt (divisorially log terminal) models along one-dimensional strata.

Keywords

mirror symmetrynon-archimedean geometryminimal model programStrominger–Yau–Zaslow conjecture

MSC classification

Primary: 14J33: Mirror symmetry
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem)
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 5 , May 2019 , pp. 953 - 972
Copyright
© The Authors 2019 

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Footnotes

Johannes Nicaise is supported by the ERC Starting Grant MOTZETA (project 306610) of the European Research Council, and by long term structural funding (Methusalem grant) of the Flemish Government. A part of the research leading to these results was carried out at the Freiburg Institute for Advanced Studies (FRIAS) with funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement number 609305. Chenyang Xu is supported by the National Science Fund for Distinguished Young Scholars (11425101), ‘Algebraic Geometry’. Tony Yue Yu is supported by the Clay Mathematics Institute.

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