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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 [email protected] Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium
Chenyang Xu
Affiliation:
Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, USA email [email protected]
Tony Yue Yu
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay, France email [email protected]
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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.

References

Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33 (American Mathematical Society, Providence, RI, 1990).Google Scholar
Berkovich, V., Vanishing cycles for formal schemes. II , Invent. Math. 125 (1996), 367–390.Google Scholar
Boucksom, S. and Jonsson, M., Tropical and non-Archimedean limits of degenerating families of volume forms , J. Éc. Polytech. Math. 4 (2017), 87–139.Google Scholar
Candelas, P., de la Ossa, X., Green, P. and Parkes, L., A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory , Nuclear Phys. B 359 (1991), 21–74.Google Scholar
Deligne, P., Théorème de Lefschetz et critères de dégénérescence de suites spectrales , Publ. Math. Inst. Hautes Études Sci. 35 (1968), 259–278.Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I , Publ. Math. Inst. Hautes Études Sci. 11 (1961), 5–167.Google Scholar
Friedman, R. and Morrison, D., The birational geometry of degenerations, Progress in Mathematics, vol. 29 (Birkhäuser, Boston, MA, 1983).Google Scholar
Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Görtz, U. and Wedhorn, T., Algebraic geometry I , in Advanced lectures in mathematics (Vieweg + Teubner, Wiesbaden, 2010).Google Scholar
Gross, M., Mirror symmetry and the Strominger–Yau–Zaslow conjecture , in Current developments in mathematics 2012 (International Press, Somerville, MA, 2013), 133–191.Google Scholar
Gross, M., Hacking, P. and Keel, S., Mirror symmetry for log Calabi–Yau surfaces I , Publ. Math. Inst. Hautes Études Sci. 155 (2015), 65–168.Google Scholar
Gross, M. and Siebert, B., From real affine geometry to complex geometry , Ann. of Math. (2) 174 (2011), 1301–1428.Google Scholar
Gross, M. and Siebert, B., An invitation to toric degenerations , in Geometry of special holonomy and related topics, Surveys in Differential Geometry, vol. 16 (International Press, Somerville, MA, 2011), 43–78.Google Scholar
Gross, M. and Wilson, P. M., Large complex structure limits of K3 surfaces , J. Differential Geom. 55 (2000), 475–546.Google Scholar
Halle, L. H. and Nicaise, J., Motivic zeta functions of degenerating Calabi–Yau varieties , Math. Ann. 370 (2018), 1277–1320.Google Scholar
Kawamata, Y., Semistable minimal models of threefolds in positive or mixed characteristic , J. Algebraic Geom. 3 (1994), 463–491.Google Scholar
Kempf, G., Knudsen, F. F., Mumford, D. and Saint-Donat, B., Toroidal embeddings. I, Lecture Notes in Mathematics, vol. 339 (Springer, Berlin, 1973).Google Scholar
Kollár, J., Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200 (Cambridge University Press, Cambridge, 2013).Google Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).Google Scholar
Kollár, J., Nicaise, J. and Xu, C., Semi-stable extensions over 1-dimensional bases , Acta Math. Sin. (Engl. Ser.) 34 (2018), 103–113.Google Scholar
Kollár, J. and Xu, C., The dual complex of Calabi–Yau pairs , Invent. Math. 205 (2016), 527–557.Google Scholar
Kontsevich, M. and Soibelman, Y., Homological mirror symmetry and torus fibrations , in Symplectic geometry and mirror symmetry (Seoul, 2000) (World Scientific, River Edge, NJ, 2001), 203–263.Google Scholar
Kontsevich, M. and Soibelman, Y., Affine structures and non-archimedean analytic spaces , in The unity of mathematics: in honor of the ninetieth birthday of I. M. Gelfand, Progress in Mathematics, vol. 244, eds Etingof, P., Retakh, V. and Singer, I. M. (Birkhäuser Boston, Boston, MA, 2006), 312–385.Google Scholar
Künnemann, K., Projective regular models for abelian varieties, semistable reduction, and the height pairing , Duke Math. J. 95 (1998), 161–212.Google Scholar
Mumford, D., An analytic construction of degenerating abelian varieties over complete rings , Compos. Math. 24 (1972), 239–272.Google Scholar
Mustaţă, M. and Nicaise, J., Weight functions on non-archimedean analytic spaces and the Kontsevich–Soibelman skeleton , Algebraic Geom. 2 (2015), 365–404.10.14231/AG-2015-016Google Scholar
Nakayama, C., Nearby cycles for log smooth families , Compos. Math. 112 (1998), 45–75.Google Scholar
Nicaise, J. and Xu, C., The essential skeleton of a degeneration of algebraic varieties , Amer. Math. J. 138 (2016), 1645–1667.Google Scholar
Nicaise, J. and Xu, C., Poles of maximal order of motivic zeta functions , Duke Math. J. 165 (2016), 217–243.Google Scholar
Grothendieck, A., Revêtements étales et groupe fondamental (SGA 1), Séminaire de géométrie algébrique du Bois Marie 1960–61, Documents Mathématiques (Paris), vol. 3, updated and annotated reprint of [Lecture Notes in Mathematics, vol. 224 (Springer, Berlin, 1971)] (Société Mathématique de France, Paris, 2003).Google Scholar
Strominger, A., Yau, S.-T. and Zaslow, E., Mirror symmetry is T-duality , Nuclear Phys. B 479 (1996), 243–259.Google Scholar
Yu, T. Y., Enumeration of holomorphic cylinders in log Calabi–Yau surfaces I , Math. Ann. 366 (2016), 1649–1675.Google Scholar
Yu, T. Y., Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. II. Positivity, integrality and the gluing formula, Preprint (2016), arXiv:1608.07651.Google Scholar