Abstract
This is an announcement of the following construction: given an integral affine manifold B with singularities, we build a topological space X which is a torus fibration over B. The main new feature of the fibration X → B is that it has the discriminant in codimension 2.
H.R. was supported by DFG grant RU 1629/4-1 and the Department of Mathematics at Universit¨at Hamburg. The research of I.Z. was supported by Simons Collaboration grant A20-0125-001.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abouzaid, M., Auroux, D. Katzarkov, L.: “Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces”, Publ.math.IHES 123, 199–282 (2016), https://doi.org/10.1007/s10240-016-0081-9.
Abouzaid, M., Ganatra, S., Iritani, H., Sheridan, N.: “The Gamma and Strominger-Yau-Zaslow conjectures: a tropical approach to periods”, https://arxiv.org/abs/1809.02177.
H¨ulya Arg¨uz, H., Siebert, B.: “On the real locus in the Kato-Nakayama space of logarithmic spaces with a view toward toric degenerations”, https://arxiv.org/abs/1610.07195.
Auroux, D.: “Mirror symmetry and T-duality in the complement of an anticanonical divisor”, J. G¨okova Geom. Topol. 1 (2007), 51–91.
Auroux, D.: “Special Lagrangian fibrations, wall-crossing, and mirror symmetry”, Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of theJournal of Differential Geometry, Surv. Differ. Geom., vol. 13, Int. Press, Somerville, MA, 2009, 1–47.
Casta˜no Bernard, R., Matessi, D.: “Lagrangian 3-torus fibrations”, J. Differential Geom. 81 (2009), 483–573. Compactifying torus fibrations 13.
Chan, K.: “The Strominger-Yau-Zaslow conjecture and its impact”, Selected Expository Works of Shing-Tung Yau with Commentary. Vol. II”, 1183–1208, Adv. Lect. Math. (ALM) 29, Int. Press, Somerville, MA, 2014.
Chan, K., Lau, S.-C., Leung, N. C.: “SYZ mirror symmetry for toric Calabi-Yau manifolds”, J. Differential Geom. 90, no. 2, (2012), 177–250.
Chan, K., Leung, N. C.: “Mirror symmetry for toric Fano manifolds via SYZ transformations”, Adv. Math. 223 no. 3, (2010), 797–839.
Evans, J., Mauri, M.: “Constructing local models for Lagrangian torus fibrations”, https://arxiv.org/abs/1905.09229.
Fang, B., Liu, C-C M., Treumann, D., Zaslow, E.: “T-duality and homological mirror symmetry for toric varieties”, Adv. Math.229, no. 3, (2012), 1875–1911.
Gross, M.: “Examples of special Lagrangian fibrations”, Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, 81–109.
Gross, M.: “Topological mirror symmetry”, Inv. Math., volume 144 (2001), 75–137.
Gross, M.: “The Strominger-Yau-Zaslow conjecture: from torus fibrations to degenerations”,Algebraic geometry, Seattle 2005. Part 1, 149–192, Proc. Sympos. Pure Math. 80, Part 1, Amer. Math. Soc., Providence, RI, 2009.
Gross, M.: “Mirror symmetry and the Strominger–Yau–Zaslow conjecture”, Current developments in mathematics 2012, Int. Press, Somerville, MA, 2013, pp. 133–191.
Gross, M., Wilson, P. M. H.: “Large Complex Structure Limits of K3 Surfaces”, J. Differential Geom., Volume 55, Number 3 (2000), 475–546.
Gross, M., Siebert, B.: “From real affine geometry to complex geometry”, Annals of Math. 174, 2011, 1301–1428.
Gross, M., Siebert, B.: “Mirror symmetry via logarithmic degeneration data I”, J. Differential Geom. 72, 2006, 169–338.
Gross, M., Siebert, B.: “Mirror symmetry via logarithmic degeneration data II”, J. Algebraic Geom. 19, 2010, 679–780.
Gross, M., Tosatti, V., Zhang, Y.: “Collapsing of abelian fibred Calabi-Yau manifolds”, Duke Math. J., Volume 162, Number 3, (2013), 517–551.
Haase, C., Zharkov, I.: “Integral affine structures on spheres: complete intersections”, Int. Math. Res. Not. 2005, no. 51, 3153–3167.
Hicks, J.: “Tropical Lagrangians and Homological Mirror Symmetry”, https://arxiv.org/abs/1904.06005.
Joyce, D.: “Singularities of special Lagrangian fibrations and the SYZ conjecture”, Comm. Anal.Geom. 11(5), 2003, 859–907.
Kontsevich, M., Soibelman Y., “Homological mirror symmetry and torus fibrations”, In: “Symplectic geometry and mirror symmetry”, Seoul, 2000, 203–263.
Kontsevich, M., Soibelman Y., “Affine structures and non-archimedean analytic spaces”, In: Etingof P., Retakh V., Singer I.M. (eds), The Unity of Mathematics. Progress in Mathematics, vol 244. Birkh¨auser Boston.
Leung, N.C.: “Mirror Symmetry Without Corrections”, Communications in Analysis and Geometry, 13(2), 2001.
Loftin, J. Yau, S.-T., Zaslow, E.: “Affine manifolds, SYZ geometry and the “Y” vertex”, J. Differential Geom.71, no. 1, (2005), 129–158.
Mak, C.Y., Ruddat, H.: “Tropically constructed Lagrangians in mirror quintic threefolds”, https://arxiv.org/abs/1904.11780.
Matessi, D.: “Lagrangian pairs of pants”, https://arxiv.org/abs/1802.02993.
Mikhalkin, G.: “Decomposition into pairs-of-pants for complex algebraic hypersurfaces”, Topology Vol. 43 (2004), Issue 5, 1035–1065.
Mikhalkin, G.: “Examples of tropical-to-Lagrangian correspondence”, European Journal of Mathematics 5, 2019, 1033–1066.
Nakayama, C., Ogus, A.: “Relative rounding in toric and logarithmic geometry”, Geom. & Topol. 14(4), 2010, 2189–2241. 14 Helge Ruddat and Ilia Zharkov.
Nicaise, J., Xu, C., Yu, T.Y.: “The non-archimedean SYZ fibration”, Compos. Math. 155, Issue 5, 2019, 953–972.
Parker, B.: “Exploded fibrations”, Proceedings of 13th G¨okova Geometry-Topology Conference, 52–90.
Prince, T.: “Lagrangian torus fibration models of Fano threefolds”, https://arxiv.org/abs/1801.02997.
Ruddat, H.: “A homology theory for tropical cycles on integral affine manifolds and a perfect pairing”, https://arxiv.org/abs/2002.12290.
Ruddat, H., Siebert, B.: “Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations”, Publ.math.IHES (2020), https://doi.org/10.1007/s10240-020-00116-y.RZ20. Ruddat, H., Zharkov, I.: “Tailoring a pair-of-pants”, https://arxiv.org/abs/2001.08267.
Ruddat, H., Zharkov, I.: “Topological Strominger-Yau-Zaslow fibrations”, in preparation.
Ruan, W.-D.: “Lagrangian torus fibrations and mirror symmetry of Calabi–Yau manifolds”, Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, 385–427.
Sheridan, N., Smith, I.: “Lagrangian cobordism and tropical curves”, https://arxiv.org/abs/1805.07924.
Strominger A., Yau S.-T., Zaslow, E.: “Mirror symmetry is T-duality”, Nuclear Phys. B 479 (1996), no. 1–2, 243-259.
Thomas, R. P.: “The Geometry of Mirror Symmetry”, Encyclopedia of Mathematical Physics. Elsevier, (2006), 439–448.
Zharkov, I.: “Torus fibrations of Calabi-Yau hypersurfaces in toric varieties”, Duke Math. J., 101:2, 2000, 237–258.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Ruddat, H., Zharkov, I. (2021). Compactifying Torus Fibrations Over Integral Affine Manifolds with Singularities. In: de Gier, J., Praeger, C.E., Tao, T. (eds) 2019-20 MATRIX Annals. MATRIX Book Series, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-030-62497-2_37
Download citation
DOI: https://doi.org/10.1007/978-3-030-62497-2_37
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-62496-5
Online ISBN: 978-3-030-62497-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)