Constructing local models for Lagrangian torus fibrations
Annales Henri Lebesgue, Volume 4 (2021) , pp. 537-570.

Metadata

KeywordsLagrangian torus, SYZ fibration, dual complex, negative vertex

Abstract

We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways:

  • We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi–Yau 3-folds with codimension 2 discriminant locus.
  • We find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.

References

[Ber90] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990 | MR 1070709 | Zbl 0715.14013

[BJ17] Boucksom, Sébastien; Jonsson, Mattias Tropical and non-Archimedean limits of degenerating families of volume forms, J. Éc. Polytech., Math., Volume 4 (2017), pp. 87-139 | Article | Numdam | MR 3611100 | Zbl 1401.32019

[CBM09] Castaño Bernard, R.; Matessi, Diego Lagrangian 3-torus fibrations, J. Differ. Geom., Volume 81 (2009) no. 3, pp. 483-573 | Article | MR 2487600 | Zbl 1177.14080

[CBM10] Castaño-Bernard, R.; Matessi, Diego Semi-global invariants of piecewise smooth Lagrangian fibrations, Q. J. Math, Volume 61 (2010) no. 3, pp. 291-318 | Article | MR 2672425 | Zbl 1197.53101

[CE12] Cieliebak, Kai; Eliashberg, Yakov From Stein to Weinstein and back, Colloquium Publications, 59, American Mathematical Society, 2012 | Article | MR 3012475 | Zbl 1262.32026

[Dan75] Danilov, Vladimir I. Polyhedra of schemes and algebraic varieties, Mat. Sb., N. Ser., Volume 97 (139) (1975) no. 1, pp. 146-158 | MR 441970 | Zbl 0321.14010

[Fri83] Friedman, Robert Global smoothings of varieties with normal crossings, Ann. Math., Volume 118 (1983) no. 1, pp. 75-114 | Article | MR 707162 | Zbl 0569.14002

[FTMZ18a] Farajzadeh Tehrani, Mohammad; McLean, Mark; Zinger, Aleksei Normal crossings singularities for symplectic topology, Adv. Math., Volume 339 (2018), pp. 672-748 | Article | MR 3866910 | Zbl 1401.53073

[FTMZ18b] Farajzadeh Tehrani, Mohammad; McLean, Mark; Zinger, Aleksei Singularities and semistable degenerations for symplectic topology, C. R. Math. Acad. Sci. Paris, Volume 356 (2018) no. 4, pp. 420-432 | Article | MR 3787532 | Zbl 1397.53089

[GH94] Griffiths, Philip; Harris, Joseph Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, 1994 (reprint of the 1978 original) | Article | MR 1288523 | Zbl 0836.14001

[GM17] Golla, Marco; Martelli, Bruno Pair of pants decomposition of 4-manifolds, Algebr. Geom. Topol., Volume 17 (2017) no. 3, pp. 1407-1444 | Article | MR 3677932 | Zbl 1376.57020

[Gro01a] Gross, Mark Special Lagrangian fibrations I: Topology, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999) (AMS/IP Studies in Advanced Mathematics), Volume 23, American Mathematical Society, 2001, pp. 65-93 | Zbl 1028.14014

[Gro01b] Gross, Mark Topological mirror symmetry, Invent. Math., Volume 144 (2001) no. 1, pp. 75-137 | Article | MR 1821145 | Zbl 1077.14052

[GW00] Gross, Mark; Wilson, Pelham Mark Hedley Large complex structure limits of K3 surfaces, J. Differ. Geom., Volume 55 (2000) no. 3, pp. 475-546 | MR 1863732 | Zbl 1027. 32021

[Hat02] Hatcher, Allen Algebraic topology, Cambridge University Press, 2002 | MR 1867354 | Zbl 1044.55001

[Joy03] Joyce, Dominic Singularities of special Lagrangian fibrations and the SYZ conjecture, Commun. Anal. Geom., Volume 11 (2003) no. 5, pp. 859-907 | Article | MR 2032503 | Zbl 1080.32501

[KNPS15] Katzarkov, Ludmil; Noll, Alexander; Pandit, Pranav; Simpson, Carlos Harmonic maps to buildings and singular perturbation theory, Commun. Math. Phys., Volume 336 (2015) no. 2, pp. 853-903 | Article | MR 3322389 | Zbl 1314.32021

[Kol13] Kollár, Janos Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, 2013 (with a collaboration of Sándor Kovács) | MR 3057950 | Zbl 1282.14028

[KS01] Kontsevich, Maxim; Soibelman, Yan Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000), World Scientific, 2001, pp. 203-263 | Article | MR 1882331 | Zbl 1072.14046

[KS06] Kontsevich, Maxim; Soibelman, Yan Affine structures and non-Archimedean analytic spaces, The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31–September 4, 2003 (Progress in Mathematics), Volume 244, Birkhäuser, 2006, pp. 321-385 | Article | MR 2181810 | Zbl 1114.14027

[KX16] Kollár, Janos; Xu, Chenyang The dual complex of Calabi–Yau pairs, Invent. Math., Volume 205 (2016) no. 3, pp. 527-557 | Article | MR 3539921 | Zbl 1360.14056

[McL16] McLean, Mark Reeb orbits and the minimal discrepancy of an isolated singularity, Invent. Math., Volume 204 (2016) no. 2, pp. 505-594 | Article | MR 3489704 | Zbl 1348.53079

[Mik04] Mikhalkin, Grigory Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology, Volume 43 (2004) no. 5, pp. 1035-1065 | Article | MR 2079993 | Zbl 1065.14056

[Nic16] Nicaise, Johannes Berkovich skeleta and birational geometry, Nonarchimedean and tropical geometry. Nonarchimedean and tropical geometry. Based on two Simons symposia, Island of St. John, March 31 – April 6, 2013 and Puerto Rico, February 1–7, 2015 (Simons Symposia), Springer, 2016, pp. 173-194 | MR 3702312 | Zbl 1349.14102

[NXY19] Nicaise, Johannes; Xu, Chenyang; Yu, Tony Yue The non-archimedean SYZ fibration, Compos. Math., Volume 155 (2019) no. 5, pp. 953-972 | Article | MR 3946280 | Zbl 1420.14093

[Pay13] Payne, Sam Boundary complexes and weight filtrations, Mich. Math. J., Volume 62 (2013) no. 2, pp. 293-322 | MR 3079265 | Zbl 1312.14049

[RSTZ14] Ruddat, Helge; Sibilla, Nicolò; Treumann, David; Zaslow, Eric Skeleta of affine hypersurfaces, Geom. Topol., Volume 18 (2014), pp. 1343-1395 | Article | MR 3228454 | Zbl 1326.14102

[Rua01] Ruan, Wei-Dong Lagrangian torus fibration of quintic hypersurfaces. I. Fermat quintic case, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999) (AMS/IP Studies in Advanced Mathematics), Volume 23, American Mathematical Society, 2001, pp. 297-332 | MR 1876075

[Rua02] Ruan, Wei-Dong Lagrangian torus fibration of quintic Calabi–Yau hypersurfaces. II. Technical results on gradient flow construction, J. Symplectic Geom., Volume 1 (2002) no. 3, pp. 435-521 | Article | MR 1959057 | Zbl 1090.14502

[Rua03] Ruan, Wei-Dong Lagrangian torus fibration of quintic Calabi–Yau hypersurfaces. III. Symplectic topological SYZ mirror construction for general quintics, J. Differ. Geom., Volume 63 (2003) no. 2, pp. 171-229 | MR 2015547

[Sal87] Salvetti, Mario Topology of the complement of real hyperplanes in n , Invent. Math., Volume 88 (1987) no. 3, pp. 603-618 | Article | MR 884802 | Zbl 0594.57009

[Sei08] Seidel, Paul A biased view of symplectic cohomology, Current developments in mathematics, 2006, International Press, 2008, pp. 211-253 | MR 2459307 | Zbl 1165.57020

[SYZ96] Strominger, Andrew E.; Yau, Shing-Tung; Zaslow, Eric Mirror symmetry is T-duality, Nucl. Phys., B, Volume 479 (1996) no. 1-2, pp. 243-259 | Article | MR 1429831 | Zbl 0896.14024

[Thu07] Thuillier, Amaury Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels, Manuscr. Math., Volume 123 (2007) no. 4, pp. 381-451 | Article | MR 2320738 | Zbl 1134.14018

[Zho20] Zhou, Peng Lagrangian skeleta of hypersurfaces in ( * ) n , Sel. Math., New Ser., Volume 26 (2020) no. 2, 26 | Article | MR 4087022 | Zbl 1439.53077

[Zun03] Zung, Nguyen Tien Symplectic topology of integrable Hamiltonian systems. II. Topological classification, Compos. Math., Volume 138 (2003) no. 2, pp. 125-156 | Article | MR 2018823 | Zbl 1127.53308