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### 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] Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990 | MR | Zbl

[BJ17] Tropical and non-Archimedean limits of degenerating families of volume forms, J. Éc. Polytech., Math., Volume 4 (2017), pp. 87-139 | DOI | Numdam | MR | Zbl

[CBM09] Lagrangian 3-torus fibrations, J. Differ. Geom., Volume 81 (2009) no. 3, pp. 483-573 | DOI | MR | Zbl

[CBM10] Semi-global invariants of piecewise smooth Lagrangian fibrations, Q. J. Math, Volume 61 (2010) no. 3, pp. 291-318 | DOI | MR | Zbl

[CE12] From Stein to Weinstein and back, Colloquium Publications, 59, American Mathematical Society, 2012 | DOI | MR | Zbl

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

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

[FTMZ18a] Normal crossings singularities for symplectic topology, Adv. Math., Volume 339 (2018), pp. 672-748 | DOI | MR | Zbl

[FTMZ18b] Singularities and semistable degenerations for symplectic topology, C. R. Math. Acad. Sci. Paris, Volume 356 (2018) no. 4, pp. 420-432 | DOI | MR | Zbl

[GH94] Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, 1994 (reprint of the 1978 original) | DOI | MR | Zbl

[GM17] Pair of pants decomposition of 4-manifolds, Algebr. Geom. Topol., Volume 17 (2017) no. 3, pp. 1407-1444 | DOI | MR | Zbl

[Gro01a] 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

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

[GW00] Large complex structure limits of $K3$ surfaces, J. Differ. Geom., Volume 55 (2000) no. 3, pp. 475-546 | MR | Zbl

[Hat02] Algebraic topology, Cambridge University Press, 2002 | MR | Zbl

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

[KNPS15] Harmonic maps to buildings and singular perturbation theory, Commun. Math. Phys., Volume 336 (2015) no. 2, pp. 853-903 | DOI | MR | Zbl

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

[KS01] Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000), World Scientific, 2001, pp. 203-263 | DOI | MR | Zbl

[KS06] 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 | DOI | MR | Zbl

[KX16] The dual complex of Calabi–Yau pairs, Invent. Math., Volume 205 (2016) no. 3, pp. 527-557 | DOI | MR | Zbl

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

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

[Nic16] 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 | Zbl

[NXY19] The non-archimedean SYZ fibration, Compos. Math., Volume 155 (2019) no. 5, pp. 953-972 | DOI | MR | Zbl

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

[RSTZ14] Skeleta of affine hypersurfaces, Geom. Topol., Volume 18 (2014), pp. 1343-1395 | DOI | MR | Zbl

[Rua01] 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 | Zbl

[Rua02] 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 | DOI | MR | Zbl

[Rua03] 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 | Zbl

[Sal87] Topology of the complement of real hyperplanes in ${\u2102}^{n}$, Invent. Math., Volume 88 (1987) no. 3, pp. 603-618 | DOI | MR | Zbl

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

[SYZ96] Mirror symmetry is $T$-duality, Nucl. Phys., B, Volume 479 (1996) no. 1-2, pp. 243-259 | DOI | MR | Zbl

[Thu07] 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 | DOI | MR | Zbl

[Zho20] Lagrangian skeleta of hypersurfaces in ${\left({\u2102}^{*}\right)}^{n}$, Sel. Math., New Ser., Volume 26 (2020) no. 2, 26 | DOI | MR | Zbl

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