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Doran–Harder–Thompson Conjecture via SYZ Mirror Symmetry: Elliptic Curves
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove the Doran–Harder–Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi–Yau manifold $X$ degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi–Yau manifold of $X$ can be constructed by gluing the two mirror Landau–Ginzburg models of the quasi-Fano manifolds. The two crucial ideas in our proof are to obtain a complex structure by gluing the underlying affine manifolds and to construct the theta functions from the Landau–Ginzburg superpotentials.
Keywords: Calabi–Yau manifolds; Fano manifolds; SYZ mirror symmetry; Landau–Ginzburg models; Tyurin degeneration; affine geometry. 
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     author = {Atsushi Kanazawa},
     title = {Doran{\textendash}Harder{\textendash}Thompson {Conjecture} via {SYZ} {Mirror} {Symmetry:} {Elliptic} {Curves}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a23/}
}
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Atsushi Kanazawa. Doran–Harder–Thompson Conjecture via SYZ Mirror Symmetry: Elliptic Curves. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a23/

[1] Aspinwall P. S., Bridgeland T., Craw A., Douglas M. R., Gross M., Kapustin A., Moore G. W., Segal G., Szendrői B., Wilson P. M. H., Dirichlet branes and mirror symmetry, Clay Mathematics Monographs, 4, Amer. Math. Soc., Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2009 | MR | Zbl

[2] Auroux D., “Mirror symmetry and $T$-duality in the complement of an anticanonical divisor”, J. Gökova Geom. Topol. GGT, 1 (2007), 51–91, arXiv: 0706.3207 | MR | Zbl

[3] Auroux D., “Special Lagrangian fibrations, mirror symmetry and Calabi–Yau double covers”, Astérisque, 321, 2008, 99–128, arXiv: 0803.2734 | MR | Zbl

[4] Cho C.-H., Oh Y.-G., “Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds”, Asian J. Math., 10 (2006), 773–814, arXiv: math.SG/0308225 | DOI | MR | Zbl

[5] Dolgachev I. V., “Mirror symmetry for lattice polarized $K3$ surfaces”, J. Math. Sci., 81 (1996), 2599–2630, arXiv: alg-geom/9502005 | DOI | MR | Zbl

[6] Doran C. F., Harder A., Thompson A., “Mirror symmetry, Tyurin degenerations, and fibrations on Calabi–Yau manifolds”, Proceedings of String-Math 2015 (to appear) , arXiv: 1601.08110

[7] Fukaya K., Oh Y-.G., Ohta H., Ono K., Lagrangian intersection Floer theory: anomaly and obstruction, v. I, II, AMS/IP Studies in Advanced Mathematics, 46, Amer. Math. Soc., Providence, RI; International Press, Somerville, MA, 2009 | MR

[8] Gross M., Siebert B., “From real affine geometry to complex geometry”, Ann. of Math., 174 (2011), 1301–1428, arXiv: math.AG/0703822 | DOI | MR | Zbl

[9] Harder A., The geometry of Landau–Ginzburg models, Ph.D. Thesis, University of Alberta, 2016

[10] Hitchin N. J., “The moduli space of special Lagrangian submanifolds”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 503–515, arXiv: dg-ga/9711002 | MR | Zbl

[11] Hori K., Vafa C., Mirror symmetry, arXiv: hep-th/0002222 | MR

[12] Kanazawa A., Doran–Harder–Thompson conjecture via integral affine geometry, in preparation

[13] Kanazawa A., Lau S.-C., Calabi–Yau manifolds of $\widetilde{A}$ and open Yau–Zaslow formula via SYZ mirror symmetry, arXiv: 1605.00342

[14] Katzarkov L., Kontsevich M., Pantev T., “Bogomolov–Tian–Todorov theorems for Landau–Ginzburg models”, J. Differential Geom., 105 (2017), 55–117, arXiv: 1409.5996 | DOI | MR | Zbl

[15] Kawamata Y., Namikawa Y., “Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi–Yau varieties”, Invent. Math., 118 (1994), 395–409 | DOI | MR | Zbl

[16] Kontsevich M., “Homological algebra of mirror symmetry”, Proceedings of the International Congress of Mathematicians (Zürich, 1994), v. 1, 2, Birkhäuser, Basel, 1995, 120–139, arXiv: alg-geom/9411018 | Zbl

[17] Leung N. C., “Mirror symmetry without corrections”, Comm. Anal. Geom., 13 (2005), 287–331, arXiv: math.DG/0009235 | DOI | MR | Zbl

[18] McLean R. C., “Deformations of calibrated submanifolds”, Comm. Anal. Geom., 6 (1998), 705–747 | DOI | MR | Zbl

[19] Seidel P., “More about vanishing cycles and mutation”, Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, 429–465, arXiv: math.SG/0010032 | DOI | MR | Zbl

[20] Strominger A., Yau S.-T., Zaslow E., “Mirror symmetry is $T$-duality”, Nuclear Phys. B, 479 (1996), 243–259, arXiv: hep-th/9606040 | DOI | MR | Zbl

[21] Tyurin A. N., “Fano versus Calabi–Yau”, The Fano Conference, University Torino, Turin, 2004, 701–734, arXiv: math.AG/0302101 | MR | Zbl