### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 024, 13 pages      arXiv:1612.04623      https://doi.org/10.3842/SIGMA.2017.024
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

### Doran-Harder-Thompson Conjecture via SYZ Mirror Symmetry: Elliptic Curves

Atsushi Kanazawa
Department of Mathematics, Kyoto University, Kitashirakawa-Oiwake, Sakyo, Kyoto, 606-8502, Japan

Received December 20, 2016, in final form April 06, 2017; Published online April 11, 2017

Abstract
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.

Key words: Calabi-Yau manifolds; Fano manifolds; SYZ mirror symmetry; Landau-Ginzburg models; Tyurin degeneration; affine geometry.

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