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

Symmetry Integrability and Geometry-Methods and Applications

article

Atsushi Kanazawa¹ 

[1] Department of Mathematics, Kyoto University

关键词: Calabi–Yau manifolds;    Fano manifolds;    SYZ mirror symmetry;    Landau–Ginzburg models;    Tyurin degeneration;    af fine geometry;   

DOI  :  10.3842/SIGMA.2017.024

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown