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

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.

References in corpus (4)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Supersymmetric partition functions on Riemann surfaces
• Local Calabi-Yau manifolds of type \tilde{A} via SYZ mirror symmetry
• Special Lagrangian fibrations, mirror symmetry and Calabi-Yau double covers

Cited by in corpus (4)

• Degenerations and Lagrangian fibrations of Calabi-Yau manifolds
• Convergence of the mirror to a rational elliptic surface
• Moduli Space in Homological Mirror Symmetry
• Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program