The Doran—Harder—Thompson conjecture and period integrals

Let X be a Calabi-Yau manifold that admits a Tyurin degeneration to a union of two quasi-Fano varieties X_1 and X_2 intersecting along a smooth anticanonical divisor D. Doran—Harder—Thompson predicted that the Landau—Ginzburg mirrors of these two quasi-Fano varieties can be glued together to obtain the mirror of the Calabi—Yau variety which admits a fibration structure over P^1. We study this conjecture in the context of enumerative mirror symmetry which relates Gromov—Witten invariants with period integrals. In this talk, I will explain how the fibration structure implies the relation among period integrals of the mirrors of X, (X_1,D) and (X_2,D). This is based on joint work with Charles Doran and Jordan Kostiuk.