On the hyperplane conjecture

Source abstract

We prove the hyperplane conjecture, proposed by Hosono, Lian, and Yau in arXiv:alg-geom/9511001 and its generalization to orbifolds. Consequently, this establishes the expected equivalence between the $A$ model (genus-zero Gromov--Witten invariants with ambient insertions) of $X$ and the $B$ model (period integrals corresponding to vanishing cohomology) of $X^{\vee}$ for Batyrev's mirror pair of Calabi--Yau hypersurfaces. As an application of our method, we also identify the residue of the Gauss--Manin connection and the cup product morphism on the $A$-side.