Toric residues and mirror symmetry

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture, which has close relations to toric mirror symmetry. Our conjecture, we call it the toric residue mirror conjecture, is that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidence for that. The proposed conjecture suggests a simple method for computing Yukawa couplings for toric mirror Calabi–Yau hypersurfaces without solving systems of differential equations. We make several explicit computations for Calabi–Yau hypersurfaces in weighted projective spaces and in products of projective spaces.