Toric Residue Mirror Conjecture for Calabi-yau Complete Intersections

The toric residue mirror conjecture of Batyrev and Materov [2] for Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties expresses a toric residue as a power series whose coefficients are certain integrals over moduli spaces. This conjecture was proved independently by Szenes and Vergne [10] and Borisov [5]. We build on the work of these authors to generalize the residue mirror map to not necessarily reflexive polytopes. Using this generalization we prove the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein toric Fano varieties [3]. We start by introducing notation and explaining the main idea of the generalization. We work over the filed K = Q. Let M ≃ Z, let ∆ ⊂ MK be a d-dimensional lattice polytope, and let T be a coherent triangulation of ∆, defined by a convex piecewise linear integral function on ∆. All lattice points in ∆ are assumed to be vertices of the simplices in T . We place ∆ in MK = (M × Z)K as ∆× {1} and let C∆ ⊂ MK be the cone over ∆ with vertex 0. Then T defines a subdivision of C∆ into a fan Σ. The idea of the toric residue mirror conjecture is to relate the semigroup ring S∆ = K[C∆ ∩ M ] to the cohomology of the fan Σ. Let I∆ ⊂ S∆ be the ideal generated by monomials t where m ∈ M lies in the interior of C∆. Given general elements f0, . . . , fd ∈ S ∆ (the superscript denotes the degree), we can construct the toric residue map [9]: Res(f0,...,fd) : (I∆/(f0, . . . , fd)I∆) d+1 ∼ → K.