Mixed Toric Residues and Tropical Degenerations

This paper is a follow-up to our paper [19], where we prove a conjecture of Batyrev and Materov, the Toric Residue Mirror Conjecture (TRMC). Here we extend our results, and show that they imply a generalization of this conjecture, the Mixed Toric Residue Mirror Conjecture (MTRMC), which is also due to Batyrev and Materov [3]. Roughly, these conjectures state that the generating function of certain intersection numbers of a sequence of toric varieties converges to a rational function, which can be obtained as a finite residue sum on a single toric variety. We first recall the TRMC in some detail. We start with an integral convex polytope Π in a d-dimensional real vector space t endowed with a lattice of full rank tZ; we assume that the polytope contains the origin in its interior. Let the sequence B = [β1, β2, . . . , βn] be the set of vertices of this polytope, ordered in an arbitrary fashion. One can associate a d-dimensional polarized toric variety (V , L) to this data in the standard fashion [19]. There is another way to obtain toric varieties from this data. Consider the sequence A = [α1, α2, . . . , αn], which is the Gale dual of B (cf. §1.3 for the construction). This is a sequence of integral vectors in the dual a∗ of a certain r = n − d-dimensional vector space a, which is also endowed with a lattice of full rank: aZ; in this setup the sequence A spans a strictly convex cone Cone(A). The simplicial cones generated by A divide Cone(A) into open chambers. Each chamber corresponds to a d-dimensional orbifold toric variety VA(c) (cf. [9]). An integral element α ∈ a∗ specifies an orbiline-bundle Lα over this variety; denote the first chern class of Lα by χ(α) ∈ H(VA(c),Q). For the purposes of this introduction we assume that this correspondence induces the linear isomorphisms a∗ ∼= H(VA(c),R) and a ∼= H2(VA(c),R) Now pick a chamber c which contains the vector κ = ∑n i=1 αi in its closure: κ ∈ c̄. To each element λ ∈ aZ, one can associate a moduli space MPλ, the so-called Morrison-Plesser space, which is a naive compactification of the space of those maps from the projective line to the variety VA(c) under which the image of the fundamental class is λ: