Toric Residue Mirror Conjecture

Let $\Delta \subset M_{\mathbb R}$ be an arbitrary reflexive $d$-dimensional polytope and let $A$ be a finite subset of $\Delta \cap M$ containing $0$ and all vertices of $\Delta$. Choose a coherent triangulation ${\mathcal T}=\{\tau_1,\ldots,\tau_k\}$ of $\Delta$ associated with $A$ such that $0$ is a vertex of every simplex $\tau_i$. Let ${\mathbb P}={\mathbb P}_{\Sigma({\mathcal T})}$ be the simplicial toric variety defined by the fan $\Sigma=\Sigma({\mathcal T})\subset M_{\mathbb R}$, whose $d$-dimensional cones are $\sigma_i=\mathbb R_{\geq 0}\tau_i$. Write $A=\{v_0=0,v_1,\ldots,v_n\}$ and set

$$f(t):=1-\sum_{i=1}^n a_i t^{v_i}.$$

For a homogeneous polynomial $P(x_1,\ldots,x_n)\in\mathbb Q[x_1,\ldots,x_n]$ of degree $d$, define the toric residue

$$R_P(a):=(-1)^d\,\operatorname{Res}_f\bigl(t_0^dP(a_1t^{v_1},\ldots,a_nt^{v_n})\bigr).$$

Toric Residue Mirror Conjecture. The Laurent expansion of $R_P(a)$ at the vertex $v_{\mathcal T}\in\operatorname{Sec}(A)$ corresponding to ${\mathcal T}$ coincides with

$$I_P(a):=\sum_{\beta\in K_{\rm eff}({\mathbb P})}I(P,\beta)a^\beta,$$

where the sum is over all integral points $\beta=(b_1,\ldots,b_n)$ of the Mori cone $K_{\rm eff}({\mathbb P})$, $a^\beta=a_1^{b_1}\cdots a_n^{b_n}$, and

$$I(P,\beta)=\int_{{\mathbb P}_\beta}P([D_1],\ldots,[D_n])\Phi_\beta=\langle P([D_1],\ldots,[D_n])\Phi_\beta\rangle_\beta.$$

Here $\Phi_\beta\in H^{2(\dim {\mathbb P}_\beta-d)}({\mathbb P}_\beta,\mathbb Q)$ is the Morrison–Plesser class of ${\mathbb P}_\beta$, and $I(P,\beta)$ is assumed to be zero when ${\mathbb P}_\beta$ is empty. The conjecture proposes a mirror correspondence between toric residues and generating functions of intersection numbers; its general status is not resolved in the supplied text.