Borisov–Horja duality conjecture for GKZ hypergeometric systems

Let $\Delta$ be a convex polytope in a lattice $N_1$, set $N=N_1\oplus\mathbb Z$, and let $C=\mathbb R_{\geq 0}(\Delta,1)$. Let $v_1,\ldots,v_n$ be lattice points of $(\Delta,1)$ containing all its vertices. The associated systems of linear partial differential equations are denoted by $bbGKZ(C,0)$ and $bbGKZ(C^{\circ},0)$, with solution families $(\Phi_c)$ and $(\Psi_d)$ indexed by $c\in C$ and $d\in C^{\circ}$, respectively. For a projective simplicial subdivision $\Sigma$ of $C$, write $K_0(\mathbb P_{\Sigma})$ and $K^c_0(\mathbb P_{\Sigma})$ for the corresponding ordinary and compactly supported $K$-groups, and let $\Gamma$ and $\Gamma^{\circ}$ denote the associated Gamma-series isomorphisms.

Borisov–Horja duality conjecture. There exists a collection of polynomials $p_{c,d}(x_1,\ldots,x_n)$, indexed by $c\in C$ and $d\in C^{\circ}$, such that only finitely many are nonzero; for every pair of solutions $(\Phi_c)$ of $bbGKZ(C,0)$ and $(\Psi_d)$ of $bbGKZ(C^{\circ},0)$, the expression

$$\sum_{c,d}p_{c,d}\Phi_c\Psi_d$$

is constant as a function of $(x_1,\ldots,x_n)$; the resulting pairing is non-degenerate; and, for every projective simplicial subdivision $\Sigma$, this pairing is the inverse of the Euler characteristics pairing between $K_0(\mathbb P_{\Sigma})$ and $K^c_0(\mathbb P_{\Sigma})$ under $\Gamma$ and $\Gamma^{\circ}$.

The conjecture describes the expected duality between the two GKZ systems and the corresponding ordinary and compactly supported $K$-theory spaces, thereby expressing the isotrivial-family prediction for complexified $K$-groups. The source paper presents it as a conjecture of Borisov and Horja and proves the duality in the two-dimensional case, while no general resolution is supplied here.