Analytic continuation of better-behaved GKZ systems and Fourier-Mukai transforms

We study the relationship between solutions to better-behaved GKZ hypergeometric systems near different large radius limit points, and their geometric counterparts given by the $K$-groups of the associated toric Deligne-Mumford stacks. We prove that the $K$-theoretic Fourier-Mukai transforms associated to toric wall-crossing coincide with analytic continuation transformations of Gamma series solutions to the better-behaved GKZ systems, which settles a conjecture of Borisov and Horja.

💡 Research Summary

The paper investigates the precise relationship between solutions of the better‑behaved GKZ (bbGKZ) hypergeometric systems at different large‑radius limit points and the K‑theoretic data of the associated toric Deligne–Mumford stacks. The authors begin by recalling the definition of bbGKZ systems, which modify the original Gel’fand‑Kapranov‑Zelevinsky construction so that the solution space always has the expected dimension, thereby avoiding the rank‑jumping phenomenon that plagues the classical GKZ equations. This property ensures that the natural “mirror‑symmetry” maps from the (complexified) K‑group of a toric stack to the space of solutions are genuine isomorphisms for all parameters.

The geometric setting is a rational polyhedral cone (C\subset N_{\mathbb R}) together with a set of degree‑one lattice generators ({v_i}{i=1}^n). From this data one builds a simplicial fan (\Sigma) and the smooth toric Deligne–Mumford stack (\mathcal P\Sigma). The paper reviews the combinatorial description of twisted sectors (Box((\Sigma))), orbifold cohomology, and the Grothendieck group (K_0(\mathcal P_\Sigma)). In particular, the K‑group is generated by classes of line bundles (L_i) (or their symbols (R_i)) subject to linear relations coming from the lattice (N^\vee) and Stanley‑Reisner type ideals.

The core of the work concerns two adjacent regular triangulations (\Sigma^+) and (\Sigma^-) of the cone (C). Their adjacency is encoded by a circuit (I) and an integral relation (h=(h_1,\dots,h_n)) satisfying (\sum_{i\in I} h_i v_i=0). The authors compute the analytic continuation of the Gamma‑series solutions of bbGKZ((C,0)) across the wall separating the two triangulations using Mellin–Barnes integrals. The continuation amounts to adding a rational multiple of the vector (h) to the exponent vectors that label the series, which precisely matches the change of variables dictated by the wall‑crossing.

On the K‑theoretic side, the wall‑crossing is realized by a toric flop: there exists a common blow‑up (\widehat{\mathcal P}) of the two stacks, and the derived equivalence between (\mathcal P_{\Sigma^+}) and (\mathcal P_{\Sigma^-}) is given by a Fourier–Mukai transform (\mathrm{FM}:K_0(\mathcal P_{\Sigma^+})\to K_0(\mathcal P_{\Sigma^-})). The transform acts on the generators (R_i) by the explicit formulas derived from the geometry of the flop, which involve inverting the classes corresponding to the rays that disappear and inserting the product of the remaining classes raised to the powers (h_j).

The main theorem (Theorem 1.2, restated as Theorems 4.5 and 5.2) asserts that the following diagram commutes:

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