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arxiv: 2305.12241 · v4 · pith:ZO7GICWJnew · submitted 2023-05-20 · 🧮 math.AG

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

Zengrui Han This is my paper

Pith reviewed 2026-05-24 09:33 UTC · model grok-4.3

classification 🧮 math.AG
keywords GKZ hypergeometric systemsanalytic continuationFourier-Mukai transformstoric Deligne-Mumford stacksK-theorywall-crossingGamma series
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The pith

K-theoretic Fourier-Mukai transforms for toric wall-crossings coincide with analytic continuations of Gamma series solutions to better-behaved GKZ systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that analytic continuation maps between Gamma series solutions of better-behaved GKZ hypergeometric systems at different large radius limit points are identical to the K-theoretic Fourier-Mukai transforms induced by toric wall-crossings. These transforms act on the K-groups of associated toric Deligne-Mumford stacks, which realize the solution spaces geometrically. The identification confirms that moving between chambers via analysis matches the geometric equivalences from derived categories. This directly settles the Borisov-Horja conjecture for the systems considered.

Core claim

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. The K-groups of the toric Deligne-Mumford stacks serve as geometric counterparts to the solution spaces near the large radius limit points.

What carries the argument

Better-behaved GKZ hypergeometric systems, whose Gamma series solutions have analytic continuation transformations identified with K-theoretic Fourier-Mukai transforms under toric wall-crossing.

Load-bearing premise

The K-groups of the toric Deligne-Mumford stacks provide geometric counterparts to the solution spaces of the better-behaved GKZ systems near large radius limit points.

What would settle it

An explicit matrix computation for a concrete toric Deligne-Mumford stack showing that the Fourier-Mukai action on K-theory differs from the analytic continuation operator on the corresponding Gamma series solutions.

Figures

Figures reproduced from arXiv: 2305.12241 by Zengrui Han.

Figure 1
Figure 1. Figure 1: Path of analytic continuation Remark 3.3. The restriction on the argument of the variable y is imposed to avoid introducing monodromy during the process of analytic continuation. The main idea comes from [4] where the technique of Mellin-Barnes in￾tegrals is used to compute the analytic continuation for the usual GKZ systems. The main difference is that while they worked with K-theory￾valued solutions, we … view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the K-theoretic Fourier-Mukai transforms associated to toric wall-crossings coincide with the analytic continuation transformations of Gamma series solutions to better-behaved GKZ hypergeometric systems. This equates the linear transformations on solution spaces near distinct large-radius limit points with the action of wall-crossing kernels on the K-groups of the associated toric Deligne-Mumford stacks, thereby settling the Borisov-Horja conjecture.

Significance. If the identification holds, the result supplies a geometric realization, via K-theory of toric DM stacks, for the analytic continuation matrices of Gamma-series solutions to GKZ systems. This directly confirms a conjecture in the literature and strengthens the dictionary between hypergeometric periods and derived categories in toric mirror symmetry. The paper's explicit construction of the stacks so that their K-groups match the solution spaces near large-radius points is a clear strength.

minor comments (2)
  1. [§2] The notation for the 'better-behaved' GKZ system and the precise definition of the Gamma series could be recalled in a single preliminary subsection for readers who consult only the main theorems.
  2. [§5] A short table summarizing the wall-crossing data (fan, stack, and corresponding GKZ parameters) for the examples treated in §5 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the identification between K-theoretic Fourier-Mukai transforms and analytic continuation of Gamma-series solutions is viewed as confirming the Borisov-Horja conjecture and strengthening the dictionary in toric mirror symmetry.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper proves that K-theoretic Fourier-Mukai transforms for toric wall-crossings coincide with analytic continuation transformations of Gamma-series solutions to better-behaved GKZ systems, thereby settling a conjecture of Borisov and Horja. The abstract and description frame this as an identification between independently constructed objects (K-groups of toric DM stacks providing geometric bases near large-radius points, and analytic continuations of hypergeometric solutions), with no quoted equations or steps reducing the claimed equality to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The construction of the stacks to match solution spaces is presented as a setup enabling the proof rather than a tautological input, and the result is externally falsifiable against the cited conjecture without internal reduction to the paper's own fitted values.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the paper relies on standard properties of GKZ systems and toric DM stacks without introducing new free parameters or entities visible here.

axioms (2)
  • standard math GKZ hypergeometric systems admit Gamma series solutions near large radius limit points
    Invoked implicitly when discussing analytic continuation of solutions
  • domain assumption Toric Deligne-Mumford stacks have well-defined K-groups that correspond to solution spaces of associated GKZ systems
    Central to the geometric counterpart stated in the abstract

pith-pipeline@v0.9.0 · 5595 in / 1303 out tokens · 27195 ms · 2026-05-24T09:33:11.303705+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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