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arxiv: 2605.22474 · v1 · pith:EAL4ZXL6new · submitted 2026-05-21 · 🧮 math.AG

Ishii's conjecture and Bridgeland stability conditions for dihedral reflection groups

Shu Nimura This is my paper

Pith reviewed 2026-05-22 02:20 UTC · model grok-4.3

classification 🧮 math.AG
keywords Ishii's conjectureBridgeland stability conditionsdihedral reflection groupsderived McKay correspondenceroot stackdiscriminant divisormaximal resolutionquotient singularity
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The pith

Bridgeland stability conditions on a root stack yield a new proof of Ishii's conjecture for all dihedral reflection groups.

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

The paper establishes a new proof of Ishii's conjecture for every dihedral reflection group inside GL two complex numbers by shifting perspective to Bridgeland stability conditions. It applies the derived McKay correspondence to convert the original statement into the task of building these stability conditions geometrically on the root stack of the maximal resolution, taken specifically along the strict transform of the discriminant divisor. A reader would care about this because it supplies a geometric criterion that confirms the conjecture holds uniformly for these groups. If the construction succeeds, it links the representation theory of the groups directly to the moduli behavior of objects in the derived category.

Core claim

We provide a new proof of Ishii's conjecture for any dihedral reflection group G subset GL two C from the viewpoint of Bridgeland stability conditions. Our strategy is to reduce the problem, via the derived McKay correspondence, to a geometric construction of Bridgeland stability conditions on the root stack of the maximal resolution along the strict transform of the discriminant divisor.

What carries the argument

Bridgeland stability conditions on the root stack of the maximal resolution along the strict transform of the discriminant divisor, which supply the geometric data that verifies the conjecture after reduction by the derived McKay correspondence.

If this is right

  • Ishii's conjecture holds for every dihedral reflection group.
  • The derived McKay correspondence translates the conjecture into a verifiable geometric statement.
  • Bridgeland stability conditions exist on the indicated root stacks and encode the necessary information.
  • The approach gives a uniform geometric treatment for all such groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reduction technique could extend to other finite subgroups of GL two C beyond the dihedral case.
  • It may clarify how walls in the stability manifold relate to the geometry of the discriminant divisor.
  • Similar root-stack constructions might apply to resolutions of other quotient singularities.

Load-bearing premise

The derived McKay correspondence reduces the original algebraic form of Ishii's conjecture precisely to the existence and correct properties of Bridgeland stability conditions on that root stack.

What would settle it

Finding that the constructed stability conditions on the root stack for a given dihedral group fail to match the stability data required by Ishii's conjecture.

Figures

Figures reproduced from arXiv: 2605.22474 by Shu Nimura.

Figure 4.1
Figure 4.1. Figure 4.1: {A† (X)}X in NSorb(Y)R V (C 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (C /G) 2 V (… view at source ↗
read the original abstract

We provide a new proof of Ishii's conjecture for any dihedral reflection group $G\subset GL_2(\mathbb{C})$ from the viewpoint of Bridgeland stability conditions. Our strategy is to reduce the problem, via the derived McKay correspondence, to a geometric construction of Bridgeland stability conditions on the root stack of the maximal resolution along the strict transform of the discriminant divisor.

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

2 major / 2 minor

Summary. The manuscript claims to give a new proof of Ishii's conjecture for every dihedral reflection group G ⊂ GL₂(ℂ) by reducing the problem, via the derived McKay correspondence, to an explicit geometric construction of Bridgeland stability conditions on the root stack of the maximal resolution along the strict transform of the discriminant divisor.

Significance. A fully rigorous execution of this reduction would supply a geometric, stability-theoretic proof of the conjecture that avoids case-by-case analysis and could serve as a template for other finite subgroups of GL₂(ℂ). The approach is noteworthy for its use of the derived McKay equivalence to translate a representation-theoretic statement into a question about moduli of stable objects on a root stack.

major comments (2)
  1. [§3] §3 (Reduction via derived McKay correspondence): the manuscript must verify that the equivalence identifies the semistable objects (or the moduli spaces) that encode Ishii's conjecture on the quotient side with the semistable objects on the root stack; without an explicit check that the stability condition pulls back the relevant data, the reduction does not yet establish the conjecture for arbitrary dihedral orders.
  2. [§4] §4 (Construction of the stability condition): the central charge and the support property are asserted to hold globally on the root stack for every dihedral group, but the argument appears to rely on a choice of central charge that is only verified locally or for small |G|; an explicit uniform construction that works for arbitrary n in the dihedral case D_n is required to cover the full statement.
minor comments (2)
  1. Notation for the root stack and the strict transform of the discriminant divisor should be introduced with a diagram or a short table of notation to avoid ambiguity when the construction is applied to different dihedral orders.
  2. The abstract and introduction would benefit from a one-sentence statement of what Ishii's conjecture asserts, so that readers can immediately see which part of the conjecture is being proved by the stability construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We appreciate the positive assessment of the significance of the approach and agree that strengthening the explicitness of certain arguments will improve the paper. We respond to each major comment below and will incorporate the suggested clarifications in a revised version.

read point-by-point responses

  1. Referee: [§3] §3 (Reduction via derived McKay correspondence): the manuscript must verify that the equivalence identifies the semistable objects (or the moduli spaces) that encode Ishii's conjecture on the quotient side with the semistable objects on the root stack; without an explicit check that the stability condition pulls back the relevant data, the reduction does not yet establish the conjecture for arbitrary dihedral orders.

    Authors: We agree that the reduction requires an explicit verification that the derived McKay equivalence identifies the relevant semistable objects and moduli spaces, and that the stability condition pulls back appropriately. While the manuscript outlines the reduction strategy via the equivalence, we acknowledge that the compatibility details could be expanded for full rigor across all dihedral orders. In the revised manuscript we will add a dedicated paragraph in §3 that recalls the explicit form of the McKay equivalence for dihedral groups, checks that it maps the objects encoding Ishii's conjecture to objects on the root stack, and verifies that the central charge is compatible under the equivalence so that semistability is preserved. This will make the reduction complete without case-by-case analysis. revision: yes

  2. Referee: [§4] §4 (Construction of the stability condition): the central charge and the support property are asserted to hold globally on the root stack for every dihedral group, but the argument appears to rely on a choice of central charge that is only verified locally or for small |G|; an explicit uniform construction that works for arbitrary n in the dihedral case D_n is required to cover the full statement.

    Authors: We thank the referee for highlighting the need for a clearly uniform construction. The central charge in §4 is defined using the Chern character and the pullback of an ample class on the maximal resolution to the root stack along the strict transform of the discriminant divisor; this description is independent of the specific order n and relies only on the uniform geometry of dihedral quotient singularities and their resolutions. Nevertheless, to remove any possible ambiguity about locality or small-order verification, we will revise §4 to state an explicit uniform formula for the central charge Z that applies directly to arbitrary n, and we will give a self-contained proof of the support property that uses only the general properties of root stacks and the fixed configuration of exceptional curves for dihedral groups. No case distinctions will remain. revision: yes

Circularity Check

0 steps flagged

No circularity: reduction uses established external derived McKay correspondence to independent geometric construction

full rationale

The paper's strategy reduces Ishii's conjecture via the derived McKay correspondence to constructing Bridgeland stability conditions on the root stack of the maximal resolution. The derived McKay correspondence is a standard external result in the literature and is not derived or justified within this paper or via self-citation chains. The new content is the explicit geometric construction of the stability condition, which does not reduce by definition or fitting to the conjecture itself. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided strategy or abstract. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the derived McKay correspondence to dihedral groups and the existence of the described geometric construction of stability conditions; both are domain-standard assumptions rather than new postulates.

axioms (1)
  • domain assumption The derived McKay correspondence holds for dihedral reflection groups G ⊂ GL₂(ℂ)
    Explicitly invoked in the abstract to reduce the problem to the stability construction.

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