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Pinot's Theorem 1.2 lists the surfaces 𝔐δ and 𝔐2δ for ten families of affine orthogonal quivers. This note compares each surface with the classical quiver variety of the same graph, and lifts each one to its link, a contact 3-manifold S³/Γ.
Whenever 𝔐δ is a surface, it equals the classical Kleinian surface of the underlying affine graph. The proof uses only Li's closed immersion (Prop. 9.2.1), not the isomorphism stated in Pinot's Remark 2.15.
Not Dn+1. For (D̃₂ₙ₋₁, a) the relation is z² = z(−x)ⁿ⁻¹ − xy²: the printed version has y in place of z in the first term. A direct numerical computation of the invariant ring confirms both rows for n = 3, 4, 5.
In three rows Γ2δ contains the classical group with index 2 (ℤ₄ₙ₋₂ ⊃ ℤ₂ₙ₋₁, ℤ₄ₙ ⊃ ℤ₂ₙ, BD₄ₙ ⊃ ℤ₂ₙ), and in the fourth they are equal. Conjecture: each 2δ surface is the classical surface divided by an involution.
The symmetric representation space is quaternionic, so the link is the Kempf–Ness / 3-Sasakian quotient of its unit sphere. Pinot's weights match the Klein invariants, so the Reeb flow is the Hopf flow: lens spaces over S²(m,m) (m odd) or S²(m/2,m/2) (m even), and prism manifolds over S²(2,2,k).
| Quiver | Sign | d | Equation | Type | Γ |
|---|---|---|---|---|---|
| (Ã₂ₙ₋₂, v-a) | +1 | δ | xy = z²ⁿ⁻¹ | A₂ₙ₋₂ | ℤ₂ₙ₋₁ |
| (Ã₂ₙ₋₂, v-a) | −1 | 2δ | xy = z⁴ⁿ⁻² | A₄ₙ₋₃ | ℤ₄ₙ₋₂ |
| (Ã₂ₙ₋₁, a-a) | (+,+) | δ | xy = z²ⁿ | A₂ₙ₋₁ | ℤ₂ₙ |
| (Ã₂ₙ₋₁, a-a) | (+,−) | 2δ | xy = z⁴ⁿ | A₄ₙ₋₁ | ℤ₄ₙ |
| (Ã₂ₙ₋₁, a-a) | (−,−) | 2δ | xy = z²ⁿ | A₂ₙ₋₁ | ℤ₂ₙ |
| (Ã₂ₙ₋₁, v-v) | δ | xy = z²ⁿ | A₂ₙ₋₁ | ℤ₂ₙ | |
| (Ã₂ₙ₋₁, c) | 2δ | xⁿ⁺¹ − y²x − z² = 0 | Dn+2 | BD₄ₙ | |
| (D̃₂ₙ₋₂, v) | δ | z² = −y(−x)ⁿ⁻¹ − xy² | D₂ₙ₋₂Dn+1 | BD8(n−2) | |
| (D̃₂ₙ₋₁, a) | +1 | δ | z² = z(−x)ⁿ⁻¹ − xy² | D₂ₙ₋₁Dn+1 | BD4(2n−3) |
Highlighted rows are corrected; the struck-through label is the one printed in arXiv:2609.39434. All other rows are reduced points.
Python 3 with numpy, scipy and sympy, and Lean 4 core (no Mathlib). The page's own check is index-verify.py: it runs in seconds and reads the saved output of the ten-minute numerical computation; add --full to recompute it.
python3 index-verify.pypython3 -m pip install numpy scipy sympypython3 index-verify.py --fulllean scripts/DM3Quivers_Check.leanindex-verify.py — the page's check of the algebra and group data above, numbered blocks, PASS/FAIL; it does not establish the geometric conclusionsscripts/verify_Dtilde_relation_numeric.py · .out — the invariant ring computed from random points of μ⁻¹(0)scripts/verify_ade_types.py — Milnor numbers of the D̃-row polynomialsscripts/verify_2delta_index2.py — the 2δ ring identitiesscripts/verify_quaternionic_SRep.py — σ commutes with the quaternionic structurescripts/verify_seifert.py — Seifert data and group ordersscripts/DM3Quivers_Check.lean — Lean 4 kernel check of the arithmetic onlyEN.tex · PT.tex · FR.tex — LaTeX sources (XeLaTeX)