Vanishing arcs for isolated plane curve singularities

Cheol-Hyun Cho; Dongwook Choa; Hanwool Bae; Pablo Portilla Cuadrado; Wonbo Jeong

REVIEW 3 major objections 5 minor 30 references

Vanishing arcs for isolated plane curve singularities

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Vanishing arcs reproduce A'Campo's vanishing cycles.

desk verdict A genuinely new arc-level refinement of vanishing cycles with a substantial A'Campo divide construction, but the main 'geometric' theorem is stated beyond the hypotheses its proof supports for A_n and D_n singularities. read the letter →

arxiv 2506.04917 v1 pith:7BX4PD77 submitted 2025-06-05 math.GT

Hanwool Bae , Cheol-Hyun Cho , Dongwook Choa , Wonbo Jeong , Pablo Portilla Cuadrado This is my paper

classification math.GT MSC 32S2557K2014H2032S40

keywords vanishingarcsvariationoperatorMilnorfiberA'Campodividegeometriccyclestopologicalexceptionalcollectionplanecurvesingularitiesarcsets

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces vanishing arcsets, finite collections of disjoint properly embedded arcs in the Milnor fiber of an isolated plane curve singularity, as geometric relative-homology counterparts of vanishing cycles. It proves that a single arc is sent to a geometric vanishing cycle by the geometric variation operator exactly when the arc and its image under geometric monodromy can be made disjoint in the interior, for singularities outside the $A_n$ and $D_n$ families with Milnor fiber genus at least 5. For collections of arcs, the obstruction disappears: an arcset's variation image is a geometric vanishing cycle precisely when it is a single non-separating simple closed curve. The paper's main construction shows that for every A'Campo divide of a totally real plane curve singularity there exists a topological exceptional collection of geometric vanishing arcsets whose variation images are isotopic to A'Campo's distinguished collection of vanishing cycles.

What carries the argument

The load-bearing objects are the geometric variation operator, which forms a closed curve from an arc and its monodromy image, and the arcset, a finite disjoint collection of properly embedded arcs whose surgery produces a simple closed curve. The combinatorial engine is the A'Campo-Gusein-Zade diagram $\mathrm{A}\Gamma(D_f)$ of a divide, whose vertices are double points and signed bounded regions of the divide and whose edges record adjacencies. For each vertex, the paper chooses a good path descending through depths to an outer vertex and assigns one basic arc to each edge of the path; the nesting and disjointness of good paths, together with intersection computations between basic arcs and their monodromy images, arrange the arcsets into a topological exceptional collection.

What would settle it

Take any A'Campo divide, draw its A'Campo-Gusein-Zade diagram, and check whether every depth-$k$ vertex is adjacent to a depth-$(k-1)$ vertex of the required sign; a single counterexample invalidates Lemma 11.2 and with it the universal form of Theorem 7.7.

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Extended reading notes

Core claim

The paper's central claim is that the classical data of vanishing cycles around an isolated plane curve singularity can be lifted to relative homology in a geometrically faithful way. The geometric variation operator $\operatorname{Var}_f(a) = \varphi_f(a) * (-a)$ sends properly embedded arcs to closed curves, and the paper characterizes when such an image is a geometric vanishing cycle purely by the intersection number $i(a, \varphi_f(a)) = 0$. For the more flexible notion of an arcset, the only requirement is that the variation image be a single non-separating simple closed curve. The article then proves that, given any A'Campo divide, one can choose good paths in the associated A'Campo-Gusein-Zade diagram and attach a basic arc to each edge so that the resulting ordered collection of arcsets is adapted to A'Campo's distinguished vanishing cycles and the geometric variation images are isotopic to them. This is the content of Theorem 7.7.

Load-bearing premise

The construction assumes a combinatorial property of A'Campo-Gusein-Zade diagrams: every vertex of depth $k$ has an edge to a vertex of depth $k-1$ with the prescribed sign ($+$ to $-$ or $-$ to $+$, and $0$ to either), and the resulting good paths are disjoint or nested; this property is stated without proof and the arcset construction collapses if it fails.

Editorial extensions

If this is right

For any A'Campo divide, the paper constructs an ordered family of disjoint arc collections, one per vertex of the A'Campo-Gusein-Zade diagram, whose geometric variation images are isotopic to A'Campo's distinguished vanishing cycles.
A single properly embedded arc is a geometric vanishing arc exactly when $i(a, \varphi_f(a)) = 0$, for singularities outside the $A_n$ and $D_n$ families with Milnor fiber genus at least 5.
Any disjoint arc collection whose variation image is one non-separating simple closed curve is a geometric vanishing arcset, and no further condition is needed.
A separating simple closed curve can never be the geometric variation image of a single arc, so the geometric variation operator is not a bijection even though the classical homology variation operator is an isomorphism.
The Seifert form is non-degenerate and triangular on the basis $\{\operatorname{Var}_f(K_1), \dots, \operatorname{Var}_f(K_\mu)\}$, matching the algebraic structure of an exceptional collection.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the same good-path combinatorics holds for diagrams arising from non-totally-real real plane curves, the construction would extend whenever a real Morsification exists; the paper notes that existence is still open in general.
The characterization turns a search problem, finding an arc whose variation image is a given vanishing cycle, into an intersection-number check on one arc against its monodromy image, which is directly computable in explicit models such as Brieskorn-Pham fibers.
The linear ordering imposed on arcsets is a topological counterpart of an exceptional collection in a triangulated category; a testable consequence is that braid group actions on distinguished vanishing cycles lift to braid actions on the corresponding arcsets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Hanwool et al. have a real new idea here: promoting the variation operator from homology to actual arcs and arcsets, and characterizing when the geometric variation image is a vanishing cycle. Theorem 5.1 is a genuine characterization under the stated hypotheses (not A_n/D_n, genus at least 5), and the depth construction for A'Campo divides in Sections 10-11 is substantial. The topological exceptional collection result (Theorem 7.7) is the kind of concrete, checkable statement that the Fukaya-Seidel/monodromy Fukaya category conjectures need.

The soft spots are real, and one is load-bearing. Theorem 5.5 is stated without the restrictions of Theorem 5.1, but its proof relies on the same PCS21a machinery that the authors themselves say fails for A_n and D_n and is technically unproven for genus < 5. Lemma 7.2 unconditionally uses Theorem 5.5, so linear arcsets are asserted geometric for all divides, including A_n and D_n. Since A'Campo divides exist for all plane curve singularities, the 'geometric' part of Theorem 7.7 is not actually proved for A_n/D_n. This is not a fatal flaw—the construction may well be correct—but the theorem as stated overreaches its proof. The fix is either to carry the PCS21a restrictions through the paper and state Theorem 7.7 for the complementary class, or to prove 5.5 without the excluded singularities using the alternative criteria in Remark 5.2.

The other gaps are smaller. The converse of Theorem 5.1 has a sketched bigon argument; I believe it, but 'can be made into a homotopy fixing the boundary pointwise' needs a few lines. Definition 11.1's good-path existence is a combinatorial assertion about AΓ(D_f) diagrams that is stated without proof in Lemma 11.2; it is probably true by nesting of a divide, but it is load-bearing for the arcset construction. And the final isotopy check in Section 11 is shown for one example and then hand-waved by 'local situations are always the same.' That may be fine for a research announcement, less so for a final version.

Bottom line: this is a strong paper that deserves a serious referee. I would send it out, but with the expectation that the A_n/D_n gap and the good-path lemma be addressed before acceptance. The intended audience is singularity theorists and symplectic geometers working on monodromy Fukaya categories. I'd cite it once the hypotheses match the statements.

Referee Report

3 major / 5 minor

Summary. The paper introduces a geometric variation operator Varf that takes properly embedded arcs in the Milnor fiber of an isolated plane curve singularity to closed curves, and defines vanishing arcsets as collections of arcs whose variation image is a geometric vanishing cycle. The main results are: (1) Theorem 5.1, a characterization, under restrictions excluding An and Dn and requiring genus at least 5, of when a single arc is a geometric vanishing arc in terms of the intersection number i(a, φf(a)); (2) Theorem 5.5, an extension to arcsets claiming that the variation image is a geometric vanishing cycle exactly when the surgered collection is a single non-separating simple closed curve; (3) Theorem 7.7, which asserts that for any A'Campo divide there is a topological exceptional collection of geometric vanishing arcsets whose variation images are isotopic to A'Campo's distinguished collection of vanishing cycles. The proof of Theorem 7.7 proceeds by choosing 'good paths' in the A'Campo–Gusein-Zade diagram, associating basic arcs to edges, grouping them into arcsets, and checking adaptedness and linearity.

Significance. If the results are correct, the paper gives a new relative-homology refinement of the classical vanishing-cycle picture: arcs and arcsets provide geometric representatives for the inverse variation operator, and the construction from A'Campo divides is explicit and potentially useful for symplectic and Fukaya-categorical applications. The intersection-number characterization in Theorem 5.1 is a clean criterion, and the higher-depth construction in Sections 10–11 is a genuine extension of the adaption framework from [BCCJ23]. The paper is also honest in Remark 5.2 about the known scope of the underlying criterion [PCS21a]. However, the proof of the main theorem currently applies Theorem 5.5 outside the cases where Theorem 5.1 is proved, and the existence of good paths is assumed without proof; these are load-bearing gaps that make the principal claim conditional.

major comments (3)

[§5, Theorem 5.5 and §7, Lemma 7.2]
[§11, Definition 11.1 and Lemma 11.2]
[§5, Theorem 5.1 proof]

minor comments (5)

[§6]
[§7, Definition 7.1]
[§8]
[§10, Definition 10.1]
[§5 and §1]

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the arcset construction is a genuinely new argument, with only minor self-citation dependencies and a non-circular correctness gap from applying Theorem 5.5 beyond its stated hypotheses.

full rationale

Central derivation is not circular. Theorem 5.1 obtains a genuine iff statement from the cited criterion [PCS21a, Theorem B] (whose restrictions the paper itself records in Remark 5.2) together with winding-number and bigon arguments; Theorem 5.5's passage from single arcs to arcsets is via the surgery identity (5.3) and the same geometric conditions, not by defining 'geometric vanishing arcset' to be the conclusion. The positive-depth construction for Theorem 7.7 is new: Proposition 11.7 verifies the adapted intersection conditions directly against the A'Campo vanishing cycles, Proposition 11.12 proves linearity from local monodromy computations (Lemma 11.8), and the final isotopy of variation images is checked by explicit surgeries. The imports from [BCCJ23] (Definition 9.1, Proposition 9.2, and the depth-zero case) and from [PCS21a] are prior stated results with their own assumptions; the present paper's conclusion is not one of those assumptions, and the main construction is then verified in this paper rather than assumed. Thus no Eq. X = Eq. Y by construction, no fitted parameter renamed as prediction, and no uniqueness theorem invoked to forbid alternatives are present. Two weaknesses are flagged but are not circularity: (a) Theorem 5.5 is stated without the A_n/D_n and genus-at-least-5 restrictions that Theorem 5.1 explicitly needs, so Lemma 7.2's unconditional use of Theorem 5.5 leaves Theorem 7.7 unproved for those divides; this is a correctness gap, not a reduction-by-definition. (b) Definition 11.1 assumes without proof the existence of good paths with adjacent opposite-sign depth-(k-1) vertices, which is a load-bearing combinatorial property of AΓ(Df) but again an omitted proof, not a circular step. The low non-zero score reflects only the non-load-bearing self-citation dependencies in the framework.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper does not fit any numerical parameters. It relies on classical theorems (Picard-Lefschetz theory, Seifert form computations) and on recent results by the same authors and collaborators. The only ad hoc assumption introduced in this paper is the unproved combinatorial property of AΓ diagrams used to define good paths. The new definitions (arcsets, linear arcsets, topological exceptional collections) are mathematical objects introduced for the construction; they do not carry independent empirical evidence.

assumptions (5)

domain assumption The classical variation operator V_f is an isomorphism for isolated hypersurface singularities.
[AGV88, Theorem 2.2], used in Section 2 and throughout (e.g., Remark 3.10).
domain assumption For f not A_n or D_n with g(Σ_f) ≥ 5, a simple closed curve is a geometric vanishing cycle iff it is non-separating and has winding number zero.
[PCS21a, Theorem B], the key external characterization used in the proof of Theorem 5.1.
domain assumption The geometric monodromy is isotopic to the identity on the boundary of the Milnor fiber.
Standard Milnor fibration fact, used in Definition 2.1 and the definition of the variation operator.
ad hoc to paper Every vertex of depth k in an A'Campo-Gusein-Zade diagram has a depth k-1 neighbor of the prescribed sign type.
Stated without proof in Definition 11.1; load-bearing for the construction of good paths.
domain assumption For totally real singularities, a real Morsification and an A'Campo divide exist.
Classical result [A'C75] and [GZ74a], used in Section 8.

invented entities (3)

geometric vanishing arcset
purpose: A finite disjoint collection of properly embedded arcs whose variation image is a geometric vanishing cycle; the paper's main object of study.
Introduced by Definition 5.6. No external falsifiable handle; its utility is established by the theorems in the paper.
topological exceptional collection of arcsets
purpose: Ordered collection of arcsets with disjointness and intersection conditions mimicking exceptional collections in categorical settings; target of the main construction.
Introduced in Definition 7.3. It is a new structural notion supported by the construction in Theorem 7.7.
linear arcset
purpose: An ordered arcset whose monodromy images intersect consecutive arcs in a single point; sufficient condition for being a geometric vanishing arcset.
Definition 7.1. New combinatorial notion used in the proof of Theorem 7.7.

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Pith. "Pith review of Vanishing arcs for isolated plane curve singularities." pith.science (2026). https://pith.science/paper/7BX4PD77

@misc{pith2026250604917,
  author       = {Pith},
  title        = {Pith review of: Vanishing arcs for isolated plane curve singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7BX4PD77}},
  note         = {Machine review of arXiv:2506.04917}
}

read the original abstract

The variation operator associated with an isolated hypersurface singularity is a classical topological invariant that relates relative and absolute homologies of the Milnor fiber via a non trivial isomorphism. Here we work with a topological version of this operator that deals with proper arcs and closed curves instead of homology cycles. Building on the classical framework of geometric vanishing cycles, we introduce the concept of vanishing arcsets as their counterpart using this geometric variation operator. We characterize which properly embedded arcs are sent to geometric vanishing cycles by the geometric variation operator in terms of intersections numbers of the arcs and their images by the geometric monodromy. Furthermore, we prove that for any distinguished collection of vanishing cycles arising from an A'Campo's divide, there exists a topological exceptional collection of arcsets whose variation images match this collection.

Figures

Figures reproduced from arXiv: 2506.04917 by the authors.

**Figure 2.2.** An relative cycle (red) and its image with reversed orientation (blue) by a diffeomorphism which is the identity on the boundary. or the more general result that there is a representative of the geometric monodromy that acts without fixed points [Tr´a78]. Relation with other invariants. Here we introduce other classical invariants that appear in the present paper and that are tightly related to the variation operato… view at source ↗

**Figure 3.5.** On the left we see the chart U. The points aj (1 − 1/n), cj and aj+1(1/n) are marked. The arc bn between aj (1 − 1/n) and aj+1(1/n) is dotted in blue. In black we see the vector field ξ which in this case is tangent to the interval aj and aj+1 at cj appropriate compact-open topology, are actually isotopic to c (see [FM12, 1.2.2]). Also, any two approximations c ′ and c ′′ to c are themselves isotopic through C 1 emb… view at source ↗

Figures from the paper (38 more)

**Figure 3.** Figure 3: fig. 3.5 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png]

**Figure 3.8.** Figure 3.8: On the left we see the nodal curve Σ1 ∪ Σ2 and in blue we see the three points Σ1 ∩ Σ2. On the right we see the result after smoothing out the three A1 points. This surface is homeomorphic to the Milnor fiber Σf . to a. Assume that Varf (a) is a simple closed curve. …

**Figure 3.9.** Figure 3.9: In red, a separating simple closed curve in Σf which, by the homological coherence property, has vanishing winding number. components using 3 cylinders as in fig. 3.8. Moreover, the core curves of each of these cylinders are geometric vanishing cycles. Therefore, usi…

**Figure 4.6.** Figure 4.6: The relative position of the curves c and c ′ . If c is a geometric vanishing cycle, then c ′ is not; but they represent the same homology class. principle, up to an element of Mod(Σ) we can assume that c is as in fig. 4.6 since all nonseparating simple closed curve…

**Figure 5.** Figure 5: fig. 5.4 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 5.4.** Figure 5.4: On the left we see a neighborhood around a point of a transverse intersection between two oriented segments belonging to a closed curve in a surface. On the right, we see the neighborhood that substitutes the previous one after surgery is performed. Remark 5.7. Note …

**Figure 6.1.** Figure 6.1: On the left we see the bipartite graph K3,5. On the right the surface Σˆ resulting from cutting Σ along the graph K3,5 ,→ Σ. On both sides, in green, a properly embedded arc transverse to the graph and its image by the geometric monodromy. Note that Theorem 5.1 does …

**Figure 6.** Figure 6: fig. 6.1 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 8.4.** Figure 8.4: An example of a divide for the plane curve defined by −x 8 − x 7 − 3x 5y + y 3 using Gusein-Zade method via Chebyshev polynomials. When we regard the order only, we omit types and denote the vanishing cycles by just (V1, . . . , Vµ). Then, the geometric monodromy φf …

**Figure 8.7.** Figure 8.7: Example of AΓ diagram and depth. Milnor fiber and vanishing cycles from an A’campo divide. A’Campo [A’C99] gave a combinatorial model of the Milnor fiber Mf from a divide (or from AΓ(Df ) diagram). For each double point v in Df (which is each 0 vertex of AΓ(Df )), co…

**Figure 8.8.** Figure 8.8: The building block Fv. them. Upper and lower ribbons are placed along f −1 (0)t0 and they cross each other at v. Regarding the + region as the first quadrant of the plane, x-axis is the upper ribbon and y-axis is the lower ribbon. Given the same orientation for each …

**Figure 8.** Figure 8: fig. 8.9 [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]

**Figure 8.9.** Figure 8.9: The building block Fv. 9. Proof of Theorem 7.7 for depth 0 cases Theorem 7.7 for the depth 0 cases was essentially proved in [BCCJ23] and we will recall the construction therein. In this case, each geometric vanishing arcset consists of a single properly embedded arc…

**Figure 8.** Figure 8: fig. 8.9 [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

**Figure 9.** Figure 9: fig. 9.4 [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]

**Figure 9.4.** Figure 9.4: K0 i in the building block. • + vertex A non-compact Lagrangian K+ i for a vanishing cycle V 0 i satisfies K+ i • V • j =    1 (• = +, i = j), 1 (• = 0, V 0 j and V + i are connected in AΓ diagram), 1 (• = −, V − j and V + i are connected in AΓ diagram), 0 (…

**Figure 9.5.** Figure 9.5: K + i in the building block. One can check that these collection of properly embedded arcs K1, · · · , Kµ are all disjoint, adapted and i(Ki , φf (Ki)) = 0 for any i. This proves Theorem 7.7 for depth 0 cases [PITH_FULL_IMAGE:figures/full_fig_p027_9_5.png]

**Figure 10.** Figure 10: fig. 10.2 [PITH_FULL_IMAGE:figures/full_fig_p028_10.png]

**Figure 10.2.** Figure 10.2: Examples of relevant vertices. Basic arcs. We now define the associated basic arc for each edge of AΓ(Df ). Lemma 10.3. For any edge e in AΓ(Df ), there exists an arc K in M such that K • V = ( −1 (V ∈ Re), 0 (V /∈ Re). These arcs are denoted by K+,0 ,K+,−, or K0,− …

**Figure 10.** Figure 10: fig. 10.4 [PITH_FULL_IMAGE:figures/full_fig_p029_10.png]

**Figure 10.4.** Figure 10.4: Description of basic arcs K+,0 and K+,−. V − V 0 K0,− [PITH_FULL_IMAGE:figures/full_fig_p030_10_4.png]

**Figure 10.5.** Figure 10.5: K0,− and corresponding V 0 , V −. Lemma 10.6. Let v be a depth 0 vertex of type + or − in AΓ(Df ). Then, there exists a properly embedded arc K in M such that K • V = ( 1 (V ∈ Rv), 0 (V /∈ Rv). Monodromy images of basic arcs. Now, we describe the monodromy images of…

**Figure 10.** Figure 10: fig. 10.7 [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]

**Figure 10.7.** Figure 10.7: Monodromy image of K+,0 . V − V + τV + (K+,−) (a) τV + (K+,−). V − V + τV − ◦ τV + (K+,−) (b) τV − ◦ τV + (K+,−) [PITH_FULL_IMAGE:figures/full_fig_p031_10_7.png]

**Figure 10.8.** Figure 10.8: Monodromy image of K+,−. Lastly, the basic arc K0,− does not intersect any + vanishing cycles and then the arc τV 0 (K0,−) meets one V − corresponding − vertex of given edge (which is not a relevant vanishing cycle). See fig. 10.9 for the monodromy image φf (K0,−). …

**Figure 10.9.** Figure 10.9: Monodromy image of K0,−. From these observations, we characterize basic arcs in the following way [PITH_FULL_IMAGE:figures/full_fig_p031_10_9.png]

**Figure 11.** Figure 11: fig. 11.5 [PITH_FULL_IMAGE:figures/full_fig_p033_11.png]

**Figure 11.5.** Figure 11.5: Kγ for depth 3 + vertex. We compute the intersection of the arcset Kγv associated to the good path γv and other vanishing cycles. Note that Kγv may have several connected components. We will partition these components into smaller groups so that each group is one of…

**Figure 11.** Figure 11: fig. 11.5 [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]

**Figure 11.** Figure 11: fig. 11.9 [PITH_FULL_IMAGE:figures/full_fig_p037_11.png]

**Figure 11.9.** Figure 11.9: Case (i) in Lemma 11.8. + − + 0 K+,− e1 K−,+ e2 ϕf (K+,− e1 ) K+,0 e3 [PITH_FULL_IMAGE:figures/full_fig_p037_11_9.png]

**Figure 11.10.** Figure 11.10: Case (ii) in Lemma 11.8. Proposition 11.12. For any good path γv, the vanishing arcset Kv is linear. Therefore, Kv is a geometric vanishing arcset. Proof. Let γv = e1 . . . em such that t(em) = v and Kv = K|s(e1)| `m i=1 Kei . We already observed in Section 10 that…

**Figure 11.11.** Figure 11.11: Case (iii) in Lemma 11.8. To show the condition (ii), assume that a good path γv is nonconstant (i.e., Kv has more than one component). We mainly use Lemma 11.8 to get intersection patterns between φf (Kei ) and Kej for any 1 ≤ i ̸= j ≤ m. Moreover, we need more in…

**Figure 11.** Figure 11: fig. 11.14 [PITH_FULL_IMAGE:figures/full_fig_p039_11.png]

**Figure 11.14.** Figure 11.14: Three disjoint copies of K+,− [PITH_FULL_IMAGE:figures/full_fig_p039_11_14.png]

**Figure 11.** Figure 11: fig. 11.15 [PITH_FULL_IMAGE:figures/full_fig_p040_11.png]

**Figure 11.15.** Figure 11.15: Three curves Varf (K−,+ e1 ), Varf (K−), and sg(Varf (K−,+ e1 ), Varf (K−)). − + − sg(Varf (K−,+ e ), Varf (K−)) Varf (K+,− e2 ) [PITH_FULL_IMAGE:figures/full_fig_p040_11_15.png]

**Figure 11.16.** Figure 11.16: Three curves sg(Varf (K−,+ e1 ), Varf (K−)), Varf (K+,− e2 ), and Varf (Kγ) [PITH_FULL_IMAGE:figures/full_fig_p040_11_16.png]

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