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Toric sheaves and polyhedra

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Pith's one-line read This paper proves that the torus-invariant cohomology of a toric sheaf equals the cohomology of a constructible sheaf built from the polytopes of its Weil decoration.

desk verdict A substantial generalization of the rank-one Klyachko dictionary to arbitrary rank toric sheaves; the main theorem is plausible and useful, but two geometric contractibility lemmas in the proof are only sketched and need completion. read the letter →

arxiv 2412.03476 v2 pith:F2UCILNL submitted 2024-12-04 math.AG

Klaus Altmann , Andreas Hochenegger , Frederik Witt This is my paper · ORCID
classification math.AG MSC 14C2014F0614F0814M2518G1052C0755N05
keywords toricsheavesreflexiveWeildecorationsconstructiblepolyhedracohomologyspectralsequencevarieties
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 establishes that the torus-invariant cohomology of a toric sheaf on a smooth projective toric variety is a polyhedral-topology invariant. Concretely, after twisting to an ample 'materialisation', the invariant piece $H^\ell(X,E)_0$ is isomorphic to the sheaf cohomology $H^\ell(\Delta, F(E))$ of a constructible sheaf built from the polytopes decorating the sheaf. It also constructs the torus-invariant universal extension of two nef line bundles through polyhedral inclusion/exclusion, and gives a spectral sequence that computes $H^\ell(X,E)_0$ from reduced cohomology groups of polyhedral subsets. A sympathetic reader would care because it converts cohomological questions about equivariant reflexive sheaves into concrete convex geometry.

What carries the argument

The central objects are Weil decorations: maps $D$ from nonzero vectors of the torus-invariant vector space $E$ to toric divisors (or polytopes), satisfying the subadditivity condition $D(e+e') \ge D(e) \wedge D(e')$. They completely encode a toric sheaf, and after an ample twist they define the constructible sheaf $F(E)$ on $\Delta$. The proof that $F(E)$ computes cohomology runs through the inverse system of local sheaves $F_\sigma$ on the fan and hinges on exactness of the augmented stalk complex $K(F)^\bullet$; the critical geometric input is Lemma 7.10, which asserts that the dual boundary subcomplex $\Delta'$ contracts to the point $u$ for every stratum and boundary position.

What would settle it

Find a smooth projective toric variety, an amply decorated toric sheaf, and a boundary point $u \in \partial \Delta$ for which the subcomplex $\Delta'$ of facets with $\min\langle D^+(S), \rho\rangle \ge_\rho \langle u, \rho\rangle$ has nonvanishing reduced homology, contradicting the contraction asserted in Lemma 7.10; then the complex $K(F)^\bullet$ becomes non-exact and Theorem 6.1 must fail.

Watch

Extended reading notes

Core claim

Theorem 6.1 states that if $X$ is a smooth projective toric variety and $E^+ = E(\Delta)$ is amply decorated for $\Delta \in \mathrm{Pol}^+(\Sigma)$, then $H^\ell(X,E)_0 \cong H^\ell(\Delta, F(E))$, where $F(E)(U) = \{e \in E \mid U \subseteq D^+(e)\}$ is a constructible sheaf on the polytope $\Delta$. Together with Theorem 8.3, this says the $T$-invariant cohomology of every toric sheaf is the abutment of a spectral sequence whose $E_1$ terms are sums of reduced cohomology groups $\tilde H^{q-1}(P(T))$ of polyhedral subsets $P(T) = \Delta \setminus \mathrm{int}_\Delta D^+(T)$.

Load-bearing premise

The theorem stands or falls on the geometric contraction claim in Lemma 7.10: for every stratum and every boundary position of $u$, the dual boundary subcomplex $\Delta'$ must be contractible to $u$; together with the deferred acyclicity of local sheaves for line bundles in Lemma 6.11, this exactness drives the entire bridge.

Editorial extensions

If this is right

  • Amply decorated toric sheaves are acyclic: $H^\ell(X,E) = 0$ for all $\ell \ge 1$ (Corollary 6.3).
  • The $T$-invariant cohomology of any toric sheaf is computable from the reduced cohomology of polyhedral subsets via the spectral sequence of Theorem 8.3.
  • The torus-invariant universal extension of two nef line bundles admits an explicit polyhedral inclusion/exclusion description (Theorem 5.2).
  • The Euler characteristic of a toric sheaf in degree zero is governed by the Möbius function of its stratum poset (Corollary 8.8).
  • Shifting $\Delta$ by $m \in M$ recovers the $m$-graded piece $H^\ell(X,E)_m$ (Remark 6.2).

Reading between the lines

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

  • The constructible-sheaf bridge suggests a practical algorithm: from a Weil decoration one can compute $H^\ell(X,E)_0$ directly by polyhedral topology, bypassing resolutions by line bundles.
  • If the local acyclicity assumptions survive without smoothness, the same polyhedral formula might extend to singular or non-complete toric varieties.
  • The spectral sequence could be compared with the coherent-constructible correspondence to translate Morse-theoretic or tropical data on $\Delta$ into sheaf cohomology.
  • Because the complex length is governed by the height of the stratification, the method is especially short for rank-two sheaves, as the paper demonstrates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a polyhedral framework for toric sheaves (equivariant reflexive sheaves) on smooth projective toric varieties. Every toric sheaf is encoded by a Weil decoration, a map from the nonzero vectors of the associated torus-invariant vector space to toric divisors, equivalently to virtual polytopes. The authors use this to construct the torus-invariant universal extension of two nef line bundles via inclusion/exclusion of polyhedra, and then to attach to each toric sheaf E a constructible sheaf F(E) on an ample polytope Δ. The central result, Theorem 6.1, asserts that H^ℓ(X,E)_0 is isomorphic to H^ℓ(Δ,F(E)); a spectral sequence in Theorem 8.3 computes these groups from reduced singular cohomology of polyhedral subsets of Δ. The paper closes with worked examples, including line bundles and the twisted tangent sheaves of the projective plane.

Significance. If the main theorem is correct, it gives a genuinely polyhedral-topological description of the T-invariant cohomology of every toric sheaf after a twist, which is a substantial generalization of the known line-bundle formula of [ABKW20] and [AP20]. The construction of the universal extension by Weil decorations and inclusion/exclusion sequences is elegant and appears to explain and extend [AFH23], including cases where the virtual intersection is not an honest lattice polytope. The spectral sequence of Theorem 8.3, with E1-terms involving only reduced cohomology of polyhedral subsets, is concrete and well suited to examples such as the tangent sheaf of P2. The paper is ambitious, mostly well organized, and the large supply of worked examples is a real strength. The main reservation is that two load-bearing geometric inputs in the proof of Theorem 6.1 are only sketched or deferred.

major comments (3)
  1. [§7.2.2, Lemma 7.10] Lemma 7.10 is the key exactness input: the subfan Čech complex C(Σ_S)• is exact because the dual boundary subcomplex Δ′ 'can be contracted to u'. The proof of this contractibility is not a complete argument: for u in the relative interior of Δ the picture in Figure 15 is plausible, but for u∈∂Δ the relations (54) and (55) reverse strictness, so Δ′ may include facets with min⟨D+(S),ρ⟩ = ⟨u,ρ⟩ that the standard case would exclude. Remark 7.8 explicitly flags that in this boundary regime a>ρ b need not imply a≥ρ b. No proof is given that the resulting larger subcomplex is still contractible. This matters because Lemma 7.10 is used to show exactness of the augmented stalk complex K(F)• in (52), which is exactly what Proposition 7.5 and hence Theorem 6.1 require. The authors should supply a rigorous proof of the contractibility of Δ′ for every stratum S and every u∈Δ, with the boundary cases handled explicitly rather than by appeal to a figure.
  2. [§6.3, Lemma 6.11 and Proposition 6.7] Lemma 6.11 asserts that for a line bundle with ample twist and Δ nef, the local sheaves Fσ have vanishing higher cohomology; the proof is only sketched and refers to [AP20, Claim 3.3.2] for the key claim that the set S(σ) is empty or retractible to rσ. This lemma is load-bearing: Proposition 6.7, which follows from it, is used in Lemma 7.3 to conclude that RΓΔ(F•) = ΓΔ(F•), and that equality is needed in equation (49) to pass from the Klyachko–Čech complex to H(Δ, R lim← F•). The manuscript should either give a complete proof of Lemma 6.11 or state precisely which assertion is imported from [AP20] and verify that the cited claim covers all cones σ∈Σ, not only maximal cones. As written, this is a second gap in the proof of the central theorem.
  3. [§8.2, Theorem 8.3, Eq. (67)] The spectral sequence in Eq. (67) is stated with sums over S<T and chains chℓ(S,T) for S<T. For ℓ=0, ch0(S,T) is empty unless S=T, so the displayed E1-term makes E^{0,q}_1 = 0. This contradicts Theorem 8.2, where G^0 = ⊕_S G_S, and also contradicts the surrounding discussion and Figure 16, which visibly uses E^{0,q}_1 from the generic and minimal strata. The sums should be over S≤T, with the ℓ=0 case giving S=T. This is a straightforward indexing correction but it affects the statement of a main theorem and therefore should be fixed in the text.
minor comments (4)
  1. [§7.2.2, Remark 7.8] Remark 7.8 acknowledges the nonstandard behavior of the relations >ρ and ≥ρ on ∂Δ but gives no concrete example where a>ρ b occurs without a≥ρ b. Since this is exactly the regime that makes Lemma 7.10 delicate, a short explicit example would substantially help the reader.
  2. [§8.5, final paragraph] The last sentence of Section 8.5 ends with 'In particular, it follows that' followed immediately by Section 9; the intended display or conclusion appears to be missing. Please complete the sentence and give the promised Euler-characteristic conclusion.
  3. [§6.2.1, Eq. (30)] The derivation of H^ℓ(Δ,F) = H̃^{ℓ−1}(Z) uses the reduced cohomology convention in which H̃^{-1}(∅) = k; this is stated in the introduction but it would be helpful to repeat the convention at first use in the Gysin sequence.
  4. [§6.2.3, Eq. (33)] The Weil decoration of the tangent sheaf is written as D(ρ_i) = Δ_1, Δ_1−[1,0], Δ_1−[0,1]; the reader has to infer that the rays ρ_i are ordered consistently with the labels in Figure 9. A one-sentence explanation of the labeling would remove ambiguity.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central isomorphism is proved via original polyhedral/constructible arguments, with prior author results serving as independent inputs rather than as the target claim.

full rationale

The paper's central isomorphism (Theorem 6.1) is proved in Section 7 by an original chain: the Klyachko–Čech complex (47) computes H•(X,E)_0 (an external theorem of Klyachko), the identification (34) rewrites the degree-zero pieces as Γ(Δ,Fσ), equation (49) converts the computation into H(Δ, R lim F•), and Proposition 7.5 is then proved by showing exactness of the augmented stalk complex K(F)• via the strata double complex and the contraction argument in Lemma 7.10. None of these steps defines the target cohomology in terms of itself; F(E) is built directly from the Weil decoration D+, which is independent of the cohomology. The paper does cite the authors' own previous work at several load-bearing points: Eq. (1)/(26) from [ABKW20]/[AP20] for line-bundle cohomology and Ext groups, Lemma 6.11 relying on [AP20, Claim 3.3.2] for local acyclicity of line bundles, and Theorem 5.2 relying on [AFH23, Theorem 19 and Section 4.2.2] for the nested case and pushout characterization. These are independent published results with stated assumptions that do not include the present theorem; the current paper's central claim is not merely a restatement of them, and the general toric-sheaf case is new. The geometric contraction claim in Lemma 7.10 is asserted from Figure 15 with the boundary cases regulated by the ad hoc relations (54)–(55) and flagged in Remark 7.8; a failure there would make the proof incorrect, but that is a correctness risk rather than circularity. The indexing issue in Theorem 8.3 (S<T instead of S≤T) is likewise an apparent error, not a circular reduction. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors is invoked to force a choice.

Assumptions & free parameters 0 free parameters · 7 assumptions · 3 invented entities

No free parameters are fitted: this is a pure mathematics derivation. The auxiliary choices (the ample polytope Δ for materialization, and writing a divisor as a difference of nef divisors) are shown to be immaterial to the main statements (Remark 2.5). The axioms are the published results the new theorems lean on: Klyachko's Čech complex, the rank-one cohomology/extension formulas from [ABKW20]/[AP20], the nested universal extension and pushout characterization from [AFH23], the derived inverse-limit representation from [BL03], and [AP20, Claim 3.3.2] for local line-bundle acyclicity. The invented entities are the paper's new objects; each carries independent evidence through proved theorems, agreement with known examples, and reduction to previously known objects in special cases.

assumptions (7)
  • standard math The Klyachko-Čech complex K(E)^•_m computes the m-graded sheaf cohomology H^•(X,E)_m.
    Cited from [Kly90] in Subsection 7.1.3; it is the bridge from global cohomology to the inverse limit over the fan in the proof of Theorem 6.1.
  • domain assumption Rank-one cohomology formula H^ℓ(X,O_X(D))_0 ≅ H̃^{ℓ-1}(∇- ∖ ∇+) for D = ∇+ - ∇-.
    Treated as known from [ABKW20] and [AP20]; used in Eq. (26) to identify Ext(∇-,∇+)_0 with reduced cohomology of ∇- ∖ ∇+ and as the rank-1 check of Theorem 6.1 in Subsection 6.2.1.
  • domain assumption For the nested case ∇+ ⊆ ∇-, the sheaf E(∇-,∇+) built from inclusion/exclusion polytopes is the universal extension ([AFH23, Theorem 19]), and the torus-invariant universal extension is a pushout ([AFH23, Section 4.2.2]).
    Used at the end of the proof of Theorem 5.2 to identify the newly constructed E(∇-,∇+) with the universal extension; prior work by the same research group.
  • domain assumption For a line bundle with E+ ample and Δ = ∇- nef, the set S(σ) = ∇- ∖ (∇+ + σ∨) is empty or retractible to r_σ, making F_σ acyclic.
    Lemma 6.11 is only sketched and is declared a direct consequence of [AP20, Claim 3.3.2]; it is load-bearing for Proposition 6.7 and hence for Step 1 of the proof of Theorem 6.1.
  • ad hoc to paper The subcomplex Δ' of ∂Δ determined by the rays with D+(S) not containing u contracts to u, so the fan subcomplex C(Σ_S)^• is exact.
    Lemma 7.10 plus Figure 15; the contractibility is asserted with a heuristic, including boundary cases regulated by (54)-(55), and carries the exactness of the strata double complex in the proof of Proposition 7.5.
  • standard math The Klyachko-Čech complex (48) represents the derived inverse limit R lim← Γ_Δ(F_•) in Db(Veck), per [BL03, Section 3.5].
    Invoked in Subsection 7.1.3 to obtain equality (49) in the proof of Theorem 6.1.
  • domain assumption Working hypotheses: k algebraically closed of characteristic 0, X smooth projective with convex full-dimensional fan support (assumption (5)).
    Assumed from Sections 2 and 6 onward; the proofs use char-0 singular cohomology with k coefficients, Gysin sequences, and local translations of σ∨ that require smoothness.
invented entities (3)
  • Weil decoration D_E: E ∖ {0} → Div_T(X) independent evidence
    purpose: Classifies toric sheaves (reflexive T-equivariant sheaves) as subsheaves of E ⊗ k[M], generalizing the Weil divisor picture from rank one to arbitrary rank.
    Recovers Klyachko filtrations via eq. (2) and is validated on known examples (line bundles, direct sums, tangent sheaf); the classification content reformulates Klyachko data, while the semilattice structure supports the paper's new constructions.
  • Constructible sheaf F(E) on the polytope Δ, with F(E)(U) = {e ∈ E | U ⊆ D+(e)} independent evidence
    purpose: Computes the T-invariant cohomology of E through H^ℓ(X,E)_0 ≅ H^ℓ(Δ,F(E)) (Theorem 6.1).
    Its cohomology is checked against known results in the examples (OP2(−3), TP2(ℓ), line bundles) and it generalizes the known rank-one formula (1) from [ABKW20]/[AP20].
  • Virtual polytope intersection ∇- ∧ ∇+ = P(D- ∧ D+) in Pol(Σ) independent evidence
    purpose: Extends the polyhedral inclusion/exclusion construction of universal extensions to the case where ∇- ∩ ∇+ is not a compatible lattice polytope.
    Specializes to the actual intersection when ∇- ∩ ∇+ is a compatible lattice polytope, and the resulting extension matches the known universal extension of [AFH23] (verified via Klyachko filtrations in Example 5.3).

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Pith. "Pith review of Toric sheaves and polyhedra." pith.science (2026). https://pith.science/paper/F2UCILNL

@misc{pith2026241203476,
  author       = {Pith},
  title        = {Pith review of: Toric sheaves and polyhedra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F2UCILNL}},
  note         = {Machine review of arXiv:2412.03476}
}
read the original abstract

Over a smooth projective toric variety we study toric sheaves, that is, reflexive sheaves equivariant with respect to the acting torus, from a polyhedral point of view. One application is the explicit construction of the torus invariant universal extension of two nef line bundles via polyhedral inclusion/exclusion sequences. Second, we link the cohomology of toric sheaves to the cohomology of certain constructible sheaves explicitly built out of the associated polyhedra. For the latter we define a concrete double complex and a spectral sequence which computes the cohomology of toric sheaves from the reduced cohomology of polyhedral subsets living in the realification of the character lattice of the toric variety.

Figures

Figures reproduced from arXiv: 2412.03476 by the authors.

Figure 1
Figure 1. The Hirzebruch surface F1 = TV(Σ) given by the fan on the left-hand side. The red dot on the right-hand side indicates the origin in MR and fixes the position of the polytopes. the inclusion/exclusion sequence of polyhedra displayed in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The polyhedral resolution of ∇− \ ∇+ defined by a split bundle) 0 → OX(∇+) → E(∇−, ∇+) := OX (∇0) ⊕ OX(∇1) → OX(∇−) → 0. (3) More generally, the same method applies for more than two components as long as ∇+ ⊆ ∇−. This viewpoint, however, breaks down if ∇+ ∩ ∇− is no longer a lattice polytope compatible with the fan (see [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The fan of P 2 . As we show in Example 4.6 below, its global canonical stratification is η = S(0) S(Dρ0 ) = k× · ρ0 S(Dρ1 ) = k× · ρ1 S(Dρ2 ) = k× · ρ2. Locally, say over Uσ0 , we find the canonical stratification η = S(0) S(div(x (1,0))) = k× · ρ1 S(div(x (0,1))) = k× · ρ2 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: The sequence (3) from a Weil decoration point of view [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: The polyhedra sit inside MR while b ∈ NR. Lemma 5.1. Let C be a connected component of ∆ \ Q and ∇ := Q ∪ C. Then [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: An example of a configuration Q ⊆ ∆. The area shaded in green represents a connected component C. The ori￾gin of MR is marked in red. Let σ be the tail cone of ∆ and Q. Next, let 0 6= b ∈ σ ∨ ⊆ NR. Then minh∆, bi ≤ minh∇, bi ≤ minhQ, bi, for ∆ ⊇ ∇ ⊇ Q. Furthermore, min…
Figure 7
Figure 7. Figure 7: The polytope ∆, the normal fan Σ = N (∆), and the polytopes ∇− and ∇+. The red dots mark the origins. To ease notation we let Di = Dρi and D± = D∇± . The divisor of ∆ is given by D∆ = 3D3 + 7D4 + 5D5 + 4D6, [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: The lattice polytopes of ∇ + + and ∇ + −, the intersection Q+ and the polytopes ∇0 and ∇1. decoration of E + := E(∇ + −, ∇ + +) is therefore given on 0 6= v ∈ V by DE+ (v) =    ∇0 if v ∈ [∇0 ] · k ∇1 if v ∈ [∇1 ] · k ∇ + + if v ∈ ([∇1 ] − [∇0 ]) · k Q+ if …
Figure 10
Figure 10. Figure 10: Using Equation (30) we compute Heℓ ((∆ + m) \ U) for m ∈ M and [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 9
Figure 9. Figure 9: The fan Σ(P 2 ). ∆4 D+(1) ∆4 + [−1, −1] D+(1) [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: The materialised Weil decoration of L = OP2 (∆1 − ∆4) with ∆ = ∆4 (left-hand side) and ∆ = ∆4 + [−1, −1] (right￾hand side). We indicate the origin by a red dot. U = int∆ D+(1). The space (∆ + m) \ U is contractible unless m = [−1, −1]. In the latter case we obtain a s…
Figure 11
Figure 11. Figure 11: The positive Weil decoration of TP2 . D+(ρ0) D+(ρ1) D+(ρ2) ∆ = ∆1 D+(ρ0) D+(ρ1) D+(ρ2) ∆1 + [1, 0] [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: The ample materialised Weil decoration of TP2 . In addition, the black triangle displays ∆ = ∆1 on the left and ∆ = ∆1 + [1, 0] on the right hand side. To capture the degree m ∈ M we consider Fm = Fm(TP2 ). For m = 0 we have F = Nk whence H 0 (∆, F) = Nk is the only n…
Figure 13
Figure 13. Figure 13: ℓ = 1 (left hand side) and ℓ = 2 (right hand side). 6.3. The local sheaves Fσ. Mutatis mutandis the definition for the k-sheaf G in Subsection 6.1 also works for the affine toric varieties Uσ, σ ∈ Σ. We define the local k-sheaves of E by Eσ := E|Uσ and Fσ(E) = F(Eσ). …
Figure 14
Figure 14. Figure 14: int∆D+ ρ1 D+ ρ1 ∆ = 4∆1 u int∆D+ ρ1 D+ ρ1 ∆ = 4∆1 − [1, 0] u [PITH_FULL_IMAGE:figures/full_fig_p035_14.png]
Figure 15
Figure 15. Figure 15: The subcomplexes ∆′ ⊂ ∂∆ and Σ S ⊆ Σ. The left￾hand side displays the standard case hu, ρi > minh∆, ρi, e.g., u is an inner point of ∆. The right-hand side corresponds to equality. ∆′ consists of the “lower boundary” of D+(S) considered as a subset of ∆+(u). Depending…
Figure 16
Figure 16. Figure 16: The spectral sequence for h = 2 Next, we define the closed subsets P(S) := ∆ \ int∆ D +(S) (65) of ∆. Let jS : int∆ D +(S) ֒→ ∆ and ιS : P(S) ֒→ ∆ be the open and closed embedding of int∆ D+(S) and P(S), respectively, with as￾sociated Gysin sequence 0 (jS)!k k (iS)∗k …
Figure 17
Figure 17. Figure 17: The Weil decoration of the twisted tangent bundle TP2 (−3) = ΩP2 with E −1,2 1 = M 2 i=0 [PITH_FULL_IMAGE:figures/full_fig_p047_17.png]
Figure 18
Figure 18. Figure 18: Positive and ample Weil decorations two polytopes has two components. However, He0 [PITH_FULL_IMAGE:figures/full_fig_p048_18.png]

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