Contact isotopies in the coherent-constructible correspondence

Harold Williams; Jishnu Bose

Contact isotopies in the coherent-constructible correspondence

T0 review · 0 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The mirror of tensoring by a toric line bundle is the zero-smoothing limit of quantized Hamiltonian flows.

desk verdict Solid, genuinely new proof that the mirror Picard action in the coherent-constructible correspondence is the nearby-cycles limit of quantized contact isotopies; deserves a serious referee. read the letter →

arxiv 2505.05012 v1 pith:GOO2AYLD submitted 2025-05-08 math.AG math.SG

Jishnu Bose , Harold Williams This is my paper · ORCID

classification math.AGmath.SG MSC 14M2514F0853D1053D37

keywords coherent-constructiblecorrespondencetoricmirrorsymmetryPicardgroupactioncontactisotopysheafquantizationtwistedpolytopenearbycyclesconstructiblesheaves

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 shows that the mirror Picard group action on the sheaf side of toric mirror symmetry is not merely an algebraic convolution: it is realized geometrically as the zero-smoothing limit of quantized Hamiltonian flows. Starting from a torus-invariant Cartier divisor $D$ on a normal toric variety $X_\Sigma$, the authors mollify the piecewise-linear support function $\varphi_D$, form the GKS kernels (sheaf quantizations of Hamiltonian isotopies) of the resulting homogeneous Hamiltonians on $T^*T^n$, and take their nearby-cycles limit. The main theorem states that this limiting kernel $\psi(K_{D,I})$ acts on any sheaf $F$ exactly as convolution with the twisted polytope sheaf $p_!P(D)$, the established mirror of $\mathcal{O}(D)$; on the category $\mathrm{Sh}^c_{\Lambda_\Sigma}(T^n)$ this makes $\psi(K_{D,I})\circ F$ isomorphic to $A(\mathcal{O}(D)\otimes B(F))$. This provides a sheaf-theoretic counterpart to the Hamiltonian-flow description of the mirror Picard group known in the symplectic setting, and it holds for every normal toric variety, with no smoothness or projectivity assumption.

What carries the argument

The proof combines two mechanisms. The first is a general fact about GKS kernels on a Lie group $G$: a kernel quantizing a $G\times G$-invariant Hamiltonian acts on sheaves by group convolution, $K\circ F \cong P*F$ with $P=K\circ C_e$, and this action commutes with nearby cycles under a compactness bound on the derivatives of the Hamiltonian (Propositions 3.5 and 3.9). The second is a convex-geometric analysis of the homogenized smoothing $\varphi_\varepsilon(\xi)=\|\xi\|(\eta_\varepsilon*\varphi_D)(\hat{\xi})$ of the support function: as $\varepsilon\to 0$ the derivatives $d\varphi_\varepsilon$ remain in an $\varepsilon R$-neighborhood of $\operatorname{Conv}(\{\chi_\sigma\})$, and their limit points over each cone face $\tau\preceq\sigma$ lie in $(\chi_\sigma,0)+\tau^\perp\times \dot{\tau}$ (Propositions 4.1, 4.7, 4.11). These bounds control the singular support of the nearby-cycles kernel, allowing a comparison (Proposition 4.13) between its action on the shard sheaves $\mathbb{C}_{\mathrm{Int}(\sigma^\vee)}$ and the \v{C}ech summands $\mathbb{C}_{\mathrm{Int}(\chi_\sigma+\sigma^\vee)}$ of the twisted polytope sheaf $P(D)$; assembling these comparisons through the equivariant-to-nonequivariant reduction $p_!$ yields Theorem 1.1.

What would settle it

Run the construction on $\mathbb{P}^2$ with $D$ a hyperplane: compute both sides of Theorem 1.1 on the skyscraper sheaf at a point of $T^2$ and compare stalks; any mismatch refutes the claim. More surgically, for a fan whose adjacent maximal cones have Cartier data far apart in $\mathbb{M}_\mathbb{R}$, numerically sample $d\varphi_\varepsilon(\xi)$ for small $\varepsilon$ at points approaching the common wall: a value outside $B_{\varepsilon R}(\operatorname{Conv}\{\chi_\sigma\})$ would break Proposition 4.11, the analytic core of the proof.

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Extended reading notes

Core claim

Theorem 1.1 asserts that for every torus-invariant Cartier divisor $D$ on a normal toric variety $X_\Sigma$, the nearby-cycles kernel $\psi(K_{D,I})$ satisfies $\psi(K_{D,I})\circ F \cong (p_!P(D))*F$ for all $F\in \mathrm{Sh}(T^n)$, where $P(D)$ is the twisted polytope sheaf mirror to $\mathcal{O}(D)$ and $*$ is convolution on the torus. In the subcategory $\mathrm{Sh}^c_{\Lambda_\Sigma}(T^n)$ identified by the coherent-constructible correspondence with $\mathrm{Coh}(X_\Sigma)$, this becomes the autoequivalence $F\mapsto A(\mathcal{O}(D)\otimes B(F))$. The substance of the result is that the $\varepsilon\to 0$ limit of the quantized flows exists as a genuine sheaf even though the limiting Hamiltonian---the pullback of $\varphi_D$---is not smooth and has no Hamiltonian flow of its own; that limit coincides with the convolution action of $p_!P(D)$. Thus the mirror of tensoring by a line bundle is a limiting contact isotopy, giving the Picard action a direct symplectic-geometric mechanism on the A-side.

Load-bearing premise

The load-bearing premise is that the gradients of the smoothed support functions remain uniformly within a shrinking neighborhood of the convex hull of the Cartier data as the smoothing radius tends to zero, with limits confined to the expected cone strata; if that analytic control failed near a fan wall, the nearby-cycles kernel would stop agreeing with the twisted-polytope convolution.

Editorial extensions

If this is right

The mirror action of $\mathcal{O}(D)$ on constructible sheaves is a genuine limiting Hamiltonian flow, giving line-bundle twists a symplectic-geometric meaning rather than only a combinatorial convolution formula.
The isomorphism $\psi(K_{D,I})\circ F\cong (p_!P(D))*F$ holds for every sheaf $F$ on $T^n$, so the geometric description is not confined to the constructible CCC category.
The theorem applies to arbitrary normal toric varieties, including singular and non-projective ones where the analogous Floer-theoretic statements are not available.
For a compact Lie group $G$, the same GKS mechanism expresses the action of a bi-invariant norm Hamiltonian as convolution with a kernel supported on the geodesic flow (Propositions 3.2 and 3.5), an interesting case in its own right.

Reading between the lines

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

The mollifier-independence suggested by the estimates points to a general construction: any piecewise-linear pullback Hamiltonian may acquire a canonical 'limiting flow' via nearby cycles of quantized smoothings, extending the method beyond the toric Cartier case.
Because the proof passes first through an equivariant statement on $\mathbb{R}^n$, it should extend to the toric stacks covered by the nonequivariant CCC, as the authors anticipate in Remark 4.23, yielding the same geometric Picard action for stacky fans.
For compact Lie groups, the convolution description of bi-invariant-norm GKS kernels hints at explicit central convolution operators acting on equivariant sheaf categories, a direction the paper does not pursue.

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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

Quick take: this is the real thing. Bose and Williams prove that the mirror Picard group action in the coherent-constructible correspondence arises as the nearby-cycles limit of quantized contact isotopies — the sheaf-theoretic analog of Hanlon's Floer-theoretic result, and it covers arbitrary normal toric varieties, not just smooth projective ones. Theorem 1.1, the identification of ψ(K_{D,I}) ∘ F with (p_!P(D)) * F, is new; nobody had connected nearby cycles of GKS kernels to twisted polytope sheaves before.

What the paper does well: the heavy lifting is in the analytic control of the mollified support-function derivatives (Propositions 4.1, 4.7, 4.11). I traced those estimates — the bounding of the non-convolution term in (4.9), the limsup argument with covectors restricted to the strata, and the uniform bound in Proposition 4.11 — and they are correct. The singular-support arguments in Proposition 4.13 are careful, and the passage from shard sheaves to the full sheaf via the Čech resolution is legitimate. The general machinery on GKS kernels for Lie groups in Section 3 (convolution description, nearby-cycles compatibility) is clean and likely to be independently useful.

Soft spots, in proportion: they are minor. The notation 'τ ≼ σ' is defined as 'τ is a facet of σ', but (4.8) and (4.15) really need it to mean 'face' including τ = σ; the arguments work with the intended reading, so it is a presentation slip, not a gap. The proof of Theorem 4.17's identification of the induced maps on the resolution is slightly compressed — it leans on a stalkwise determination that is fine but could be spelled out. The paper is honest about two places it does not fully cover: Remark 4.20 admits cohomological constructibility is needed for the clean nearby-cycles-of-the-action formulation, only conjecturing the singular weakly-constructible case, and Remark 4.23 explicitly declines to extend to toric stacks. Neither undermines the stated theorems.

The citation pattern is clean: GKS, FLTZ, Kuwagaki, Zhou, Hanlon/Hicks, KS, NS20 are all used appropriately, with no self-citation leverage.

Who this is for: anyone working on toric mirror symmetry, sheaf quantization, or the CCC. It deserves a serious referee — the analytic estimates in §4 take time but check out, and the referee should focus there. Send it to review; accept after the notation slip is fixed.

Referee Report

0 major / 4 minor

Summary. The paper proves Theorem 1.1, which states that for a torus-invariant Cartier divisor D on a normal toric variety X_Σ, the nearby cycles kernel ψ(K_{D,I}) obtained from a family of quantized contact isotopies acts on sheaves by convolution with the twisted polytope sheaf p_!P(D). In the category of constructible sheaves realizing the coherent-constructible correspondence, this action is shown to coincide with the mirror action of the line bundle O(D). The proof combines the GKS quantization of Hamiltonian isotopies with a detailed asymptotic analysis of the derivatives of mollified support functions, proves singular support bounds for the action on shard sheaves, and passes to the full kernel via Čech resolutions. The non-complete fan case is reduced to the complete case by a fan refinement argument.

Significance. The result is a significant contribution to toric mirror symmetry: it provides a purely sheaf-theoretic, Floer-theory-independent realization of the mirror Picard group action as the ε→0 limit of quantized contact isotopies, and it holds for arbitrary normal toric varieties, going beyond the smooth projective hypotheses of prior Floer-theoretic results. The proof is well-structured and detailed: the analytic estimates in Propositions 4.1, 4.7, and 4.11 are explicit and checkable, the singular support bounds are applied carefully, and the passage from shard sheaves to the full kernel via the resolution of C_0 is legitimate. The paper is honest about the scope of its methods, including the remark that the toric stack case is not treated.

minor comments (4)

[Section 4, after Proposition 4.1] The notation 'τ ≼ σ indicates that τ is a facet of σ' is inconsistent with the use of ≼ in (4.8), (4.15), and the proof of Proposition 4.7, where τ ranges over all faces of σ including τ = σ; under the literal definition, a point in the relative interior of σ would not be covered. The authors should change the definition to mean 'face' (including equality) and then reconcile the union in (4.15) by taking closures if needed.
[Section 4, proof of Proposition 4.13] The step from the limsup bound on the singular support of ~K_{D,ε,1} ∘ C_{Int(σ^∨)} to the singular support bound (4.16) for the nearby cycles is justified by a reference to [NS20, Lem. 3.16]; since this is a delicate point, a sentence explaining why the hypotheses of that lemma hold would be helpful.
[Section 3, Proposition 3.9] The statement of Proposition 3.9 restricts to cohomologically constructible F for the full diagram, but the proof notes that the vertical isomorphisms are defined without any constructibility hypotheses; stating this explicitly in the proposition would avoid confusion about the scope of the theorem that uses it.
[Throughout] There are a few typographical errors that should be corrected: the abstract contains 'to ric mirror symmetry' instead of 'toric mirror symmetry', and in the proof of Proposition 3.2 the expression 'ad^*_{dϕ(ξ)}ξ' appears without a space in the subscript; these do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the nearby-cycles kernel action is derived from the Hamiltonian flow and matched against the independently defined twisted polytope sheaf; no fitted parameters or load-bearing self-citations.

full rationale

The derivation chain is self-contained and noncircular. The input is a toric Cartier divisor D with support function phi; the paper builds homogeneous Hamiltonians H_epsilon by mollifying phi, quantizes their flows via GKS kernels K_{D,I}, and then proves psi(K_{D,I}) is isomorphic to convolution by p_!P(D). The twisted polytope sheaf P(D) and its Cech summands C_{Int(chi_sigma+sigma^vee)} are defined independently from the Cartier data chi_sigma (Zhou, following FLTZ11), and the proof does not assume the target isomorphism. The load-bearing analytic estimates (Propositions 4.1, 4.7, 4.11) are proved in the text: they control the limiting derivatives dphi_epsilon, giving the singular support bound (4.16) and the stalk computation in Proposition 4.13. Proposition 4.13 then identifies each nearby-cycles summand with the corresponding polytope-sheaf summand, and Propositions 3.5/3.9 (proved for GKS kernels on any Lie group) allow the kernel action to be converted into convolution. The only issue a reader might flag is notational: the text says 'tau ≼ sigma indicates that tau is a facet of sigma' while (4.8) and (4.15) need 'face'; with 'face' the arguments are consistent, and this is a presentation slip rather than a circular step. There are no fitted parameters, no benchmark quantity is used in its own proof, and the cited results (GKS12, FLTZ11, Kuw20, Zhou) are external and not from the present authors. Accordingly the paper scores 0 for circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper builds on a standard toolkit: six-operation sheaf theory, GKS quantization, the coherent-constructible correspondence, and the combinatorial description of twisted polytope sheaves. No new freely chosen parameters or invented entities appear; the construction of H_epsilon from the support function is canonical up to the irrelevant extension in the noncomplete case. The main analytic burden is carried by the derivative estimates in Propositions 4.1, 4.7, and 4.11, which are proved in the text.

assumptions (6)

standard math Six-operation formalism for derived infinity-categories of sheaves, including proper base change, projection formula, and smooth base change.
Invoked throughout Section 2; references [KS94], [Jin24], [Vol21].
domain assumption GKS sheaf quantization theorem: a homogeneous Hamiltonian isotopy of T*M is quantized by a unique kernel, with singular support tracking the flow.
Core tool in Section 2.4; used in Propositions 3.5, 3.9 and Section 4; reference [GKS12].
domain assumption Coherent-constructible correspondence for toric varieties (equivariant and nonequivariant), intertwining tensor with convolution.
Used to translate the sheaf-theoretic equality into the mirror statement; see Section 2.6 and equation (2.4); references [FLTZ11], [Kuw20].
domain assumption Twisted polytope sheaf P(D) admits a Cech resolution by shard sheaves C_{Int(sigma^vee + chi_sigma)}.
Used in the proof of Theorem 4.17 to compare psi(K tilde_{D,I} circ C_0) with P(D); references [Zho19], [Zho20].
domain assumption Singular support bound for nearby cycles from [NS20, Lem. 3.16].
Used in Proposition 4.13 to bound ss(psi(K tilde_{D,I} circ C_{Int(sigma^vee)})) from the limit of the singular supports of the finite-epsilon actions.
standard math Standard microlocal facts: singular support of integral transforms and pullbacks along smooth maps ([GKS12, Prop. 3.2(i)], [KS94, Prop. 5.4.5]).
Used in Propositions 3.2, 3.5 and throughout Section 4 for computing singular supports.

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Cite this review

Pith. "Pith review of Contact isotopies in the coherent-constructible correspondence." pith.science (2026). https://pith.science/paper/GOO2AYLD

@misc{pith2026250505012,
  author       = {Pith},
  title        = {Pith review of: Contact isotopies in the coherent-constructible correspondence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GOO2AYLD}},
  note         = {Machine review of arXiv:2505.05012}
}

abstract

The coherent-constructible correspondence is a realization of toric mirror symmetry in which the A-side is modeled by constructible sheaves on $T^n$. This paper provides a geometric realization of the mirror Picard group action in this correspondence, characterizing it in terms of quantized contact isotopies and providing a sheaf-theoretic counterpart to work of Hanlon in the Fukaya-Seidel setting. Given a toric Cartier divisor $D$, we consider a family of homogeneous Hamiltonians $H_\varepsilon$ on $T^* T^n$. Their flows act on sheaves via a family of kernels $K_\varepsilon$ on $T^n \times T^n$. The nearby cycles kernel $K_0$ corresponds heuristically to the Hamiltonian flow of the non-differentiable function $\lim_{\varepsilon \to 0} H_\varepsilon$, which is the pullback of the support function of $D$ along the cofiber projection. We show that the action of $K_0$ coincides with the convolution action of the associated twisted polytope sheaf, hence mirrors the action of $\mathcal{O}(D)$ on coherent sheaves.

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom

M. Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol. 10 (2006), 1097--1156

work page 2006

[2] [2]

(N.S.) 15 (2009), no

, Morse homology, tropical geometry, and homological mirror symmetry for toric varieties, Selecta Math. (N.S.) 15 (2009), no. 2, 189--270

work page 2009

[3] [3]

Achar, Perverse sheaves and applications to representation theory, Mathematical Surveys and Monographs, vol

P. Achar, Perverse sheaves and applications to representation theory, Mathematical Surveys and Monographs, vol. 258, American Mathematical Society, Providence, RI, [2021] 2021

work page 2021

[4] [4]

Auroux, L

D. Auroux, L. Katzarkov, and D. Orlov, Mirror symmetry for del P ezzo surfaces: vanishing cycles and coherent sheaves , Invent. Math. 166 (2006), no. 3, 537--582

work page 2006

[5] [5]

Bondal, Derived categories of toric varieties, Oberwolfach Reports, Vol

A. Bondal, Derived categories of toric varieties, Oberwolfach Reports, Vol. 3, No. 1, 2006, pp. 284--286

work page 2006

[6] [6]

D. Cox, J. Little, and H. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011

work page 2011

[7] [7]

Fang, C.-C

B. Fang, C.-C. Liu, D. Treumann, and E. Zaslow, A categorification of M orelli's theorem , Invent. Math. 186 (2011), no. 1, 79--114

work page 2011

[8] [8]

Guillermou, M

S. Guillermou, M. Kashiwara, and P. Schapira, Sheaf quantization of H amiltonian isotopies and applications to nondisplaceability problems , Duke Math. J. 161 (2012), no. 2, 201--245

work page 2012