Density of fibers for the filtered Fukaya category of $T^*N$

Bingyu Zhang; Claude Viterbo; St\'ephane Guillermou

REVIEW 1 major objections 5 minor 14 references

Density of fibers for the filtered Fukaya category of $T^*N$

T0 review · 1 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper proves that in the filtered Fukaya category of a cotangent bundle, the iterated cones of cotangent fibers form a dense subcategory in the interleaving distance: every closed exact Lagrangian is approximated arbitrarily well by suc

desk verdict The density theorem is genuinely new and answers Biran's question, but the proof leans hard on very recent, partly unpublished inputs—most of all [Amb25]—and the abstract promises an appendix that is missing from the body. read the letter →

arxiv 2602.21759 v2 pith:N6QAIRFP submitted 2026-02-25 math.SG math.CT

Stéphane Guillermou , Claude Viterbo , Bingyu Zhang This is my paper

classification math.SGmath.CT MSC 53D3753D4035A27

keywords filteredFukayacategorycotangentbundleinterleavingdistanceiteratedconessheafquantizationTamarkinCechresolutionRouquierdimension

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

The paper proves a density theorem: in the filtered Fukaya category of a unit cotangent bundle, the Yoneda images of cotangent fibers generate, under iterated cones, a subcategory that is dense with respect to the interleaving distance. Equivalently, any closed exact Lagrangian can be approximated to any prescribed accuracy by an iterated cone of finitely many fibers. The proof works through sheaf quantization: it first establishes a stronger density statement in the Tamarkin category of sheaves on N×R, then transfers the approximation back to the Fukaya side via a quantization functor. A Cech resolution of the constant sheaf on the base manifold N is the engine: it writes any sheaf as an iterated cone of pieces supported near points, which are then approximated by projected fibers. The paper also gives a sharper version: if direct sums are allowed, only dim N + 1 iterated cones are needed, bounding the interleaving Rouquier dimension.

What carries the argument

The carrying object is the pair consisting of the sheaf quantization functor Q and the projector P'_{DT*N}. Q sends a closed exact Lagrangian brane L to a sheaf Q(L) on N×R with reduced microsupport in the unit cotangent bundle; the projector P', a right adjoint to the embedding of the Tamarkin subcategory, produces objects W(x,a) from the skyscraper sheaves k_{x}×[a,∞). The distance-comparison result shows that applying P' does not increase the interleaving distance, so a Cech-resolution approximation in the sheaf category — where sheaves are built from constant sheaves on small balls, then from point-fibers, then from W(x,a) — descends to an approximation in the filtered Fukaya category vi

What would settle it

Exhibit a single closed exact Lagrangian in DT*N whose Yoneda module stays at positive interleaving distance from every iterated cone of cotangent fibers — or, more minimally, find a pair of non-transverse Lagrangians for which the continuation map FC^*(L, Γ_{d f0}) → FC^*(L, Γ_{d f n}) raises the action filtration, contradicting Proposition 4.2. Either observation would disprove the density theorem as stated.

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Extended reading notes

Core claim

The central discovery is that the question of whether cotangent fibers generate a dense subcategory has a positive answer, and that the obstruction to an exact generation statement is purely metric, not algebraic. In the filtered Fukaya category, a Hamiltonian perturbation of a Lagrangian is isomorphic to the original only when they coincide geometrically, so generation cannot be exact; the paper shows that the next best thing holds — density in the interleaving distance. The mechanism is a transfer from sheaves: the sheaf quantization functor sends Lagrangians to sheaves on N×R with reduced microsupport in the unit cotangent bundle, and in that sheaf category a Cech resolution of the consta

Load-bearing premise

The theorem collapses if the cited construction of the filtered Fukaya category fails to provide regular Floer data with filtration-preserving A_infinity operations, particularly for non-transverse intersections where holomorphic clusters must be counted instead of disks; the paper explicitly outsources that verification.

Editorial extensions

If this is right

For every closed exact Lagrangian and every ε>0, some iterated cone of finitely many cotangent fibers lies within ε of it in the interleaving distance.
The subcategory generated by the fibers is dense in the filtered Fukaya category, settling the density question affirmatively.
Allowing direct sums, dim N + 1 iterated cones suffice; consequently the interleaving Rouquier dimension of the relevant category is at most dim N.
Because the sheaf-side density statement is independent of perturbation data, the Fukaya-side approximation bound holds uniformly across all perturbation data for which the filtered category is defined.
The sheaf-side theorem applies to sheaves associated with immersed or C^0 Lagrangians, so the density phenomenon extends to those settings.

Reading between the lines

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

A natural next step is to extract explicit quantitative bounds: the proof's constants (10 × 8^n ε) suggest an algorithmic procedure that, given ε, outputs the number and location of fibers; the paper does not optimize these constants.
If the density theorem composes with the quotient that kills Tamarkin torsion, it should recover, and refine, the known generation of the wrapped Fukaya category by a single fiber; the paper sketches this via interleaving Rouquier dimension but leaves the idempotent-completion equivalence as a claim.
The same Cech-approximation strategy may adapt to other generating sets (e.g., graphs of functions or conormal bundles) as long as one can approximate the local pieces by the chosen generators; this is an extension the paper does not pursue.
One could test the sharpness of dim N + 1 by asking whether dim N cones suffice generically; the paper only proves an upper bound.

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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

The density theorem is real. Theorem 1.1 answers Biran's question, and the sheaf-side version is actually stronger. The proof strategy is coherent: Cech descent on the base, a wrapping projector, and a clean reduction from filtered Fukaya to Tamarkin sheaves. The lemmas on interleaving distances and iterated cones are carefully done and I did not find an internal contradiction. The paper also introduces interleaving Rouquier dimension, which looks like a useful invariant, and the bound IRdim ≤ dim N is a natural structural result.

The new content is not recycled: Abouzaid's theorem is unfiltered, Biran's question was posed in a talk, and the concurrent [ABC26] is acknowledged. The self-citations are tool citations—quantization, projectors, persistence—not disguised versions of the main claim. No circularity.

Now the soft spots, in proportion. First, the abstract promises an appendix proving that exact Lagrangians are never totally bounded for the spectral distance, but no such appendix appears. That is an editorial defect, not a mathematical one, but it should be fixed before publication. Second, and more substantively, the paper outsources the hardest analytical step. Proposition 4.2 is explicit: the filtered A-infinity operations that preserve filtration are the main contribution of [Amb25], and the non-transverse intersection counts are left to that paper's holomorphic cluster machinery. If [Amb25] is wrong, Theorem 1.1 is unsupported. The authors disclose this, and the reduction to sheaves is independent enough that a referee can check the logic without redoing the Floer analysis, but the foundation is external. The imported projector from [KSZ23]/[Kuo23] and the persistence structure from [BCZ] are also recent, mostly by the same group. That is not by itself a flaw—these are tools proved elsewhere—but it does mean the paper's solidity is inherited.

The structural logic holds up at the level I can verify. The distance estimates in Section 7 are plausible and the role of the projector is clear. I would want a referee who knows both filtered Fukaya categories and microlocal sheaf theory, and I would ask the authors to either supply the missing appendix material or remove the abstract claim, and to state explicitly which parts of Proposition 4.2 are imported verbatim from [Amb25].

This deserves a serious referee, not a desk reject. It is an important step for the filtered Fukaya category and worth the refereeing effort.

Referee Report

1 major / 5 minor

Summary. The paper proves a density theorem for the filtered Fukaya category of a cotangent bundle: every closed exact Lagrangian L is arbitrarily close (in the interleaving distance on the Yoneda module category) to an iterated cone of cotangent fibers V(x_i,a_i). The proof uses Viterbo's sheaf quantization functor, enhanced to the filtered A_∞ setting, and reduces the problem to a sheaf-theoretic density result in the Tamarkin category. The sheaf result is proved by a Čech cover of the base N, approximating F⊗k_{U_J^cl} by F⊗k_{x_J}, and then approximating the one-dimensional factors by cones of sheaves k_{x_J}×[a,∞). A projector P'_DT*N transfers the approximation to the wrapped sheaf objects W(x,a), and the Lipschitz property of the induced functor Q_* transfers it back to the Fukaya module category. An appendix introduces an interleaving Rouquier dimension and gives a bound IRdim≤dim N.

Significance. If correct, the main theorem answers a question of Biran and Cornea and provides the expected filtered analogue of Abouzaid's generation theorem, with quantitative estimates. The sheaf-theoretic density statement (Theorem 5.2) is natural and likely of independent interest; it also gives a perturbation-independent bound. The paper is structurally well organized: the reduction from Fukaya to sheaves is coherent, the main proof is explicit, and the quantitative constants are tracked. A notable strength is that the sheaf-side result is stronger than the Fukaya statement and yields a uniform bound for all perturbation data, assuming the comparison-functor compatibility promised in Remark 4.1. However, the central theorem is conditional on the filtered Fukaya category construction of [Amb25] and other external tools, several of which are preprints; this external dependence is the main risk to correctness.

major comments (1)

[§7.3, proof of Theorem 5.2] The step 'By our hypotheses F_J is a limit of constructible sheaves and, by Lemma 7.13, we deduce that there exists D_J ... such that γτ(F_J,D_J)<ε' is not immediate: Lemma 7.13 applies to constructible sheaves, not to limits. One must first approximate F_J by a constructible sheaf within ε/2 and then apply Lemma 7.13 to that constructible approximation. This is a local gap that is easily repairable, but as written the proof skips a necessary argument.

minor comments (5)

[Theorem 5.2 and §6] The distance γ_s used in Theorem 5.2 is not defined before that statement; the proof uses γτ. Please define γ_s explicitly as the Tamarkin interleaving distance γτ.
[Introduction and §5] The subcategory ⟨V(x1,a1),...,V(xl,al)⟩ is described as consisting of 'iterated cones on the generators,' but the proof uses direct sums of generators at each stage. Remark 1.3 mentions direct sums, but the formal definition should include them or state that the iterated cones are taken on direct sums.
[§5, diagram before Proposition 5.1] The text says 'we let i_* be the functor induced on the filtered derived categories,' but the diagram uses i^* for the restriction functor. This is a notational inconsistency that should be fixed.
[Proposition 4.8 proof] The convergence of F(φ_n(L1),L2) to F(L1,L2) is attributed to '[BCZ, Theo. 3.-4-(i)]'; the theorem number seems malformed. Please give a precise reference or statement.
[Appendix A] The claim Idem((C(DT*N))∞)≃Perf(WF(T*N)) is presented only as an idea, with the endomorphism-algebra computation deferred to other papers. Since this is not needed for the main theorem, it would be helpful to mark it as a conjecture or to give a complete proof if it is to remain a claim.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the density theorem is proved from sheaf-theoretic Cech descent and one-dimensional constructible approximation, not from its own statement; self-citations and the [Amb25] dependency are conditional inputs but not circular reductions.

full rationale

The claimed derivation does not reduce to its inputs by construction. Theorem 1.1 is transferred to Theorem 5.2 through the quantization functor Q and the 1-Lipschitz functor Q*, with V(x,a) identified to Q*W(x,a) in (5.4); this identification is proved in Prop. 5.1 from the sheaf-quantization property Cor. 4.5, not assumed as the conclusion. The sheaf density theorem is then proved by choosing an epsilon-cover, writing F as a Cech iterated cone, approximating F⊗k_U by F⊗k_x (Lemmas 7.7, 7.11), and approximating each one-dimensional constructible sheaf by iterated cones of half-lines (Lemma 7.13); no parameter is fitted and no target object is inserted into the hypotheses. The final application to Lagrangians uses only the constructibility of Q(L) restricted to fibers (Rem. 4.6). The main caveats are external conditionality and self-citation: the filtration-preserving Floer data are explicitly outsourced to [Amb25] in Prop. 4.2 ('This is the main contribution of [Amb25]'), and the projector/quantization tools come from [Vit19], [KSZ23], [KZ25], [Zha23], mostly overlapping with the authors. These are load-bearing but independent tools, not statements equivalent to the density theorem. The manuscript itself flags omitted verifications (e.g., 'we lack references' in Section 4; 'we will not verify it here' in Appendix A; Remark 4.1 on perturbation-data compatibility), which are completeness concerns, not circularity. Hence score 2: minor-to-moderate self-citation and external conditionality, but no circular reduction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 2 invented entities

The central claim rests on the filtered Fukaya category of [Amb25], the projector of [KSZ23]/[Kuo23], and the quantization functor of [Vit19] as upgraded here. No data-fitted parameters appear: the free parameters listed are auxiliary geometric choices (metric, covering scale, wrapping radius) that do not encode the target result.

free parameters (3)

Riemannian metric on N
All interleaving estimates (Lemmas 6.11–6.13, 7.4, 7.7) use the induced distance and injectivity radius; the theorem's statement is metric-independent but the quantitative constants are not.
Covering scale ε < inj(N)/3 = arbitrary ε (quantification variable)
Each approximating error in Section 7.3 is linear in ε and the cover is chosen ε-small; this is the error tolerance of the density statement, not a data fit.
Wrapping radius s and Hamiltonian scale r = s < inj(N); limit r → 1
Auxiliary parameters in the small-wrapping computation (Lemma 6.13, Lemma 7.4). The final object W(x,a) is independent of r via the limit r → 1.

assumptions (6)

domain assumption Filtered Fukaya category F(T*N) exists as a strict unital (filtered) A_∞ category with action-filtration-preserving operations (cluster counting).
Invoked in §1 ¶2 and throughout §4; the entire theorem is stated for this category, and Prop 4.2 delegates regularity/filtration preservation to [Amb25].
domain assumption Projector P'_Z = right adjoint to i_Z: T_Z(N) → T(N) exists (wrapping functor).
Used to define W(x,a) and Prop 7.5; existence is cited to [Kuo23] and [KSZ23, Prop 6.5], not proved here.
domain assumption The two translation functors on Mod_fil(F) (from F's own action and from Ch^fil) coincide.
Remark 3.1(2); cited to [BCZ, Remark 3.8] (preprint). Required for γ to be the same on modules as on objects.
domain assumption Sheaf quantization Q of [Vit19] extends to the ∞-categorical filtered setting and is fully faithful on Yoneda modules (Prop 4.8).
Section 4 constructs the extension, but full faithfulness relies on [Vit19, Prop 8.1, Coro 8.2, Prop 9.9] and on convergence arguments from [AI20].
standard math Standard microlocal sheaf calculus (Kashiwara–Schapira): microsupport bounds, microlocal cut-off, microlocal Morse lemma, sheaf quantization of Hamiltonians (GKS12).
Used throughout Sections 6–7 (Lemmas 6.2, 6.11, 6.12–6.13, 7.7); treated as background.
standard math Cech resolution: constant sheaf on N is an n-step iterated cone of k_{U_J} for a good cover (Lemma 7.12); constructible sheaves on R are iterated cones of k_{[a,∞)} (Lemma 7.13).
Standard; Lemma 7.13 requires a coefficient ring with finite free resolutions for finite type modules (Z/2 satisfies it).

invented entities (2)

W(x,a) = P'_DT*N(k_{x}×[a,∞)) — wrapped fiber in the Tamarkin category
purpose: Approximating objects in Sh_DT*N whose Q* images are the Yoneda modules V(x,a); the bridge that lets sheaf density imply Fukaya density.
Defined via the projector; its properties (Lemma 7.4, Prop 7.5) are proved inside this paper; no external evidence outside this paper yet.
Interleaving Rouquier dimension IRdim (with solo-approximators)
purpose: Complexity measure for stable ∞-categories with R-actions; captures the number of iterated cones needed up to ε.
New invariant (Appendix A, Def A.1); Prop A.3 bounds IRdim ≤ dim N. The definition is checkable, but no independent computation by other groups exists yet.

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Pith. "Pith review of Density of fibers for the filtered Fukaya category of $T^*N$." pith.science (2026). https://pith.science/paper/N6QAIRFP

@misc{pith2026260221759,
  author       = {Pith},
  title        = {Pith review of: Density of fibers for the filtered Fukaya category of $T^*N$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N6QAIRFP}},
  note         = {Machine review of arXiv:2602.21759}
}

abstract

We answer a question of Biran and Cornea about the density of iterated cones of fibers in the Fukaya category of a cotangent bundle. We prove that indeed if we take a dense set of basepoints, the iterated cones of the cotangent fibres are dense in the Filtered Fukaya category. In an appendix we prove that the space of exact Lagrangians in a symplectic manifold is never totally bounded for the spectral distance (unless it is empty). This was implicit in \cite{MCA-VH-CV} for $n=1$ and proved for cotangent bundles of negatively curved manifolds in \cite{A-B-C}.

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Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Abouzaid.A cotangent fibre generates the Fukaya category.Advances in Math- ematics228.2 (2011), pp

[Abo11] M. Abouzaid.A cotangent fibre generates the Fukaya category.Advances in Math- ematics228.2 (2011), pp. 894–939.DOI:10.1016/j.aim.2011.06.007. [Amb25] G. Ambrosioni.Filtered Fukaya categories.Journal of Symplectic Geometry23.2 (2025), pp. 423–509.DOI:10.4310/jsg.250624013956. [ABC26] G. Ambrosioni, P . Biran, and O. Cornea.Approximability for Lagra...

doi:10.1016/j.aim.2011.06.007 2011

[2] [8]

Kashiwara and P

[KS18] M. Kashiwara and P . Schapira.Persistent homology and microlocal sheaf theory. 2018.DOI:10.1007/s41468-018-0019-z. arXiv:1705.00955 [math.AT]. [Kuo23] C. Kuo.Wrapped Sheaves.Advances in Mathematics415 (2023), p. 108882.DOI: 10.1016/j.aim.2023.108882. [KSZ23] C. Kuo, V . Shende, and B. Zhang.On the Hochschild cohomology of Tamarkin cat- egories

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[Lur17] J

arXiv:2503.15933. [Lur17] J. Lurie.Higher Algebra. Preprint, available athttps : / / www . math . ias . edu / ~lurie/papers/HA.pdf

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arXiv:2510.26269 [math.AG]. REFERENCES 34 [Tam08] D. Tamarkin.Microlocal condition for non-displaceablility. Ed. by M. Hitrik et al. 2008.DOI:10.1007/978-3-030-01588-6_3. arXiv:0809.1584 [math.SG]. [Vit19] C. Viterbo.Sheaf Quantization of Lagrangians and Floer cohomology

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arXiv:2306.12316. UMR CNRS 6629DUCNRS, LABORATOIRE DEMATHÉMATIQUESJEANLERAY, 2 CHEMIN DE LAHOUSSINIÈRE, BP 92208, F-44322 NANTES CEDEX3 FRANCE Email address:Stephane.Guillermou@univ-nantes.fr UNIVERSITÉPARIS-SACLAY, CNRS UMR 8628, LABORATOIRE DE MATHÉMATIQUES D’ORSAY, 91405, ORSAY, FRANCE. Email address:claude.viterbo@universite-paris-saclay.fr CENTRE FOR...

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Springer Verlag, 2014, pp

Lecture Notes of the Unione Matem- atica Italiana. Springer Verlag, 2014, pp. 43–85.DOI:10 . 1007 / 978 - 3 - 319 - 06514-4_3. [GVit22a] S. Guillermou and C. Viterbo.The support of sheaves isγ-coisotropic.Geometric and Functional Analysis34 (2024), pp. 1052–1113.DOI:10.1007/s00039- 024- 00682-x. arXiv:2203.12977 [math.SG]. [Ike19] Y. Ike.Compact Exact Lag...

arXiv 2014

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Wissenschaften

Grundlehren der Math. Wissenschaften. Springer-Verlag, 1990.DOI:10.1007/978-3-662-02661-

work page doi:10.1007/978-3-662-02661- 1990

[8] [2018]

A ∞-bimodules and SerreA ∞-functors

[Kerodon] J. Lurie et al.An online resource for homotopy-coherent mathematics.2025.URL: https://kerodon.net. [LM08] V . Lyubashenko and O. Manzyuk. “A ∞-bimodules and SerreA ∞-functors”. In:Ge- ometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov. Basel: Birkhäuser Basel, 2008, pp. 565–645.ISBN: 978-3-7643-8608-5.DOI:10 . 1007/978-3-7...

2025