Fourier transforms and a filtration on the Lagrangian cobordism group of tori

\'Alvaro Mu\~niz-Brea

REVIEW 2 major objections 4 minor 30 references

Fourier transforms and a filtration on the Lagrangian cobordism group of tori

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For symplectic tori built from polarized tropical affine tori, the fibered Lagrangian cobordism group carries a finite geometric filtration of length n+1.

desk verdict A substantial paper whose Fourier transform half is solid, but whose main filtration theorem currently rests on an unproved immersed-to-embedded passage. read the letter →

arxiv 2411.16543 v2 pith:D4PYQM2W submitted 2024-11-25 math.SG math.AG

Álvaro Muñiz-Brea This is my paper · ORCID

classification math.SGmath.AG MSC 53D1253D3714C1514T05

keywords LagrangiancobordismsymplectictoritropicalaffineparallelotopefiltrationPontryaginproductFouriertransformhomologicalmirrorsymmetryChowgroups

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 studies the group generated by Lagrangian torus fibers of a symplectic torus modulo Lagrangian cobordism, the equivalence relation generated by Lagrangian submanifolds of $X\times\mathbb{C}$ with prescribed cylindrical ends. For symplectic tori built from a polarized tropical affine $n$-torus, it proves that the parallelotope filtration, defined by Pontryagin products of differences of fibers, terminates after $n+1$ steps. Equivalently, any $n+1$-fold Pontryagin product of fiber differences is Lagrangian cobordant to zero. Earlier work had shown these cobordism groups are infinite-dimensional, so the result establishes that infinite generation can coexist with a finite, geometrically meaningful filtration. The paper also constructs a Fourier transform between the Fukaya categories of dual tori and shows it is mirror to the algebraic Fourier transform, connecting the filtration to a classical vanishing statement for zero-cycles on abelian varieties.

What carries the argument

The central object is the parallelotope filtration $F^i\mathrm{Cob}_{\mathrm{fib}}(X)=\mathrm{Cob}_{\mathrm{fib}}(X)_{\mathrm{hom}}^{\star i}$, where $\mathrm{Cob}_{\mathrm{fib}}(X)_{\mathrm{hom}}$ is generated by differences of Lagrangian torus fibers and $\star$ is the Pontryagin product induced by the group law on the base. The argument is carried by three mechanisms: the duality between the two Lagrangian torus fibrations on $X(B)$, which turns the Pontryagin product into fiberwise addition; the use of polarizations to write every constant section, up to Hamiltonian isotopy, as the graph of a tropical rational function; and Lagrangian surgery along tropical hypersurfaces, which converts differences of sections into lifts of tropical subvarieties. A transversality statement for the tropical hypersurfaces $V(f_{k,\delta})$ ensures that an $(n+1)$-fold fiberwise sum is empty. In the second half, the key object is the graded symplectomorphism $\iota(q,p)=(-p,q)$ between $X(B)$ and $X(B^\vee)$, which induces the symplectic Fourier transform and is proved to be the mirror of the algebraic Fourier transform.

What would settle it

For $n=1$, Theorem A asserts that $(F_a-F_b)\star(F_c-F_d)=0$ in the fibered Lagrangian cobordism group of the $2$-dimensional symplectic torus. Writing down the corresponding embedded Lagrangian in $X(B)\times\mathbb{C}$, or showing by a direct topological obstruction that no embedded Lagrangian with those ends exists, would settle whether the immersed-to-embedded reduction is valid.

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Extended reading notes

Core claim

The central claim, Theorem A, is that for a polarized tropical affine torus $B$ of dimension $n$, with $X(B)$ the associated symplectic torus, the parallelotope filtration satisfies $F^{n+1}\mathrm{Cob}_{\mathrm{fib}}(X(B))=0$. Here $F^i$ is the subgroup generated by $i$-fold Pontryagin products of elements $F_p-F_q$, where $F_p$ and $F_q$ are Lagrangian torus fibers. The proof reinterprets fibers of the fibration $X(B)\to B$ as constant sections of the dual fibration $X(B)\to F$, so the Pontryagin product becomes fiberwise addition. In the polarized case, constant sections are Hamiltonian isotopic to graphs of tropical rational functions, and differences of such sections are cobordant to lifts of tropical hypersurfaces of codimension at least one. Choosing the tropical functions generically makes the hypersurfaces transverse, so a fiberwise sum of $n+1$ of them is empty; the paper concludes that the corresponding element of the fibered Lagrangian cobordism group vanishes. A second theorem identifies the symplectic Fourier transform between Fukaya categories of dual tori with the functor induced by the graded symplectomorphism $(q,p)\mapsto(-p,q)$, and shows that under homological mirror symmetry it matches the algebraic Fourier transform; this is then used to present the filtration as the mirror of the classical Pontryagin-power filtration on zero-cycles of an abelian variety.

Load-bearing premise

The proof of the vanishing in the embedded cobordism group relies on the assertion that, although the intermediate cobordisms have immersed ends, fiberwise summing $n+1$ of them produces a valid relation in the original group; the paper states this but does not supply a formal embedding or gluing argument.

Editorial extensions

If this is right

The subgroup generated by differences of Lagrangian fibers is nilpotent under the Pontryagin product, with nilpotency index at most $n+1$, so high powers of fiber differences carry no Lagrangian-cobordism information.
The first two graded pieces are explicit: $F^0/F^1\cong H^0(B;\mathbb{Z})$ and $F^1/F^2\cong \mathrm{Alb}(B)$, the Albanese torus of the base, giving computable invariants of the otherwise infinite-dimensional group.
The symplectic Fourier transform between dual tori is induced by an honest graded symplectomorphism, not by a kernel object, so Lagrangian submanifold invariants can be transported between dual tori by a coordinate change.
Under homological mirror symmetry the filtration corresponds to the classical Pontryagin-power filtration on zero-cycles of the dual abelian variety, placing the symplectic cobordism result inside the theory of algebraic cycles.
The polarization hypothesis is essential: for non-polarizable bases the filtration can be unbounded rather than terminating, by previously known results on Lagrangian cobordism of torus fibers.

Reading between the lines

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

The paper does not determine sharpness of the filtration length; since $F^1/F^2$ is nonzero, at least the first step carries information, but whether lower graded pieces can be nonzero for $n>1$ is left open. Computing explicit examples for $2$- and $3$-dimensional polarized tori would test this.
The main unresolved technical step is the passage from immersed intermediate Lagrangians to embedded cobordisms after summing $n+1$ of them; if that passage fails, the immersed version of Theorem A would stand while the embedded statement would not. This could be settled by writing down explicit regularizations for small $n$.
Because the symplectic Fourier transform is a symplectomorphism, it should commute with all Lagrangian cobordism invariants, not just the objects of the Fukaya category; a concrete consequence would be a coordinate-swap relation for Lagrangian Floer cohomology groups of dual tori, up to the grading shift computed in the paper.
The transversality machinery for tropical hypersurfaces is not obviously restricted to tori; it may extend to other affine manifolds with Lagrangian torus fibrations, yielding finite filtrations on cobordism groups of a broader class of mirror-symmetric spaces if the embeddedness gap can be closed.

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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

Here's my take. The paper has two real results. Theorem B, the symplectic Fourier transform, is clean: the map ι is a graded symplectomorphism, the grading computation is explicit, and the identification with Mukai's transform via Abouzaid's HMS is a nice piece of work. That half deserves to be published. Theorem A, the finite parallelotope filtration, is the headline and it's a good idea—the switch from Pontryagin product on fibers to fiberwise addition on the dual fibration is elegant, and the use of tropical hypersurfaces is genuinely new. The first two graded pieces being H^0(B) and Alb(B) is a real structural observation.

The soft spot is exactly where the stress-test puts it. Lemma 3.10 proves a cobordism relation in a group where immersed Lagrangians are allowed as generators. The paper then uses Lemma 3.17 and Theorem 3.18 to conclude vanishing in the original embedded cobordism group. The paragraph after Lemma 3.10 says that after fiberwise summing n+1 such cobordisms one gets a valid relation in the embedded group, but that is asserted, not proved. Proposition 3.15 arranges transversality of the tropical hypersurfaces; it does not construct an embedded cobordism or prove that the map from the embedded group to the immersed group is injective on the relevant subgroup. Since that map can have kernel, vanishing in the immersed group does not automatically imply vanishing in Cob_fib(X). This is a load-bearing gap, not a cosmetic one.

A couple of smaller points. Assumption 5.6 is flagged, and the paper is honest that it is only needed for the Chow-group comparison, so I wouldn't call that a flaw, just a limitation. The references are fair: Sheridan-Smith, Hicks, Abouzaid, Subotic are the right names, and the author says where each tool comes from. No circularity: the filtration theorem isn't assumed anywhere.

Who should read this? Symplectic topologists and anyone working on mirror symmetry for tori. The Fourier transform section is a good read now. Theorem A is worth serious referee time, but only with a request to close the immersed-to-embedded gap: either an explicit embedded cobordism realizing the relation after taking n+1 fiberwise sums, or a proof that the relevant map from Cob_fib(X) to the immersed group is injective. I'd accept it for review, with the clear expectation of a major revision.

Referee Report

2 major / 4 minor

Summary. The paper studies the fibered Lagrangian cobordism group Cob_fib(X) of a symplectic torus X = X(B) associated to a polarized tropical affine torus B. It defines a decreasing parallelotope filtration by powers of the ideal generated by differences F_p - F_q under the Pontryagin product, and claims Theorem A: this filtration terminates after n+1 steps. The proof strategy is to exchange Pontryagin products of fibers for fiberwise addition of flat sections of the dual fibration, replace flat sections by tropical Lagrangian sections using a polarization, and then use Hicks's surgery cobordisms to convert differences of sections into lifts of tropical hypersurfaces. Transversality of the tropical hypersurfaces is then used to conclude that n+1-fold fiberwise additions vanish. The second half constructs a symplectic Fourier transform on Fukaya categories induced by the graded symplectomorphism (q,p) -> (-p,q), proves that under Abouzaid's homological mirror symmetry it corresponds to Mukai's Fourier-Mukai transform, and connects the filtration to the Bloch filtration on Chow groups of abelian varieties.

Significance. If Theorem A is established, it is a genuine structural result: previous work of Sheridan-Smith and the author showed that these cobordism groups are infinite-dimensional, so a natural finite geometric filtration with understood first graded pieces is new and interesting. The Fourier transform theorem is elegant, explicit, and independent of the main gap in Section 3; identifying the symplectic Fourier transform with a graded symplectomorphism is a clean geometric statement that should be of independent value. The paper is also unusually honest about its conditional inputs: Remark 1.11 and Assumption 5.6 explicitly flag statements that have not appeared in the literature, and Remark 1.6 states that the Fourier transform material is not needed for Theorem A. The central proof, however, currently passes through a cobordism group with immersed generators without a rigorous return to the embedded group, so Theorem A is not yet established as written.

major comments (2)

[§3.3 (Theorem 3.18), §3.1 (Lemma 3.10), §2.3 (Definition 2.8)] The proof of the main theorem proves a statement in the wrong cobordism group at a load-bearing step. Lemma 3.10 concludes Eq. (14) only in "a cobordism group of immersed Lagrangians modulo immersed unobstructed cobordisms (although not in our original cobordism group)"; Lemma 3.17 then states that its representatives are "possibly immersed tropical Lagrangians". The paragraph immediately before Lemma 3.10 says that after fiberwise summing n+1 of these cobordisms one obtains a valid relation in the original group, but no construction or proof of this assertion is given. Proposition 3.15 proves only transversality of the tropical hypersurfaces, not that the iterated fiberwise compositions are embedded or that a projected cobordism can be chosen embedded. Remark 3.11 ensures only that each L_{φ±} is individually embedded. Therefore the final step of Theorem 3.18—"lives over tropical subvarieties of codimension at least n+1, hence vanishes"—establishes vanishing of the image of L in the immersed cobordism group. Since the map from Cob_fib(X), whose generators are embedded by Definition 2.8, to the immersed cobordism group is not shown to be injective, Theorem A does not follow as written. The missing ingredient is either an explicit embedded cobordism realizing the asserted n+1-fold fiberwise sum, or a proof of injectivity of the embedded-to-immersed map on the filtration under consideration.
[§1.4, Remark 1.11; §5.1, Assumption 5.6; Proposition 5.8] The paper's advertised connection to the Bloch filtration is conditional rather than a proved theorem. Proposition 5.8 uses the homomorphism Cob(X) → K0(Fuk(X)) whose existence Remark 1.11 says "has not yet appeared in the literature", and Assumption 5.6 states an unobstructed-immersed-cobordism-to-cone-decomposition principle that the paper also says has not appeared. Neither input is needed for Theorem A or Theorem B, and the paper is transparent about this, but the abstract and §1.4 present the mirror statement to Bloch's theorem as one of the main outcomes. This should be clearly labelled as conditional, or the missing functor and assumption should be proved.

minor comments (4)

[Lemma 3.4] The statement reads "A tropical affine torus B is z if and only if..."; the missing word is presumably "polarized".
[§4.1, Equations (19)–(20)] In Equation (19), the right-hand side "a" should presumably be the skyscraper sheaf O_a; as written it is not of the same type as the right-hand side of Equation (20).
[Corollary 3.8] The phrase "δ± satisfying Equation (12)" appears to be a cross-reference error: the admissibility condition on δ is Equation (10), while Equation (12) is the quasi-periodicity of f_{k,δ}.
[§1.3, Definition 4.6 and §1.2, Definition 1.4] The algebraic Fourier transform, the symplectic Fourier transform, and individual Lagrangian fibers are all denoted F. This is manageable in context but could be confusing; a distinct notation for the algebraic versus the symplectic functor would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A is proved via tropical surgery and transversality results, and Theorem B via Abouzaid's HMS plus a standard Fourier–Mukai uniqueness lemma; the caveat about embedded versus immersed cobordisms is a rigor gap, not a circular step.

full rationale

The paper's main results are not obtained by assuming their own conclusions. Theorem A (Theorem 3.18) reduces the vanishing of F^{n+1}Cob_fib(X) to Lemma 3.17, whose inductive proof uses Corollary 3.8 and Lemma 3.10 to replace differences of flat sections by lifts of tropical hypersurfaces, and Proposition 3.15 to arrange transverse intersections. The input hypotheses—polarized tropical affine torus, constant sections, Hicks surgery cobordisms, and Sheridan–Smith regularity results—do not include the target vanishing, so the derivation is self-contained modulo external results. Theorem B identifies the homological-mirror conjugate of Mukai's Fourier–Mukai functor with the functor induced by the coordinate swap ι(q,p)=(-p,q). The proof uses the standard fact (Lemma 4.8, citing Huybrechts) that an autoequivalence preserving skyscrapers and the structure sheaf is a line-bundle twist, then verifies the two characterizing equations. Definition 4.6 does define the symplectic Fourier transform as the HMS-conjugate of the algebraic Fourier transform, so the 'correspondence to Mukai' is partly by construction; the nontrivial content is the symplectomorphism description, and that is proved rather than assumed. The only caution is an exposed gap surrounding Lemma 3.10: the relation (14) is said to hold in an immersed cobordism group, and the text asserts that after fiberwise summing n+1 such cobordisms one obtains a valid relation in the embedded group, citing Proposition 3.15, but Proposition 3.15 only establishes tropical transversality and does not explicitly construct the required embedded cobordism. This is a correctness/rigor concern about the reduction from immersed to embedded cobordism classes, not a circularity: no fitted parameter is renamed a prediction, no theorem is assumed in its own proof, and the author's self-citation [MB24] is used only as a contrast regarding infinite-dimensionality and is not load-bearing. External citations—Abouzaid, Hicks, Sheridan–Smith, Fukaya—are used as genuine ingredients. Hence the circularity score is 0.

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

The central claim of Theorem A rests on standard-domain assumptions from tropical geometry and Lagrangian surgery, all cited or developed in the paper. The only clearly ad hoc inputs are Assumption 5.6 and the existence of the K0-map, which are explicitly flagged by the author as not yet in the literature and are not needed for the main theorem. No fitted parameters or invented physical entities are used.

assumptions (6)

domain assumption Abouzaid's homological mirror symmetry: the family Floer functor gives a fully faithful embedding Fuk(X(B)) → D^bCoh(Y(B)), which is an equivalence when B is polarized.
Used for Theorem B and the Chow-group comparison. Not needed for Theorem A.
domain assumption Hicks' Lagrangian surgery cobordism: for a tropical polynomial φ there is a Lagrangian L_φ lifting V(φ) and a Lagrangian cobordism between (Γ0, Γ(dφ)) and L_φ (Proposition 2.10).
Central tool in the proof of Theorem A, used to turn differences of sections into tropical Lagrangians.
domain assumption Sheridan-Smith's tropical regularity results: for large k and generic δ, the tropical hypersurfaces V(f_{k,δ}) are regular, and a polarization determines a Riemannian metric with integer periods (SS21 Lemma 3.24 and Lemma 3.9).
Used to express constant sections as tropical rational functions (Corollary 3.9) and to ensure embedded lifts of tropical hypersurfaces.
domain assumption Fukaya's result that geometric composition of unobstructed immersed Lagrangian correspondences is unobstructed.
Used to define fiberwise addition and the Pontryagin product in settings where immersed Lagrangians appear.
ad hoc to paper Assumption 5.6: unobstructed immersed cobordisms between weakly-exact embedded Lagrangians induce cone decompositions in the Fukaya category. The paper states this has not appeared in the literature.
Needed only for the map Cob(X) → K0(Fuk(X)) and the ring homomorphism to Chow groups (Proposition 5.8). Not needed for Theorem A or Theorem B.
ad hoc to paper Existence of the map Cob(X) → K0(Fuk(X)) (Remark 1.11), which the paper says has not yet appeared in the literature.
Used to construct the composite map from the cobordism group to Chow groups.

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Pith. "Pith review of Fourier transforms and a filtration on the Lagrangian cobordism group of tori." pith.science (2026). https://pith.science/paper/D4PYQM2W

@misc{pith2026241116543,
  author       = {Pith},
  title        = {Pith review of: Fourier transforms and a filtration on the Lagrangian cobordism group of tori},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D4PYQM2W}},
  note         = {Machine review of arXiv:2411.16543}
}

read the original abstract

Given a polarized tropical affine torus, we show that the fibered Lagrangian cobordism group of the corresponding symplectic manifold admits a natural geometric filtration of finite length. This contrasts with results of Sheridan-Smith in dimension four and the present author in higher dimensions, who showed that such group is infinite-dimensional. In the second half of this paper, we construct a Fourier transform between Fukaya categories of dual symplectic tori. We show that, under homological mirror symmetry, it corresponds to the Fourier transform between derived categories of coherent sheaves of dual abelian varieties due to Mukai. We use this to show how our filtration is mirror to the Bloch filtration on Chow groups of abelian varieties, but the results may be of broader interest.

Figures

Figures reproduced from arXiv: 2411.16543 by the authors.

**Figure 1.** Lagrangian cobordism between L− and L+. The shaded region (including the two horizontal lines) depicts a typical projection of a two-ended Lagrangian cobordism V ⊂ X × C to C. We have included (in dashed lines) an example of a compact region K ⊂ C outside which V is product type: in C \ K the projection looks like two straight lines, and living over them in X ×C we have the cylidrical Lagrangians L− × R<−a− and L+ ×… view at source ↗

**Figure 2.** Family of polarised tropical affine tori parametrized by (α1, α2, α3) ∈ R 3 >0 . The labels αi indicate the length of each edge, which have directions (1, 0),(−1, −1) and (0, 1) respectively. Remark 1.12. Up to now we have only used the fact that B is a tropical affine torus, and not that it is polarized. The second idea is to study these flat sections using the tropical geometry of the base F. We show in Corollary … view at source ↗

**Figure 3.** Hick’s surgery inside T 2 = X(S 1 ). Left: the tropical section Γ(dφ) (in red) intersects the zero-section Γ0 (in blue) everywhere except where the smoothing has been applied. Right: the surgery Lφ = Γ0#Γ(dφ) (in orange) agrees with Γ0 and Γ(dφ) outside a neighborhood of their intersection. More generally, global sections φ ∈ H0 (C trop/ Aff) define Lagrangian sections Γ(dφ) ⊂ T ∗B/T ∗ ZB. Associated to φ there is … view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

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