Lagrangian skeleta of very affine complete intersections

Danil Ko\v{z}evnikov

REVIEW 3 major objections 5 minor 19 references

The paper computes the Lagrangian skeleton of an open Batyrev–Borisov complete intersection from toric data and derives homological mirror symmetry for the pair in the large-volume limit.

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2026-08-04 07:51 UTC pith:3YSOIUTD

load-bearing objection Serious conditional preprint: new skeleton computation for BB complete intersections, but the S^{n-r} factor rests on an unproved external sphere theorem and an acknowledged MPCS assumption. the 3 major comments →

arxiv 2510.23418 v2 pith:3YSOIUTD submitted 2025-10-27 math.SG math.AG

Lagrangian skeleta of very affine complete intersections

Danil Ko\v{z}evnikov This is my paper

classification math.SG math.AG MSC 53D3714J3314M25

keywords Lagrangian skeletonvery affine complete intersectionsBatyrev–Borisov mirror pairsnef partitionFLTZ skeletonhomological mirror symmetrywrapped Fukaya categorytropical geometry

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper aims to show that the symplectic geometry of an open Batyrev–Borisov complete intersection—a complete intersection inside (C*)^n compactifying to a smooth Calabi–Yau in a Fano toric variety—is fixed by toric combinatorics. The main claim is that the Lagrangian skeleton of the associated Weinstein domain is built from standard mirror-to-toric pieces, the FLTZ skeleta, glued according to the nef partition data; explicitly, the skeleton is Φ^{-1}(LΣ)∩Bβ. From this description the paper derives a quasi-equivalence between the wrapped Fukaya category of the open variety and the derived category of a toric boundary subvariety, establishing homological mirror symmetry for Batyrev–Borisov pairs in the large-volume limit. If correct, wrapped Floer-theoretic invariants of these Calabi–Yau complete intersections reduce to finite triangulation data, and the computation is independent of choices of Kähler potential.

Core claim

The central claim is that for a centred refined triangulating function on the dual nef partition, and under the MPCS smooth-compactification hypothesis, the skeleton of the open Batyrev–Borisov complete intersection is exactly Λ = Φ^{-1}(LΣ)∩B_β: the standard FLTZ skeleton LΣ intersected with the preimage of the boundary of the joint amoeba complement under the Liouville isomorphism. This singular Lagrangian is ambient diffeomorphic to (M_{S^1} × i(S^{n−r}))∩LΣ for a suitable embedding of an (n−r)-sphere, and it admits an open cover by FLTZ pieces L_{Σ/σ}×C_σ with standard inclusions. The paper then proves Perf W(Z)^{op} ≃ D^b(Ẑ), where Ẑ = ∂^{trans}X_Σ is the transversal boundary of the tor

What carries the argument

The load-bearing identity is Theorem 6.1: skel(Ẑβ)=Φ^{-1}(LΣ)∩B_β. Here LΣ is the FLTZ skeleton ∪_{σ∈Σ} σ^⊥×σ inside T*M_{S^1}, the standard singular Lagrangian mirror to a toric variety; Φ is the Liouville isomorphism determined by a strongly adapted Kähler potential; and B_β is the joint boundary of the tailored amoeba complements. The proof localizes the complete intersection by tailoring, i.e., cutting off non-dominant monomials with compatible cut-off functions, so that near each tropical stratum the defining equations become monomial; adapted potentials and smoothing arguments then control how Φ(B_β) meets the fan Σ. The nef partition supplies the transversal-cone poset that indexes wh

Load-bearing premise

The load-bearing premise is the MPCS condition (Definition 3.31): the centred refined triangulation of the canonical compactification must avoid all orbifold strata—essentially the simplices in each polytope summand must be unimodular—so that smooth toric compactifications exist; the proof also imports, without proof, the statement that the boundary of the tropical Batyrev–Borisov complete intersection is a topological (n−r)-sphere, and if either condition fails the skeleton

What would settle it

Compute the wrapped Fukaya category of the paper's r=2 example (an elliptic curve with eight punctures) from the claimed cover by eight FLTZ pieces and compare any Hom-space with the derived category of the transversal toric boundary; a mismatch refutes the main HMS theorem. Alternatively, exhibit a nef partition with a centred refined triangulation violating MPCS whose open complete intersection still has a smooth toric compactification and satisfies the same skeleton formula, which would show the stated hypotheses are stronger than necessary.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The wrapped Fukaya category W(Z) of an open Batyrev–Borisov complete intersection is equivalent to the derived category of the transversal toric boundary Ẑ, so it is computed by finite toric data.
The skeleton's open cover by FLTZ pieces with standard inclusions lets local wrapped-Fukaya computations be reduced to homological mirror symmetry for lower-dimensional toric varieties.
The stabilisation A×C^{r−1} embeds into the cosphere bundle of the n-torus, placing these open complete intersections inside the microlocal-sheaf framework and making the HMS equivalence compatible with toric HMS.
Natural inclusions of open BBCIs obtained by summing defining equations induce inclusions of mirror subvarieties inside the toric boundary, giving functoriality of the correspondence.
Strongly adapted potentials exist for every simple polytope, so the skeleton computation is independent of earlier perfectly-centredness restrictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if the identity skel(Ẑβ)=Φ^{-1}(LΣ)∩B_β holds, the wrapped Fukaya category is an invariant of the nef partition alone, so mirror symmetry for these varieties is a finite polyhedral computation and can be deformed by changing the triangulating function.
Editorial inference: because only transversal cones contribute to the skeleton, the non-transversal part of the FLTZ skeleton is invisible to the wrapped category of the open variety; this predicts a clean separation between wrapped invariants (which see only the transversal boundary) and compact invariants (which see the whole toric boundary).
Editorial inference: the MPCS unimodularity hypothesis is the main obstacle to generalising the result; if the paper's sketched orbifold-chart version can be carried out, the same formula would likely extend to complete intersections beyond nef/reflexive settings, including rigid mirrors.
Editorial inference: the recursive FLTZ cover gives a direct way to write the wrapped Fukaya category as a cosheaf on the skeleton, potentially making symplectic cohomology and Hochschild homology of these open Calabi–Yaus cellular and computable.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Serious conditional preprint: new skeleton computation for BB complete intersections, but the S^{n-r} factor rests on an unproved external sphere theorem and an acknowledged MPCS assumption.

read the letter

Serious preprint, roughly 100 pages, computing Lagrangian skeleta of open Batyrev–Borisov complete intersections and using them to prove HMS at the large-volume limit. If the main results hold, this is the r>1 generalization of the hypersurface story from Gammage–Shende and Zhou, and Theorem 1.1 is genuinely new: the complete-intersection skeleton, the cover by FLTZ pieces, and the categorical equivalence with the transversal boundary of the toric variety.

The visible core is in good shape. The tropicalisation of complete intersections (Theorem 3.15, Lemma 3.16) is spelled out; the tailoring for complete intersections (Proposition 5.10) reduces to explicit estimates; the skeleton computation in Section 6 is organized; and Section 7 gives a concrete cover of the skeleton by standard FLTZ pieces using a nef partition scaling action. The paper also scopes itself honestly: Remark 1.2 flags the reliance on [HZ05], the unused Lemma 6.9 is noted, and the mirror map is deferred.

The soft spots are real but conditional, not fatal. The S^{n-r} factor in Theorem 1.1(iii) depends entirely on [HZ05, Thm 2.5], which says the bounded complex of a tropical BBCI is a topological sphere. The paper cites it without proof, and the stress-test note is right: this is a global PL input, not a consequence of the local cell complex description, and the smoothing results in Appendix B cannot create the sphere factor. The paper itself calls this the main bottleneck. A referee needs to verify that the hypotheses of [HZ05] match this setting, including the non-unimodular simplices allowed by Definition 2.13. The MPCS standing assumption is the second load-bearing point: if the projective simplicial refinement cannot avoid the orbifold strata, the smooth toric compactification and the Liouville-domain argument collapse. That restriction is acknowledged, so it is a scope condition rather than a hidden flaw.

I could not check the sharpest parts: the four-case zero-escape lemma, the smoothing machinery, and the HMS proof in Section 9, which is truncated in the review copy. No error surfaced in what I could read, so CONDITIONAL/MODERATE is the right verdict. This deserves a serious referee. Send it out, and ask specifically for verification of the HZ05 applicability and the MPCS/compactification step.

Referee Report

3 major / 5 minor

Summary. The paper claims to compute the Lagrangian skeleton of open Batyrev–Borisov complete intersections (BBCIs) inside (C*)^n in the large-β limit. Theorem 1.1 asserts that the skeleton Λ is a singular Lagrangian ambient diffeomorphic to (M_{S^1} × i(S^{n-r})) ∩ L_Σ, with an open cover by standard FLTZ pieces L_{Σ/σ} × C_σ indexed by transversal cones. Theorem 1.3 then uses this skeleton description, together with the stabilization and microlocal sheaf framework, to prove a quasi-equivalence Perf W(Z)^op ≅ D^b(Ẑ), where Ẑ is the transversal boundary of the toric compactification, and Theorem 1.4 states compatibility with toric HMS. Sections 3–5 develop the necessary tropical, potential-theoretic, and tailoring machinery; Section 6 computes the skeleton via the Liouville flow; Sections 7–9 analyse its topology and derive the HMS consequences.

Significance. If the main results hold, this is a substantial advance: it extends the known hypersurface skeleton computations of Nadler, Gammage–Shende, and Zhou to a large class of complete intersections and provides an explicit homological mirror symmetry statement for Batyrev–Borisov pairs in the large-volume limit. The paper makes real technical contributions: a detailed tropicalization and smoothing theory for complete intersections, strongly adapted potentials removing perfect-centredness assumptions, a complete-intersection tailoring statement with explicit estimates, and a careful cover of the skeleton by FLTZ pieces. The recognition that the global topology of the skeleton depends on an external sphere theorem is honest, but the manuscript does not fully verify that theorem's hypotheses in its setting, which leaves a load-bearing point unresolved.

major comments (3)

[Remark 6.2 and Section 3.3 (Prop. 3.21)] Theorem 1.1(iii) asserts that the skeleton is ambient diffeomorphic to (M_{S^1} × i(S^{n-r})) ∩ L_Σ. The factor S^{n-r} is justified only by citing [HZ05, Thm 2.5] in Remark 6.2. Proposition 3.21 identifies the relevant complex |T^∇| combinatorially as ∂(r∆∨) ∩ ∇, but it does not prove that this complex is a topological (n−r)-sphere, and sphericality is a global PL fact not implied by the local cell description. Since the manuscript allows refined but not necessarily unimodular triangulations (Definition 2.13), the precise hypotheses of [HZ05, Thm 2.5] must be checked in this setting. The paper itself flags this as a bottleneck in Remark 1.2. Please either include a proof of the applicability of [HZ05, Thm 2.5] to the present data, state it as a clearly identified assumption in Theorem 1.1, or supply a self-contained proof of the sphericality of the skeleton factor.
[Section 3.4, Definition 3.31, Corollaries 3.33 and 5.28–5.30] The passage from the very affine complete intersection Z_β to a Liouville domain A, and hence to the HMS equivalence in Theorem 1.3, relies on the existence of a smooth toric compactification under the MPCS condition. The manuscript notes (Remark 3.32) that MPCS essentially requires unimodularity of the simplices in ˇT_j, but it does not prove that any nontrivial family of BBCIs beyond the toy Example 2.8 satisfies this condition, nor does it provide a substitute construction if MPCS fails. Because Corollary 3.33 and Theorem 5.27 feed directly into Corollaries 5.28–5.30, this is load-bearing for the main theorem. Please state the scope of the MPCS assumption more precisely, give a concrete class of examples where it holds, or explain how the orbifold-chart route mentioned in Remark 3.32 would repair the argument.
[Section 6.3, proof of Lemma 6.12 (Cases 2 and 3)] The zero-escape argument is the most delicate step in the computation of the skeleton. In Cases 2 and 3, the proof constructs a corrected tangent vector v′ using an inverse of the rescaled differential ℓ(z), claiming that ℓ(z) takes values in a fixed compact subset V⊂GL(r,C). The text does not specify the compact set or give the uniform bounds that ensure the inverse ℓ(z)^{-1} is bounded independently of z, u, and β. Since the whole contradiction in these cases depends on this uniform estimate, please spell out the compactness argument and the relevant constants.

minor comments (5)

[Section 6.2] In the outline, the text says the computational details are 'filled in in Section 6.2', but the calculations appear in Section 6.3. Please correct this cross-reference.
[Theorem 1.1] The phrase 'for all β > 0 large enough' is grammatically awkward; it should read 'for all sufficiently large β > 0' or similar.
[Definition 4.8 and Lemma 4.11] The definition calls potentials 'smooth away from the origin', while Lemma 4.11 speaks of smooth strictly convex functions without specifying the behavior at the origin. Please clarify the regularity convention at 0, especially since strong convexity on lines through the origin is used in Lemma 4.26.
[Lemma 5.18] The proof refers to 'Theorem 7.13' to justify the indexing of cells by mixed cells, but Theorem 7.13 is proved later. A forward reference is acceptable, but a short pointer to the relevant statement would improve readability.
[References] The full bibliographic data for [HZ05], [Dob17], and a few other references is not given in the supplied text. Please ensure the published version includes complete references and, where possible, precise theorem numbers for [HZ05, Thm 2.5].

Circularity Check

0 steps flagged

No significant circularity: the skeleton and HMS statements are derived from the Liouville flow, tailoring, and external toric/HMS inputs; the only load-bearing external input is explicitly cited.

full rationale

The paper's central derivation is not circular. Theorem 6.1 identifies skel(eZβ) with Φ^{-1}(LΣ)∩Bβ by a direct flow calculation on the tailored complete intersection (Lemmas 6.5–6.12), not by assuming the FLTZ answer; adapted potentials are auxiliary choices whose existence is proved in Sections 4.2–4.4, and the skeleton inclusion in LΣ is a proved consequence of adaptedness rather than an input. The FLTZ cover in Theorem 7.13 is obtained from strong adaptedness and the smoothing results of Appendix B (Corollary B.42, Corollary B.48), which provide ambient isotopies to standard stratified models; this is a genuine topological argument, not the renaming of a known result. On the B-side, the mirror Ẑ := ∂trans XΣ is defined combinatorially so that its toric strata correspond to the FLTZ pieces of the skeleton cover, but Theorems 1.3–1.4 are then proved by invoking GPS24a and toric HMS, so the categorical equivalence is assembled from external theorems rather than being true by definition. The paper explicitly flags its one fragile external input in Remark 1.2 and Remark 6.2: the S^{n-r} factor of Theorem 1.1(iii) relies on [HZ05, Thm 2.5] that ∂C_{0,trop} is a topological (n−r)-sphere, together with smoothing results from Appendix B. That is an unproved imported topological theorem and a genuine correctness risk if its hypotheses do not match, but it is not a circular reduction: the paper does not define the skeleton to contain that sphere, nor does it fit any parameter to the claimed answer. There is no fitted input renamed as a prediction, no self-citation chain carrying the argument, and no ansatz smuggled in via citation; the author's own limitation statements are consistent with an honest, non-circular derivation modulo explicitly cited external results.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 3 invented entities

The paper is a pure-mathematics theorem paper. No numbers are fitted to data; the auxiliary choices (triangulating functions h, ẖ, constants c_{α,j}, the potential φ, smoothing parameters) are existence/construction data, not fitted parameters. The load-bearing input that the reader 'did not pay for' consists mostly of external results: the nef-partition duality of [Bor93]/[BB96], the [HZ05] combinatorial model for tropical BBCI's, and the toric-HMS/microlocal-sheaf framework ([Fan+12], [GS22], [GPS24a]). New constructs introduced here (transversal boundary, strongly adapted potentials, nef scaling action) are internal tools with proofs, not postulates with independent physical handles.

axioms (5)

standard math Duality of nef partitions and Borisov's theorem (Theorem 2.6): ∇ = ∇_1 + ⋯ + ∇_r is reflexive, ∇∨ = Conv{∆_1,…,∆_r}, ∆∨ = Conv{∇_1,…,∇_r}
Cited from [Bor93]/[BB96], used throughout §3.3 to construct the tropical BBCI and the fan Σ; not reproved in the paper.
domain assumption [HZ05] Theorem 2.5 / Propositions 2.2–2.4: combinatorial description of tropical BBCI cells and the fact that ∩_j ∂C_{0,trop,j} is a topological (n−r)-sphere
Imported from [HZ05]; the paper flags (Remark 1.2) that its methods 'heavily rely' on this. Used for the bounded-cell combinatorics (Proposition 3.23) and the sphere factor in Theorem 1.1(iii) (Remark 6.2).
domain assumption MPCS condition holds (Definition 3.31): smooth toric compactification avoiding orbifold strata, equivalently unimodularity of simplices in ˇT_j
Standing hypothesis of the main theorems; needed for Corollary 3.33 and Theorem 5.27 to construct the Liouville domain. The paper notes (Remark 3.32) it is 'essentially an extra restriction' beyond MPCP.
domain assumption Toric HMS and microlocal sheaf cosheaf correspondence: Perf W(T*M_S1, L_Σ) ≅ D^b(X_Σ) and the GPS24a equivalence between wrapped Fukaya categories and microlocal sheaves on skeleta
External results ([Fan+12], [GS22], [GPS24a]) used in §9 to convert the skeleton computation into Theorems 1.3 and 1.4. Not machine-checked, but published and widely used in the field.
standard math Ehresmann's theorem for manifolds with corners (Theorem 4.19) and the stratification-relative smoothing theorems (Appendix B, after [Whi61])
The paper labels Theorem 4.19 'essentially folklore'; Corollary B.42 is proved in the appendix. Load-bearing for Proposition 4.12 (existence of strongly adapted potentials) and for the diffeomorphism type of Λ in Theorem 1.1(iv).

invented entities (3)

Transversal boundary Ẑ = ∂trans X_Σ = ∪_{σ∈Σtrans} O(σ) independent evidence
purpose: B-side mirror of the open Batyrev–Borisov complete intersection; defined in §1.1 and used in Theorem 1.3
Defined purely combinatorially from the fan Σ; the claimed HMS equivalence (Theorem 1.3) is a checkable statement giving an external handle, and the cover by FLTZ skeleta is shown to match D^b(Ẑ) via toric HMS.
Strongly adapted potentials (Definition 4.8) no independent evidence
purpose: Technical tool to control the smooth topology of the skeleton (Section 7.3)
New notion introduced in this paper, extending Zhou's adapted potentials; not a physical entity, but its existence (Proposition 4.12) and isotopy properties (Lemma 4.24) are load-bearing for Theorem 1.1(iv).
Nef partition scaling action R^r ↷ |Σtrans| (Definition 7.8) no independent evidence
purpose: Replaces the radial scaling of the hypersurface case to construct the cover of Λ by FLTZ pieces
New construction; its smoothings (Lemma 7.11, Corollary 7.12) are what make Theorem 7.13 (the cover Λ(σ) ≅ L_{Σ/σ} × relint(σ∞_1) × ⋯ × relint(σ∞_r)) hold.

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read the original abstract

Let $Z^\circ$ be a complete intersection inside $(\mathbb{C}^*)^n$ that compactifies to a smooth Calabi-Yau subvariety $Z$ inside a Fano toric variety. We compute the Lagrangian skeleton of $Z^\circ$ and describe its decomposition into standard pieces that are mirror to toric varieties. This set-up was first considered by Batyrev and Borisov, who used combinatorial techniques to construct a mirror pair $(Z,\check{Z})$ of Calabi-Yau complete intersections in Fano toric varieties. We apply our main result to establish homological mirror symmetry for Batyrev-Borisov pairs in the large-volume limit. We also prove that the equivalence is compatible with toric HMS, along with further functoriality properties with respect to certain natural inclusions of very affine complete intersections.

Figures

Figures reproduced from arXiv: 2510.23418 by Danil Ko\v{z}evnikov.

**Figure 1.** Figure 1: Dual nef partitions (∆ = ∆1 + ∆2, ∇ = ∇1 + ∇2) from Example 2.8 . Sometimes, nef partitions can be decomposed into smaller pieces, which leads to the following notion: Definition 2.9. We say that a nef partition ∆ = ∆1 + · · · + ∆r is irreducible if there is no proper subset {i1, . . . , ik} ⊊ {1, . . . , r} such that the polytope ∆i1 + · · · + ∆ik contains 0 in its relative interior. Otherwise, the nef pa… view at source ↗

**Figure 2.** Figure 2: Tropical Batyrev–Borisov complete intersection from Example 3.20 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Barycentric subdivision of the fan of the first Hirzerbruch surface positive-dimensional cones σ ∈ Σ, pick an arbitrary point xσ ∈ relint(Fσ). Then we can observe that P(σ) := Conv({xτ : τ ⊂ σ}) will be a cube of the same dimension as σ bar. Clearly, this gives a decomposition of P into a complex of cubes that is combinatorially isomorphic to the decomposition of Σbar. By gluing homeomorphisms between the … view at source ↗

**Figure 4.** Figure 4: Level set of a tented potential ϕ1 before smoothing and convexifying for P dual to the fan from [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗

**Figure 5.** Figure 5: Deformations of the piecewise linear functions (from the setting associated to [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗

**Figure 6.** Figure 6: The skeleton associated to the open BBCI Zβ = {z ∈ (C ∗ ) 3 : e −β (z1 + z −1 1 + z3) − 1 = 0, e−β (z2 + z −1 2 + z −1 3 ) − 1 = 0} from Example 3.20 6.2. Outline of the proof. We give an outline of the proof of Theorem 6.1, highlighting where some of the assumptions from the set-up are used. Note that it is also instructive to see how our arguments work in the case r = 1 where the entire setting is more … view at source ↗

**Figure 7.** Figure 7: Idea behind our proof of HMS illustrated on the running example Since our definition of Zˇ is based on a somewhat ad hoc combinatorial gluing construction intended to match the description of the skeleton of Z, we finish the section by explaining how it fits within some well-known frameworks for producing mirror pairs. Remark 9.3. Since Zˇ is a singular variety obtained by gluing toric varieties along the… view at source ↗

**Figure 8.** Figure 8: The refined stratification S t Σ associated to the fan of the first Hirzerbruch surface (as considered in [PITH_FULL_IMAGE:figures/full_fig_p086_8.png] view at source ↗

discussion (0)

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Pith/arXiv arXiv 2024