Normal Crossings Singularities for Symplectic Topology, II

Aleksey Zinger; Mark McLean; Mohammad Farajzadeh Tehrani

REVIEW 3 major objections 2 minor 20 references

Normal Crossings Singularities for Symplectic Topology, II

T0 review · 3 major / 2 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any normal crossings symplectic variety, the paper claims regularizations exist virtually: the space of regularizations is weakly homotopy equivalent to the space of symplectic structures.

desk verdict Extends their SC symplectic divisor/variety program to arbitrary normal crossings with real new equivalences, but the central homotopy theorem still rests on an unproved transfer from their earlier paper. read the letter →

arxiv 1908.09390 v1 pith:U7FHZWOU submitted 2019-08-25 math.SG hep-thmath.AG

Mohammad Farajzadeh Tehrani , Mark McLean , Aleksey Zinger This is my paper · ORCID

classification math.SGhep-thmath.AG MSC 53D3553D4514J17

keywords normalcrossingssingularitiessymplecticdivisorsvarietiesregularizationsweakhomotopyequivalencetransverseimmersionsGromov-Wittentheoryalmostcomplexstructures

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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 extends the authors' earlier topological treatment of simple normal crossings (SC) symplectic divisors and varieties to arbitrary normal crossings (NC) singularities. It gives two equivalent descriptions: local models that look like SC configurations in charts, and global models as images of transverse immersions with compatible involutions. The central reach is Theorem 4.5, which claims that for any NC symplectic variety, the space of regularizations maps to the space of symplectic structures as a weak homotopy equivalence. If correct, every NC symplectic variety admits the regularized almost complex structures needed for Gromov-Witten type constructions, generalizing the simple-crossings theorem.

What carries the argument

The load-bearing object is a regularization: a compatible system of tubular-neighborhood diffeomorphisms, one for each stratum of the divisor or variety, equipped with Hermitian line-bundle data that turns the model normal-bundle symplectic form into the ambient symplectic form. For NC objects the regularization is assembled from local SC regularizations via chart compatibility, or globally from reified regularizations of the transverse immersion. A deliberately weaker version, weak regularizations, is the proof device: Section 5 argues that the SC proof of the weak homotopy equivalence goes through for weak regularizations in the NC setting, then cites the SC lemmas saying weak regularizations can be strengthened to genuine ones.

What would settle it

Take the 3-fold folding example of Section 4.5 and explicitly compute the normal-bundle isomorphisms that the overlap of two NC charts induces; if a positive-area triple point forces the product Hermitian structure on one chart to differ from the pulled-back structure on the other by a nonzero multiple of the symplectic class, then the weak regularizations cannot be strengthened and Theorem 4.5 would fail for that family.

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Extended reading notes

Core claim

The paper's central claim is that the topological SC symplectic divisor and variety notions, and their regularization machinery, survive passage to arbitrary normal crossings. An NC symplectic divisor is defined locally as an SC symplectic divisor, and equivalently as the image of a closed transverse immersion of codimension two whose multiple-point strata inherit symplectic forms with matching intersection orientations. An NC symplectic variety is defined by an NC atlas whose local models are SC symplectic configurations, and equivalently as a quotient of a symplectic manifold by a compatible involution on its normalization. The main theorem, Theorem 4.5, asserts that the projection from the space of regularizations for an NC symplectic variety to the space of its symplectic structures is a weak homotopy equivalence; the proof is sketched in Section 5 by reducing to weak regularizations and transferring the SC proof, rather than carried out in full here.

Load-bearing premise

The result rests on the assertion that weak regularizations, which are easier to glue in the general normal-crossings setting, can be cut down to genuine regularizations exactly as in the simple-crossings case; the paper sketches this transfer rather than proving it in the NC context.

Editorial extensions

If this is right

If Theorem 4.5 holds, the projection from regularizations to symplectic structures is a weak homotopy equivalence, giving a virtual existence of regularized almost complex structures for every NC symplectic variety.
The local and global perspectives on NC symplectic divisors and varieties are interchangeable, so a construction can be checked either chart-by-chart or via transverse immersions.
The divisor-level analogue, Theorem 3.4, follows from the variety-level theorem, matching the situation for SC singularities.
The regularization framework extends symplectic sum and degeneration techniques from SC to arbitrary NC symplectic divisors and varieties.
The examples show that genuinely non-SC NC configurations, including folded varieties and 3-fold identifications, are admitted by the new definitions.

Reading between the lines

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

If the weak homotopy equivalence is established, Gromov-Witten invariants for NC symplectic varieties can likely be constructed by the same regularization route used for SC varieties, without adding smoothness assumptions on the singularities.
The global immersion viewpoint suggests a computational extension: many NC varieties can be encoded by finitely many local charts and an involution, reducing regularization questions to patching Hermitian bundles over the multiple-point strata.
The weak-regularization strategy, if it succeeds, may also apply to log or exploded variants of Gromov-Witten theory, where only local compatibility of almost complex structures is required.
A natural test beyond the paper is whether the weak homotopy equivalence remains true for families of NC varieties with varying symplectic structures; the paper's local inductive proof structure indicates this should be checked stratum by stratum.

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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 real news here is the extension of the authors' SC framework to arbitrary normal crossings, and the local/global equivalence is actually worked out rather than hand-waved. Proposition 3.6 identifying NC symplectic divisors with transverse immersions whose k-fold pullbacks are symplectic, and Lemma 4.9 matching the global quotient picture with NC atlases, are genuine contributions. The examples at the end (the fold construction, the 3-fold with the non-trivial ψ action) are useful and show the new definitions are not vacuous. I also appreciate that the paper is honest about what is and isn't proved.

The soft spot is exactly where the stress test points: Theorem 4.5 is the main result, and its proof is not actually carried out. Section 5 reduces it to cutting weak regularizations down to strong ones, citing Lemmas 5.8 and 5.9 from [5], and then says 'the same reasoning applies' for arbitrary NC viewed from the global perspective. The concern is not minor. In the arbitrary NC case the involution ψ can act nontrivially on higher strata and on normal bundles, as in their own examples, and the cut-down in [5] was not proved ψ-equivariantly. Proposition 4.12 shows that ψ-equivariance is equivalent to conditions (4.30)–(4.31), but it does not prove those conditions survive the cut-down. So the weak homotopy equivalence is plausible but unsupported in this text. I do not see circularity or invented entities: the paper is open about relying on [5], and the new equivalence theorems are benchmarked against classical geometry. The gap is a proof-transfer gap, not a disguised fitting problem.

For whom? Symplectic topologists working on Gromov–Witten degeneration formulas, symplectic sums, or log-style structures in the symplectic category. They will want the definitions and equivalences, and they will want to know whether Theorem 4.5 really holds. The paper deserves a serious referee, but the referee should be asked to demand a fuller proof of the transfer step—either written out or with a precise reference to a companion paper where the ψ-equivariant cut-down is proved. If the authors cannot supply that, Theorem 4.5 should be downgraded to a conjecture with strong evidence.

Referee Report

3 major / 2 minor

Summary. The paper extends the authors' previous work on simple crossings (SC) symplectic divisors and varieties to arbitrary normal crossings (NC) symplectic divisors and varieties. It gives local and global definitions of NC divisors and NC varieties, proves that these perspectives are equivalent (Propositions 3.6 and 4.12, and the correspondence in Section 4.3), and states Theorem 4.5 as the NC analogue of [5, Theorem 2.17], asserting that the projection from the space of regularizations to the space of symplectic structures is a weak homotopy equivalence. Section 5 sketches the proof of Theorem 4.5 by reducing to a weaker notion of regularization and asserting that the cut-down lemmas of [5] transfer to the NC setting.

Significance. If Theorem 4.5 is established, the paper provides a foundational framework for regularized almost complex structures on normal crossings symplectic varieties, with direct applications to Gromov-Witten theory and symplectic sum constructions. The manuscript is careful and detailed in its definitions, and it contains substantial new material: the local-global equivalence for NC varieties, the resolution uniqueness result (Proposition 3.14), the alternative global characterization of regularizations (Proposition 4.12), and a range of clarifying examples, including non-SC NC varieties. However, the load-bearing existence theorem is not proved in the manuscript: the proof in Section 5 depends on an unproven transfer of cut-down lemmas from the SC setting, and the required equivariance is precisely the new difficulty in the NC case. The central claim is therefore currently conditional on a nontrivial missing argument.

major comments (3)

[Section 5, final two paragraphs] The reduction of Theorem 4.5 to weak regularizations is not complete without a proof of the cut-down step in the NC setting. The manuscript states: 'By Lemma 5.8 and Corollary 5.9 in [5], weak regularizations and equivalences between them in the simple NC setting can be cut down to regularizations and equivalences between regularizations. The same reasoning applies in the arbitrary NC setting viewed from the global perspective of either Section 4.2 or 4.4.' This is the only justification for passing from weak to strong regularizations. In the SC setting of [5], the normalization is a disjoint union of smooth components and the involution ψ only interchanges branches along double loci; in the NC setting, ψ acts on higher strata and on normal bundles, as illustrated by Example 4.17 and Figure 4. The compatibility conditions (4.17)–(4.18), or equivalently (4.30)–(4.31) via Proposition 4.12, require the cut-down construction to be ψ-equivariant. The manuscript does not prove that [5, Lemma 5.8] and [5, Corollary 5.9] admit such an equivariant extension. Since the weak-homotopy-equivalence conclusion of Theorem 4.5 depends exactly on this step, the central claim is currently unsupported.
[Section 5, weak regularization in the global perspective] The global notion of a weak ω-regularization for the pair (ι,ψ) is introduced only informally. The paragraph beginning 'Suppose (~X,~ω) and (ι,ψ) are as in Definition 4.8' describes a tuple satisfying (4.21) and a list of conditions, with the first condition in (3.23) 'may not hold' and the second holding only over an intersection of domains. No explicit space of weak regularizations, no topology, and no equivalence relation for weak regularizations in the global perspective is defined. The inductive proof later uses equivalences of weak regularizations over open sets, so these missing definitions are not merely cosmetic. Theorem 4.5 is a statement about families and homotopy equivalences; without a precise global notion of weak regularization, the argument in Section 5 cannot be verified.
[Theorem 3.4 and its relation to Theorem 4.5] Theorem 3.4, the NC divisor analogue of [5, Theorem 2.13], is stated without a direct proof; the text says it 'is implied by Theorem 4.5' and refers to Example 4.15. Since the proof of Theorem 4.5 contains the gap described above, the status of Theorem 3.4 is also unresolved. The authors should either give a direct proof of Theorem 3.4 or complete the proof of Theorem 4.5 so that the implication is valid.

minor comments (2)

[Section 5, first paragraph after Definition 5.2] The sentence 'The last task is readily accomplished by combining the proof of [5, Theorem 2.7] with the local perspective of Section 4.1' appears to contain a typo: the reference is likely to [5, Theorem 2.17] or [5, Theorem 2.13]. Please check the intended theorem.
[Title page] The title reads 'Normal Crossings Singularities for Symplectic Topology, I I' with an extra space; this is a formatting typo.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 4.5 rests on an unproved self-cited transfer of the weak-to-strong regularization cut-down; the core geometric equivalences remain independent content.

self citation load bearing [Section 5 (proof of Theorem 4.5), paragraph following Definition 5.2]
"By Lemma 5.8 and Corollary 5.9 in [5], weak regularizations and equivalences between them in the simple NC setting can be cut down to regularizations and equivalences between regularizations. The same reasoning applies in the arbitrary NC setting viewed from the global perspective of either Section 4.2 or 4.4. Thus, it is sufficient to establish Theorem 4.5 with regularizations replaced by weak regularizations everywhere."

The proof of the paper's central claim, Theorem 4.5, is reduced to two tasks: constructing weak regularizations and cutting them down to genuine regularizations. The second task is the load-bearing step that actually yields the theorem's conclusion, but it is not proved for arbitrary NC varieties. The paper asserts it follows from the authors' own prior [5, Lemma 5.8, Corollary 5.9] by 'the same reasoning applies,' without proving the needed ψ-equivariant transfer to higher strata or normal-bundle actions (cf. Examples 4.16-4.17).

full rationale

The paper is not circular in the definitional sense: the local and global notions of NC divisors and varieties (Definitions 3.2, 4.3, Lemma 3.5, Proposition 3.6, Corollary 4.10) are genuine mathematical definitions and equivalences, not fitted to the desired conclusions. The new definitions are benchmarked against classical geometric notions, and Examples 4.13-4.17 provide independent content. However, the proof of Theorem 4.5 is not carried out here. Section 5 reduces the theorem to a 'weak regularization' version and then asserts, by appeal to the authors' own [5], that weak regularizations can be cut down to strong regularizations in the arbitrary-NC setting. The quoted passage does not prove this transfer; it declares that the same reasoning applies. This is a load-bearing self-citation gap: if the transfer fails, the weak-homotopy-equivalence conclusion is unsupported. That is a correctness/proof-completeness concern rather than a constructed equivalence or fitted parameter, so the score is moderate rather than high. The independent geometric equivalences and examples prevent the paper from being wholly circular, but the central theorem's proof is not self-contained.

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

No fitted constants or invented physical entities appear. The paper introduces definitions and proves structural theorems; the only external inputs are standard mathematics and the authors' previously established SC framework.

assumptions (3)

standard math Standard results in differential topology (tubular neighborhood theorem, Inverse Function Theorem, submanifold transversality) are assumed.
Used throughout for regularizations, transverse immersions, and the canonical resolution arguments.
domain assumption The prior theorems and lemmas of the authors' paper [5] (Theorems 2.13, 2.17, Lemmas 5.8, 5.9, Proposition 5.3) are established and valid.
The present paper relies on these to prove Theorems 3.4 and 4.5 and to transfer the weak-regularization equivalence to the NC setting.
domain assumption An NC variety is required to be Hausdorff, paracompact, and second-countable, with overlap maps that are diffeomorphisms in the sense of Definition 4.1.
These conditions are part of the definition of NC variety and are needed for the normalization construction in Lemma 4.9 and for the patching arguments in Section 5.

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Pith. "Pith review of Normal Crossings Singularities for Symplectic Topology, II." pith.science (2026). https://pith.science/paper/U7FHZWOU

@misc{pith2026190809390,
  author       = {Pith},
  title        = {Pith review of: Normal Crossings Singularities for Symplectic Topology, II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7FHZWOU}},
  note         = {Machine review of arXiv:1908.09390}
}

read the original abstract

In recent work, we introduced topological notions of simple normal crossings symplectic divisor and variety, showed that they are equivalent, in a suitable sense, to the corresponding geometric notions, and established a topological smoothability criterion for them. The present paper extends these notions to arbitrary normal crossings singularities, from both local and global perspectives, and shows that they are also equivalent to the corresponding geometric notions. In subsequent papers, we extend our smoothability criterion to arbitrary normal crossings symplectic varieties and construct a variety of geometric structures associated with normal crossings singularities in algebraic geometry.

Figures

Figures reproduced from arXiv: 1908.09390 by the authors.

**Figure 1.** A 3-fold NC variety such that ωi1 |Xi1i2 =ωi2 |Xi1i2 for all i1, i2∈[N]. For an N-fold transverse configuration X as in Definition 2.3, we define the spaces X∅ ⊃X∂ as in (1.2) and (1.3). If in addition Xij is a closed submanifold of Xi of codimension 2 for all i, j ∈[N] distinct, let Symp(X) denote the space of all symplectic structures on X and Symp+ [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** The solid lines in the bottom row represent the imag [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** The normalization Xe ≡X⊔(Ve×C) of the NC variety Xι ′ ,ψ associated with an NC divisor V ⊂X as in Example 4.15. Example 4.16. A generalization of the 2-fold SC symplectic configuration of Example 4.14 is obtained by taking two disjoint copies, V1 and V2, of a smooth symplectic divisor V in the same symplectic manifold (X, e ωe). Let ψ : V1 −→ V2 be a symplectomorphism and ψ : V2 −→ V1 be its inverse; thus, ψ is an i… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The normalization of an NC variety with ι(veij )=xei and ψ(veij )=veji. Definition 5.1. Suppose X ≡ {XI}I∈P∗(N) is a transverse configuration as in Definition 2.3, I ∗∈P∗ (N), and U ⊂ XI ∗ is an open subset. A regularization for U in X is a tuple (Ψi)i∈I ∗ , where Ψi i…

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

Farajzadeh Tehrani, M

M. Farajzadeh Tehrani, M. McLean, and A. Zinger, Normal crossings singularities for sym- plectic topology, Adv. Math. 339 (2018), 672-748

work page 2018

[2] [1]

Abramovich and Q

D. Abramovich and Q. Chen, Stable logarithmic maps to Deligne-Faltings pairs II , Asian J. Math. 18 (2014), no. 3, 465–488 39

work page 2014

[3] [2]

Caporaso and J

L. Caporaso and J. Harris, Counting plane curves in any genus , Invent. Math. 131 (1998), no. 2, 345–392

work page 1998

[4] [3]

Atiyah-Floer Conjecture: a Formulation, a Strategy to Prove and Generalizations

A. Daemi and K. Fukaya, Atiyah-Floer Conjecture: a formulation, a strategy to prove , and generalizations, math/1707.03924

work page Pith review arXiv

[5] [4]

Farajzadeh Tehrani, M

M. Farajzadeh Tehrani, M. McLean, and A. Zinger, Singularities and semistable degenerations for symplectic topology , C. R. Math. Acad. Sci. Paris 356 (2018), no. 4, 420-432

work page 2018

[6] [6]

The Smoothability of Normal Crossings Symplectic Varieties

M. Farajzadeh Tehrani, M. McLean, and A. Zinger, The smoothability of normal crossings symplectic varieties , math/1410.2573v2

work page Pith review arXiv

[7] [7]

Farajzadeh Tehrani and A

M. Farajzadeh Tehrani and A. Zinger, Normal crossings singularities for symplectic topology: structures, in preparation

work page

[8] [8]

Gompf, A new construction of symplectic manifolds , Ann

R. Gompf, A new construction of symplectic manifolds , Ann. of Math. 142 (1995), no. 3, 527–595

work page 1995