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Cohomologous symplectic forms with different Gromov widths

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Pith's one-line read Cohomologous symplectic forms can have different Gromov widths.

desk verdict A clean answer to two open problems via a clever transplant of a symplectic class from a rational model to an exotic 4-manifold, then stabilizing; the proof is sound and the main stress-test worry dissolves on close reading. read the letter →

arxiv 2505.09550 v1 pith:E6PVKCP7 submitted 2025-05-14 math.SG

Shengzhen Ning This is my paper · ORCID
classification math.SG MSC 53D0553D3557R17
keywords symplecticformsGromovwidthcohomologousfirstChernclassexotic4-manifoldspackinguniruled6-manifolds
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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 provides examples, in dimension at least 6, of cohomologous symplectic forms—forms representing the same cohomology class—with different Gromov widths, the invariant measuring the area of the largest standard ball that can be symplectically embedded. The concrete claim: if $X$ is any symplectic 4-manifold that is homeomorphic but not diffeomorphic to $\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$ ($k\geq 1$), then $X\times S^2$ carries a symplectic form cohomologous to a product form $\omega_X\oplus\omega_{S^2}$ whose Gromov width is strictly smaller, and whose first Chern class is different. This answers an open problem: cohomologous symplectic forms need not have equal width or equal first Chern class once the dimension is at least 6. The construction compares an exotic 4-manifold with its rational model and uses a diffeomorphism of the stabilized 6-manifolds to transplant a form.

What carries the argument

The construction is carried by three pieces working together. A classification theorem for simply-connected torsion-free 6-manifolds lets an intersection-form-preserving isomorphism $H^2(X')\to H^2(X)$ be realized by an orientation-preserving diffeomorphism $X\times S^2\to X'\times S^2$, so a form on the rational model $X'=\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$ can be pulled back to a cohomologous form on $X\times S^2$. On the exotic side, Corollary 2.4—a width formula from a ball-packing theorem in dimension 4—gives $w_G(X,\omega_X)=\sqrt{[\omega_X]^2}$ because $X$ is not of SW-simple type and not rational or ruled; once the $S^2$ factor has large area, the product form has width at least $\sqrt{[\omega']^2}$. The key point here is that every exceptional sphere class of the non-minimal $X$ has first Chern number 1, so blowing them down reaches the unique minimal model. On the rational side, the paper produces a uniruled class $A$ (a class with a non-trivial genus-zero Gromov-Witten invariant with a point constraint) with $A^2=0$ and $\omega'(A)<\sqrt{[\omega']^2}$; the standard width bound for uniruled classes plus the fact that $A$ lifts to the product gives the upper bound for the transported form. The strict inequality between these bounds is the width gap, and the Chern-class difference follows from the sign of $[\omega_X]\cdot K_{\omega_X}$ combined with a theorem that excludes such signs on non-rational, non-ruled 4-manifolds.

What would settle it

On a concrete candidate, say $X$ homeomorphic but not diffeomorphic to $\mathbb{CP}^2\#2\overline{\mathbb{CP}}^2$, compute the Gromov widths of the two forms on $X\times S^2$; a symplectic embedding of a 6-ball whose area exceeds the claimed upper bound, or an equality of the two widths, would refute the claim. More directly, exhibiting a non-rational, non-ruled symplectic 4-manifold with an exceptional sphere of Chern number not equal to 1 would break the width formula on which the lower bound rests.

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Extended reading notes

Core claim

The central claim is Theorem 1.3: for any symplectic 4-manifold $X$ homeomorphic but not diffeomorphic to $\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$ with $k\geq 1$, there are symplectic forms $\omega_X$ on $X$ and $\omega_{S^2}$ on $S^2$ such that the 6-manifold $M=X\times S^2$ has a symplectic form cohomologous to $\omega_X\oplus\omega_{S^2}$ with strictly smaller Gromov width; the two cohomologous forms also have different first Chern classes (the characteristic class $c_1$ of the tangent bundle). Because explicit exotic 4-manifolds homeomorphic to $\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$ are known for every $k\geq 2$, all smooth 6-manifolds $(\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2)\times S^2$ admit such pairs, and taking further products with copies of $(S^2,\omega_{S^2})$ produces the same phenomenon in every dimension at least 6. The paper also points to a distinct route for the Chern-class question: forms on $K3\times S^2$ built from homotopy $K3$ surfaces have different first Chern classes among cohomologous forms, so the Chern-class failure is not tied only to the width construction.

Load-bearing premise

The load-bearing assumption is the exact width formula for these 4-manifolds: every exceptional sphere behaves like a standard blow-up class, so $w_G(X,\omega)=\sqrt{[\omega]^2}$; if some exceptional sphere behaved differently, the lower width bound could fail and the two widths could end up equal.

Editorial extensions

If this is right

  • For every $k\geq 2$, the smooth 6-manifold $(\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2)\times S^2$ admits cohomologous symplectic forms with different Gromov widths and different first Chern classes.
  • Taking further products with copies of $(S^2,\omega_{S^2})$ gives such pairs in every dimension at least 6.
  • The open question about whether a closed manifold can carry cohomologous symplectic forms with different Gromov widths is settled: it can.
  • The related first-Chern-class question is also settled: a fixed 6-manifold can carry cohomologous symplectic forms whose first Chern classes differ, and a separate $K3\times S^2$ construction reaches the same conclusion.
  • Rational and ruled 4-manifolds do not exhibit the width phenomenon: cohomologous forms there are symplectomorphic, so the width difference is tied to the exotic or otherwise non-rational side.

Reading between the lines

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

  • The theorem is stated for 4-manifolds homeomorphic to $\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$, but the width formula it uses is broader; a natural test is whether other non-rational, non-ruled 4-manifolds not of SW-simple type give the same stabilization phenomenon.
  • The upper bound comes from a uniruled fiber class on the rational model, so replacing the $S^2$ factor by another symplectic manifold carrying a uniruled class is a plausible way to obtain analogous width gaps in other stabilized products.
  • The proof does not compute the maximal possible width gap or the threshold on the area of the $S^2$ factor; determining how much smaller the second width can be made is a quantitative question left open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper studies McDuff-Salamon's Problem 46 and Salamon's Discussion 4.6. The main result (Theorem 1.3) states that for any symplectic 4-manifold X homeomorphic but not diffeomorphic to CP^2#kCP^2 with k≥1, there is a symplectic form ωX on X and a form ωS2 on S2 such that the 6-manifold X×S2 carries two cohomologous symplectic forms ωX⊕ωS2 and f^*(ω'⊕ωS2) with different Gromov widths and different first Chern classes. The proof combines Li-Liu's symplectic cone theorem, Biran's ball-packing formula, a uniruled-class upper bound from Gromov, and Jupp's 6-manifold classification to transplant the comparison from a rational manifold. The paper also remarks that K3×S2 gives examples for the Chern-class question.

Significance. If the proof is correct, this appears to be the first affirmative answer to Problem 46 in dimension at least 6 and to Salamon's Chern-class question, both of which have been open for some time. The construction is elegant and largely assembles published theorems: Li-Liu's symplectic cone, Biran's packing theorem, Gromov's uniruled width bound, and Jupp's classification. The main new input, Proposition 2.9, is a period computation for rational 4-manifolds that is explicit and checkable. The width comparison is falsifiable and the examples are concrete. The paper is not machine-checked, but the cited tools are appropriate and the steps are standard; no circularity is apparent.

major comments (2)
  1. [Section 2.4, Proposition 2.9] The displayed identity in the proof of Proposition 2.9 is algebraically incorrect: from [ω]^2−(ω(A))^2 = a^2−Σ b_i^2−(a−b1)^2, the right-hand side is 2b1(a−b1)−Σ_{i=2}^k b_i^2, not 2b1(1−b1)−Σ_{i=2}^k b_i^2. As printed, the subsequent inequality and positivity claim do not follow. The intended computation is clear from the reduced condition a≥b1+b2+b3, and the proposition is salvageable, but the equation must be corrected.
  2. [Section 2.3, Corollary 2.4] The deduction that d'_ω=∞ for non-minimal non-rational or ruled manifolds is compressed and notationally delicate. Since E'_ω is defined as pr(E_tildeω)\setminus{0} with E_tildeω the set of symplectically exceptional classes, the one-point blowup contributes a new exceptional class whose projection is zero, while the projections of the old classes are exactly E_ω and each has c1(E)=1, so the denominator c1(B)−1 vanishes. If a reader instead reads E'_ω as coming from the smooth exceptional set E_tildeX, the classes −E_i would appear and d'_ω would be finite, destroying the lower bound in Theorem 1.3. The proof should state this explicitly to prevent the misreading, since this step is load-bearing.
minor comments (4)
  1. [Section 2.3, definition of d'_ω] In the definition of d'_ω, the case c1(B)=1 should be explicitly interpreted as +∞ so that the quotient ω(B)/(c1(B)−1) is unambiguous.
  2. [Section 4, lower bound step] The line 'Since ωS2 has large enough area, we see that wG(X×S2,ωX⊕ωS2) ≥ sqrt([ω']^2)' is terse; it would be clearer to say that a 4-ball in X of area arbitrarily close to sqrt([ω']^2) embeds, and its product with a sufficiently large 2-ball in S2 gives a 6-ball embedding into X×S2.
  3. [Throughout] There are several typos: 'Fianlly' in Section 4, 'J-holomoprhic' in Remark 2.11, and 'vanishi' in the final paragraph.
  4. [Section 2.2, discussion after Theorem 2.2] The invocation of the light cone lemma is implicit; adding a one-sentence statement of the lemma would help the reader verify that a∈C_{K+}∪C_{K-}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chains through external published theorems (Li-Liu, Biran, Jupp, Gromov, Liu-Ohta-Ono) and does not fit parameters or define its conclusions into its hypotheses.

full rationale

The paper's central construction in Theorem 1.3 proceeds by applying externally proven results to a new product manifold X x S^2. The lower Gromov width bound uses Biran's ball-packing theorem (Theorem 2.3) together with Li's identity (1) for exceptional classes; the upper bound uses Gromov's uniruled class width bound and a lifting proposition from Li-Ruan. The diffeomorphism between X x S^2 and X' x S^2 is supplied by Jupp's classification and Wall's h-cobordism result. The final Chern-class contradiction invokes Liu-Ohta-Ono's theorem. None of these ingredients is defined in terms of the target conclusion, and no parameter is fitted to then be renamed as a prediction. The author's advisor Tian-Jun Li appears among the cited authors (e.g., Li-Liu, Li 99), but those cited results are independent published theorems with their own proofs; they are not outputs of this paper and are not invoked as unverified self-supplied assumptions. A possible concern that Biran's d'_omega may not be infinite for non-minimal non-rational/ruled manifolds is a correctness risk about the interpretation of exceptional classes, not a circularity: even if Corollary 2.4 were wrong, the error would be a misapplication of an external theorem rather than a reduction of the conclusion to an assumed equivalent statement. Therefore the derivation chain is self-contained against external benchmarks and no circular step is present.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

No free parameters beyond a sufficiently large choice of the S2 area; no invented entities. The central claim rests on several cited theorems from the symplectic topology literature, listed as axioms. These are external results, not assumptions tailored to this paper, so they do not create circularity.

free parameters (1)
  • Area of the S2 factor (and extra sphere factors) = not fixed; chosen sufficiently large
    The lower bound wG(X times S2, ωX ⊕ ωS2) ≥ sqrt([ω']^2) requires the S2 factor to contain a standard ball of radius matching the 4-manifold width; the theorem asserts existence of such a choice.
assumptions (8)
  • domain assumption Li and Liu's symplectic cone theorem [LL01, Theorem 2.2]: for b2+=1, C_K = {a in the forward cone with a·E > 0 for all E in E_K}, and membership in C_{K+} ∪ C_{K-} for classes positive on exceptional classes.
    Used in Section 4 to conclude the class φtilde([ω']) is realized by a symplectic form ωX on X.
  • domain assumption Biran's ball packing theorem [Bir97] and its Corollary 2.4: for 4-manifolds not of SW-simple type and not rational or ruled, wG(X,ω) = sqrt([ω]^2).
    Used to compute wG(X,ωX) and the lower bound for the product width.
  • domain assumption Li's exceptional sphere theorem [Li99, Cor 3]: for blowups of non-rational or ruled minimal 4-manifolds, symplectic exceptional classes are exactly the line classes.
    Identifies E_{K+} = {E_i} and E'_ω in Biran's formula.
  • domain assumption Jupp's classification [Jup73] via Wall h-cobordism and Smale: an isometry φ extending to the product cohomology is realized by an orientation-preserving diffeomorphism of X times S2.
    Provides the diffeomorphism ftilde used to pull back ω' ⊕ ωS2 to a cohomologous form on X times S2.
  • domain assumption Gromov's uniruled bound [Gro85, Thm 2.8] and Li-Ruan lifting [LR13, Prop 2.10]: a uniruled class A caps the Gromov width and remains uniruled after product with S2.
    Gives the upper bound wG(X' times S2, ω' ⊕ ωS2) ≤ ω'(A).
  • domain assumption Liu-Ohta-Ono theorem [Liu96, OO96]: a symplectic 4-manifold with b2+=1 and [ω]·K_ω < 0 is rational or ruled.
    Used to force first Chern classes to differ.
  • domain assumption Existence of symplectic 4-manifolds X homeomorphic but not diffeomorphic to CP2#kCP2 for k at least 2, as cited from Kotschick, Park, Stipsicz-Szabo, Reyes-Urzua, Akhmedov-Park, Baldridge-Kirk.
    Supplies the starting 4-manifold; the theorem is conditional on their existence.
  • domain assumption Salamon's symplectic deformation results [Sal13, Examples 3.9 and 3.10] certify that the exceptional sphere areas b5,...,bk can be made arbitrarily small by deformation on CP2#kCP2.
    Underpins the strict inequality in Proposition 2.9 for k at least 5.

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Pith. "Pith review of Cohomologous symplectic forms with different Gromov widths." pith.science (2026). https://pith.science/paper/E6PVKCP7

@misc{pith2026250509550,
  author       = {Pith},
  title        = {Pith review of: Cohomologous symplectic forms with different Gromov widths},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6PVKCP7}},
  note         = {Machine review of arXiv:2505.09550}
}
abstract

We study McDuff-Salamon's Problem 46 by showing that there exist closed manifolds of dimension $\geq 6$ admitting cohomologous symplectic forms with different Gromov widths. The examples are motivated by Ruan's early example of deformation inequivalent symplectic forms in dimension $6$ distinguished by Gromov-Witten invariants. To find cohomologous symplectic forms and compare their Gromov width, we make use of Li-Liu's theorem of symplectic cone for manifolds with $b_2^+=1$ and Biran's ball packing theorem in dimension $4$. Along the way, we also show that these cohomologous symplectic forms can have distinct first Chern classes, which answers another question by Salamon.

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