{"id":"ad34827c-35ee-4918-b827-ed0955dbee98","arxiv_id":"2505.09550","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Closed 6-manifolds built as X times S2, with X an exotic CP2#kCP2, carry cohomologous symplectic forms with different Gromov widths and different first Chern classes.","lead":"This paper constructs six-dimensional manifolds that carry two symplectic forms with the same cohomology class but different Gromov widths, answering an open problem from McDuff and Salamon's book. The same examples also distinguish first Chern classes, settling a related question of Salamon.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Width lower bound in Theorem 1.3 rests on Corollary 2.4, whose proof conflates smooth and symplectic exceptional classes; if Biran's d'_omega includes classes with c_1=-1, d'_omega is finite and the bound fails.","rationale":"The reader's verdict is CONDITIONAL, with Corollary 2.4 correctly identified as the weakest assumption. My independent reading agrees that this is the hinge of the proof, but I sharpen the concern: even if Li's identity (1) holds and the exceptional classes are exactly the line classes E_i, the proof's use of E_{Xtilde} (smooth exceptional classes) introduces the sign-reversed classes -E_i with c_1=-1. Since d'_omega is defined as an infimum over this set with denominator c_1(B)-1, including -E_i gives finite positive values b_i/2. The paper's conclusion d'_omega = infinity requires the set to contain only classes with c_1=1, which is the symplectic exceptional convention. Biran's theorem is almost certainly stated for symplectic exceptional classes; if so, the argument can be repaired by replacing E_{Xtilde} with E_omega and excluding sign-reversed classes, so this is a checkable citation-level issue rather than a structural flaw. Other potential issues, including the apparent missing normalization a=1 in Proposition 2.9 and the use of deformation equivalences from [Sal13], are presentation-level or can be absorbed by scaling and do not threaten the main construction. Because the central geometric construction (product with S^2, diffeomorphism realizing a cohomological isometry, uniruled upper bound, and Chern-class contradiction) appears sound conditional on Corollary 2.4, I would keep the verdict at CONDITIONAL rather than reject; the reader's CONDITIONAL verdict is unchanged.","tokens_in":9325,"tokens_out":26057,"duration_ms":264486,"concrete_test":"Check Biran's published Theorem 6.A (Geom. Funct. Anal. 7 (1997), 420-437) and the exact definition of the exceptional set appearing in the formula for d'_omega. Determine whether the set consists of symplectic exceptional classes (which satisfy c_1(B)=1) or all smooth exceptional classes. Then recompute d'_omega for a non-minimal symplectic 4-manifold (X,omega) with one exceptional class E of omega-area b>0, not diffeomorphic to a rational or ruled manifold: if -E is included, d'_omega = b/2, so Corollary 2.4 is false and the lower Gromov-width bound of Section 4 fails; if -E is excluded, d'_omega = infinity and Corollary 2.4 stands, with the paper needing only a notational correction from E_{Xtilde} to E_{omega}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing step is Corollary 2.4, used in Section 4 to obtain the lower bound w_G(X x S^2, omega_X + omega_S^2) >= sqrt([omega']^2). The proof derives Corollary 2.4 from Biran's Theorem 2.3 and Li's identity (1). As written in Section 2.3, E'_omega is defined as pr(E_{Xtilde}) \\ {0}, where E_{Xtilde} is the set of classes represented by smoothly embedded (-1)-spheres. By (1), for a one-point blowup of a non-minimal X, E_{Xtilde} = {+-E_1,...,+-E_l, +-E_new}; projecting gives +-E_i. For B = -E_i, one has c_1(B) = -1, so the quotient omega(B)/(c_1(B)-1) equals (-b_i)/(-2) = b_i/2 > 0. If such classes are admitted, d'_omega = min_i b_i/2 is finite, not infinite, and Biran's formula yields w_G(X,omega) = min{sqrt([omega]^2), d'_omega} rather than sqrt([omega]^2). The lower bound in Theorem 1.3 would then collapse, and the two Gromov widths could coincide. The argument works only if Biran's exceptional set consists of symplectic exceptional classes, which satisfy c_1(B)=1 and make the denominator in d'_omega vanish; the manuscript does not state this and in fact uses the notation E_{Xtilde} for smooth exceptional classes. This is an internal ambiguity in the width computation, not a matter of differing consensus with Biran's theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9678,"tokens_out":45772,"duration_ms":439504,"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":[{"comment":"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.","section":"Section 2.4, Proposition 2.9"},{"comment":"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.","section":"Section 2.3, Corollary 2.4"}],"minor_comments":[{"comment":"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.","section":"Section 2.3, definition of d'_ω"},{"comment":"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.","section":"Section 4, lower bound step"},{"comment":"There are several typos: 'Fianlly' in Section 4, 'J-holomoprhic' in Remark 2.11, and 'vanishi' in the final paragraph.","section":"Throughout"},{"comment":"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-}.","section":"Section 2.2, discussion after Theorem 2.2"}],"recommendation":"minor_revision","confidential_remarks":"The reader's stress test focused on Corollary 2.4, but on reading the manuscript the supposed conflation of smooth and symplectic exceptional classes is not present: E'_ω is defined via E_tildeω, not E_tildeX. I do not see a circularity issue; the proof applies external theorems to a new construction. The main problems are the algebraic typo in Proposition 2.9 and the need for a clearer statement of the Corollary 2.4 argument. Both are local and fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This note answers McDuff–Salamon's Problem 46 and Salamon's Chern-class question in dimension ≥ 6, using a genuinely new combination of known tools. The construction is straightforward once you see it: take an exotic CP2#kCP2, use Li–Liu's symplectic cone to find a symplectic form on it that is cohomologous (after stabilizing with S^2) to a pullback of a form on the rational model, then compare Gromov widths via Biran's ball packing and Gromov's uniruled bound. The key width inequality works because the lower bound on the product comes from the 4-manifold itself, and the upper bound comes from a uniruled fiber class on the rational side. The Chern-class contradiction via Liu–Ohta–Ono is a nice touch.\n\nThe paper is honest about its reliance on external theorems, and the logic is coherent. I checked the stress-test concern about Corollary 2.4: it is a misreading. The set E'_ω is defined using E_{\\tilde ω}, the symplectic exceptional classes of the blowup, not the smooth ones. So after projection you get only classes with c1(B)=1, the denominator in d'_ω is zero, and d'_ω is indeed infinite. Corollary 2.4 stands. The noted typo in Proposition 2.9 is real but harmless: the expression \"2b1(1-b1)\" should be something like \"2b1(a-b1)\"; the intended inequality follows directly from the reduced condition, and the rest of the proof is unaffected.\n\nSoft spots are minor: the paper leans heavily on Biran's theorem and Li–Liu's cone description, so a referee should check the exact hypotheses in each cited result. The lower bound wG(X×S^2) ≥ wG(X) is used without proof, but that is standard and easily filled in. The list of exotic 4-manifolds is up to date, and the extension to higher dimensions is routine.\n\nIn short: a well-written, important-for-the-subfield research note, not a revolution. It deserves serious peer review and should be published after the typo is fixed. I would bring it to a reading group as a good example of combining classification results with packing obstructions.","headline":"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.","tokens_in":10208,"tokens_out":10284,"would_cite":true,"duration_ms":92043,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D05","53D35","57R17"],"pacs":[],"model":"deepseek-v4-flash","headline":"Cohomologous symplectic forms can have different Gromov widths.","keywords":["symplectic forms","Gromov width","cohomologous symplectic forms","first Chern class","exotic 4-manifolds","symplectic packing","uniruled class","6-manifolds"],"falsifier":"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.","tokens_in":9081,"feed_emoji":"🏀","tokens_out":20711,"duration_ms":175141,"temperature":0.7,"pith_summary":"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.","feed_headline":"Cohomologous symplectic forms can have different Gromov widths","feed_subtitle":"Some 6-manifolds carry cohomologous symplectic forms whose largest embeddable balls differ, answering an open question.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Theorem 2.3 gives the ball-packing width formula for 4-manifolds not of SW-simple type; with [Li99] it yields Corollary 2.4, the exact width $w_G(X,\\omega)=\\sqrt{[\\omega]^2}$ used for the lower bound.","marker":"[Bir97]"},{"why":"Theorem 2.2 describes the K-symplectic cone when $b_2^+=1$, used to assert that the transported class $[\\omega_X]$ is represented by an actual symplectic form on $X$.","marker":"[LL01]"},{"why":"Corollary 3 identifies the exceptional classes of a non-minimal non-rational/ruled 4-manifold as exactly the blow-up classes $E_i$ and their negatives, which makes the width formula exact.","marker":"[Li99]"},{"why":"Theorem 3.1 classifies simply-connected torsion-free 6-manifolds and is used in Corollary 3.2 to realize the cohomological identification as a diffeomorphism $X\\times S^2\\to X'\\times S^2$.","marker":"[Jup73]"},{"why":"Provides the reduced period coordinates on $\\mathbb{CP}^2\\#k\\overline{\\mathbb{CP}}^2$; Proposition 2.9 uses them to produce a uniruled class $A$ with $\\omega'(A)<\\sqrt{[\\omega']^2}$.","marker":"[KK17]"},{"why":"States the Chern-class question (Discussion 4.6) and supplies symplectic deformation equivalences used to arrange $[\\omega']\\cdot K_{\\omega'}<0$ when $k\\geq 5$.","marker":"[Sal13]"},{"why":"Proposition 2.10 says a uniruled class in the 4-manifold lifts to a uniruled class of the product with $S^2$, transferring the width upper bound to dimension 6.","marker":"[LR13]"},{"why":"The theorem that a symplectic 4-manifold with a form satisfying $[\\omega]\\cdot K_\\omega<0$ must be rational or ruled; it rules out equal first Chern classes in the construction.","marker":"[Liu96, OO96]"},{"why":"Theorem 2.1: every symplectic 4-manifold has a minimal model, unique away from rational or ruled manifolds up to symplectomorphism; this produces the model $\\hat{X}$ from which both $X$ and the rational comparison manifold are built.","marker":"[McD90, McD92]"}],"fun_headline_variants":["Same cohomology, different Gromov widths","Cohomologous symplectic forms: widths can differ","Distinct Gromov widths within a symplectic cohomology class","6-manifolds with cohomologous symplectic forms of unequal width"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Same cohomology, different Gromov widths","Cohomologous symplectic forms: widths can differ","Distinct Gromov widths within a symplectic cohomology class","6-manifolds with cohomologous symplectic forms of unequal width"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000646,"raw_usage":{"total_tokens":2996,"prompt_tokens":998,"completion_tokens":1998,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":1923}},"tokens_in":614,"tokens_out":1998,"duration_ms":15541,"temperature":1.0,"reasoning_tokens":1923,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:31:51.700997+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}