{"id":"9da02a43-09d8-42eb-8d58-f69da5091dbf","arxiv_id":"2412.19375","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"Infinite generation of some higher homotopy groups of symplectomorphism groups is proved for n-point Kahler blowups of tori, K3 surfaces, and Enriques surfaces with non-resonant Kahler classes.","lead":"This paper claims that certain even-dimensional higher homotopy groups of symplectomorphism groups of many-point blowups of tori, K3 surfaces, and Enriques surfaces are infinitely generated. It extends gauge-theoretic invariants from Kronheimer and Smirnov to higher homotopy, using a blowup formula for family Seiberg-Witten equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.9, the blowup transversality formula carrying all q-invariant computations, is not established: the key identification KS_0(T_0D^2) ≅ ker(γ) is asserted by dimension counting, and Lemma 3.10's rank computations are inconsistent.","rationale":"The reader identified Theorem 3.9 as the load-bearing premise, and my reading agrees. The proof of Theorem 3.9 contains a genuinely unsupported step: the assertion KS_0(T_0D^2) ≅ ker(γ) is justified only by 'dimension reason', with no computation of the relevant cohomology groups or ranks. The surrounding Lemma 3.10 is internally inconsistent in its rank notation and does not repair the gap. Since every q-invariant computation in Sections 4.2 and 4.3 depends on this blowup transversality formula, the claimed surjectivity of q_{k,n} — and hence the infinite generation statements in Theorem A — is not established by the text. I also agree with the reader's secondary concern about the infinitude of Δ_k: the conclusion requires infinitely many summands, but no proof or citation for this is given. These are gaps in the argument, not disagreements with the expected truth of the theorem; the strategy may well be salvageable. Because these gaps are load-bearing and central, the REJECT verdict is appropriate for the manuscript as written, and my stress-test does not change that verdict.","tokens_in":27366,"tokens_out":5684,"duration_ms":56313,"concrete_test":"Independently re-derive Theorem 3.9 in the simplest nontrivial case: take X = T^4, C an elliptic fiber in a primitive class δ, choose p∈C, and let X~→X be the blowup at p with C~ the proper transform. Compute h^1(X,Θ_X), h^1(C,N_{C|X}), h^1(X~,Θ_{X~}), h^1(C~,N_{C~|X~}) and the ranks of KS, β, and γ using the long exact sequence (3.16) and Riemann–Roch. If ker(γ) is not 2-dimensional, or if the connecting homomorphism in (3.22) has rank c≠0, then Lemma 3.12 and Theorem 3.9 fail, and the induction in Sections 4.2 and 4.3 breaks.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem A rests on the q-invariant evaluations in Sections 4.2 and 4.3, and every such evaluation reduces to Theorem 3.9: after blowing up a Kähler family along a disk transverse to the unique curve C, the proper transform C~ is claimed to be the unique regular solution of the family Seiberg–Witten equation. The proof of Theorem 3.9 ends by asserting KS_0(T_0D^2) ≅ ker(γ) purely by dimension reasons. But ker(γ) is never identified, and the supporting Lemma 3.10 does not supply the needed rank bound. In Lemma 3.10, equation (3.24) writes h^0(X,π_*Θ_{X~}) + c = H^0(X,Θ_X) (as printed) and h^1(X,π_*Θ_{X~}) = (2−c)+h^1(X,Θ_X), then derives 2−2c=2 from the Euler characteristic while the h^0 and h^1 terms are not tracked consistently. The equality h^2(X~,Θ_{X~}(−log C~)) = h^2(X,Θ_X(−log C)) is quoted from Lemma 3.11, but the preceding inequality chain (3.20)–(3.23) forces this equality only by combining the very assertion to be proved with a dimension bound. Since the induction over k in Sections 4.2 and 4.3 invokes Theorem 3.9 to establish regularity of the unique FSW solution, surjectivity of q_{k,n} is unsupported. A separate, structurally necessary premise is that the index sets Δ_k are infinite: the conclusion 'infinitely generated' requires an infinite direct sum in the image, yet the paper never proves infinitude of Δ_k, and it does not follow from non-resonance alone without a counting argument in the relevant indefinite lattice.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that for multiple-point Kähler blowups of a torus T^4, a K3 surface, or (under a density assumption) an Enriques surface, certain even-dimensional homotopy kernels of the map from the symplectomorphism group to the identity component of the diffeomorphism group are infinitely generated. The proposed mechanism extends Kronheimer's family Seiberg-Witten invariant and Smirnov's q-invariant to higher homotopy groups, constructs spherical families from links of discriminant loci in period domains, and evaluates the invariants using a blowup transversality formula for isolated regular solutions of family Seiberg-Witten equations. The main theorem, Theorem A, asserts surjectivity of homomorphisms q_{k,n} onto infinite direct sums of Z/2, which would imply the claimed infinitely generated kernels.","tokens_in":27734,"tokens_out":1996,"duration_ms":18990,"significance":"If correct, the result would be a substantial extension of known techniques for detecting non-trivial higher homotopy groups of symplectomorphism groups, and would provide the first examples of infinitely generated higher homotopy groups for symplectic manifolds with b^+=1 (Remark 10). The paper honestly borrows the gauge-theoretic framework from Kronheimer and Smirnov and aims to add a deformation-theoretic blowup formula; this is a reasonable strategy, and the spherical-family construction in Sections 4.2 and 4.3 is a genuine attempt to turn period-domain geometry into concrete invariant evaluations. The machine-checkable parts are limited, but the paper does give explicit classes and index sets. However, the central load-bearing computation is not convincingly proved, and the paper's own arguments contain internal inconsistencies, so the claimed significance is not currently supported.","major_comments":[{"comment":"The proof of Theorem 3.9 ends with the assertions 'KS_0(T_0D^2) ≅ ker(γ)' and 'because of the dimension reason, since we take disk intersecting C transversally, we can see KS_0(T_0B) ≅ ker(γ)'. The kernel ker(γ) is never identified, and no computation of its dimension or its intersection with the image of the Kodaira-Spencer map is supplied. Since every q-invariant evaluation in Sections 4.2 and 4.3 reduces to this statement, the surjectivity claim for q_{k,n} is unsupported.","section":"§3.4, proof of Theorem 3.9"},{"comment":"The proof of Lemma 3.10 contains an inconsistent rank computation. Equation (3.24) writes h^0(X,π_*Θ_{X~}) + c = h^0(X,Θ_X) and h^1(X,π_*Θ_{X~}) = (2−c) + h^1(X,Θ_X), then derives 2−2c = 2 from the Euler characteristic. The h^0 and h^1 terms are not tracked consistently: if c is the rank of the kernel of the connecting homomorphism, the claimed conclusion c=0 does not follow from the displayed equations without additional information about the h^0 and h^1 terms. This lemma is the support for the key rank bound in Theorem 3.9.","section":"§3.4, Lemma 3.10, Eq. (3.24)"},{"comment":"The equality h^2(X~,Θ_{X~}(−log C~)) = h^2(X,Θ_X(−log C)) is quoted from Lemma 3.11, but the preceding inequality chain (3.20)–(3.23) forces this equality only by combining the very assertion to be proved with a dimension bound. The chain concludes h^2(X~,Θ_{X~}) ≥ h^2(X,Θ_X), and the proof then uses the converse inequality that is equivalent to the surjectivity being proved. The argument is therefore circular at this point.","section":"§3.4, Lemma 3.11 and Eqs. (3.20)–(3.23)"},{"comment":"The conclusion 'infinitely generated' requires the index sets Δ_k to be infinite, because the image is an infinite direct sum of Z/2. The paper never proves infinitude of Δ_k; it only states that Δ_k = {δ ∈ Δ | (k−1)λ < ⟨δ,κ⟩ < kλ}. Non-resonance of κ does not by itself guarantee that infinitely many such δ exist without a counting argument in the relevant indefinite lattice. The same gap appears for K3 in §4.3 and for Enriques in §4.4.","section":"§4.2, proof of part 1 of Theorem A; §4.3, proof of part 2"},{"comment":"The base case of the induction for T^4 relies on 'the result of Smirnov [32]' for transversality of FSW_{δ−e} over F^δ_κ(1). The cited result is not stated or restated in the paper, and no precise statement or proof is given, so the induction's foundation is not self-contained.","section":"§4.2, proof of part 1 of Theorem A, base case"}],"minor_comments":[{"comment":"There are numerous typographical and grammatical issues, including 'gauge' consistently misspelled as 'guage', 'geometors', 'defiend', and incomplete sentences such as 'We take a smooth family {pXt, ωtq} of Kahler manifolds parametrized byF δ κpi1, ..., ikqs.t. rωts= ...' in the introduction; the paper would benefit from careful editing.","section":"Abstract and Introduction"},{"comment":"Theorem A is stated for 'n–point blow up' with 'the sizes of exceptional divisors are equal and their sum is small enough', but the precise dependence of the result on λ (the common size) and the exact smallness condition are not stated quantitatively; this makes the theorem difficult to verify.","section":"§1, Theorem A"},{"comment":"The proof of Theorem 2.17 cites Buse's theorem but does not explain how the tamed almost-complex structures on the blowup are obtained from those on the base manifold; the map S_{κ,λ_1,...,λ_n} → S_{κ,(λ_1−μ_1),...} is asserted up to homotopy equivalence without a complete argument.","section":"§2.4, Theorem 2.17"},{"comment":"The proof of Lemma 3.8 asserts that 'F and F1 together define family Seiberg-Witten equation over a (2k+2)-dimensional closed chain F0' and that 'for the similar reason, FSW(F0, s_{K−ϵ}) = Q^−(F1, ϵ)'; the notation is confusing and the argument for identifying the two invariants is only sketched.","section":"§3.3, Lemma 3.8"},{"comment":"The density assumption for Enriques surfaces is stated as 'txπ˚κ, δy|δ is a root in E⊕2_8 ⊕ H⊕3u is a dense subset of some neighborhood of 0 ∈ R', but the meaning of τπ˚κ in the quotient and the exact role of the density condition in guaranteeing that Δ_{k,ρ} is infinite are not explained.","section":"§4.4"}],"recommendation":"reject","confidential_remarks":"The central claims of the paper are not established because the key blowup transversality theorem (Theorem 3.9) is proved by an incomplete and apparently circular argument, and the supporting rank computation in Lemma 3.10 is inconsistent. In addition, the paper does not prove that the index sets Δ_k are infinite, which is necessary for the 'infinitely generated' conclusion. These are load-bearing issues that cannot be fixed by minor edits within the scope of the current manuscript. The author's strategy is creative and the topic is timely, but the manuscript in its present form does not meet the standards of a mathematical journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This one is a serious attempt with a genuine idea, but as written the main theorem is not proved. The paper extends Smirnov's q-invariant from π_0 to higher even homotopy groups of symplectomorphism groups of blowups of T^4, K3, and Enriques surfaces. The overall architecture—constructing spherical families from links of period-domain strata and evaluating a family Seiberg-Witten count—is sensible and worth taking seriously. The parameter-space construction in Section 4 is detailed and mostly solid.\n\nThe problem is Theorem 3.9, the blowup transversality formula, which carries every q-invariant computation. Its proof ends by asserting KS_0(T_0D^2) ≅ ker(γ) \"by dimension reasons,\" but ker(γ) is never identified, and Lemma 3.10 does not supply the needed rank bound. The rank equations in Lemma 3.10 do not parse: from the long exact sequence (3.22) the correct relation is h^1(X,π_*Θ_\\tilde X) = h^1(X,Θ_X)+c, not h^1 = (2−c)+h^1 as printed, and the derived 2−2c=2 does not follow from the Euler characteristic. The equality h^2(\\tilde X,Θ_{\\tilde X}(−log \\tilde C)) = h^2(X,Θ_X(−log C)) is quoted before the inequality chain (3.20)–(3.23) finishes, so the induction is circular at that point.\n\nThere is also a structurally separate gap: the conclusion \"infinitely generated\" requires Δ_k to be infinite, and the paper never proves or cites that. For irrational non-resonant κ this is probably true, but it is not in the text. A smaller issue: the formal dimension expression (3.10) has a sign that should be checked, and (3.24) writes H^0 where h^0 belongs.\n\nNone of this makes the strategy wrong. Extending Smirnov's homomorphism to higher homotopy groups is a natural and valuable goal, and the author has clearly thought hard about the deformation theory. But the proof of the main new ingredient is a sketch with a load-bearing gap, so the central theorem is unsupported in the present text.\n\nI would send it to a serious referee, expecting major revision. The referee should demand a complete proof of Theorem 3.9, a clean Lemma 3.10, and a proof or citation for the infinitude of Δ_k. Not a desk reject—there is real content here—but it needs substantial work before it is citable.","headline":"A credible extension of Smirnov's q-invariant to higher homotopy groups, but the load-bearing blowup transversality formula is not proved and needs substantial revision before the main theorem is supported.","tokens_in":28328,"tokens_out":6114,"would_cite":false,"duration_ms":52346,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57R17","53D35","14J28","14D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that n-point Kähler blowups of a 4-torus, a K3 surface, or an Enriques surface have infinitely generated kernels in higher symplectomorphism-group homotopy.","keywords":["family Seiberg-Witten invariant","symplectomorphism groups","higher homotopy groups","Kähler blowups","K3 surfaces","Enriques surfaces","space of symplectic forms","q-invariant"],"falsifier":"A direct check would be to take a K3 surface with a smooth rational curve $C$ of self-intersection $-2$, blow up a family of such surfaces along a disk transverse to $C$, and compute the log-tangent cohomology before and after the blowup; if the equality $h^2(\\tilde X,\\Theta_{\\tilde X}(-\\log \\tilde C)) = h^2(\\tilde X,\\Theta_{\\tilde X})$ used in Lemma 3.10 fails, Theorem 3.9 collapses. A second quantitative check is whether the index set $\\Delta_k$ is infinite for the chosen non-resonant $\\kappa$; if some $\\Delta_k$ were finite, the direct-sum image would have finite rank and infinite generation would not follow.","tokens_in":27066,"feed_emoji":"♾️","tokens_out":17908,"duration_ms":151974,"temperature":0.7,"pith_summary":"This paper tries to establish that blowing up a Kähler surface of Calabi-Yau type at n points creates infinitely many independent higher homotopy classes in its symplectomorphism group. For n-point Kähler blowups of a 4-torus, a K3 surface, or an Enriques surface, the claimed conclusion is that, under a non-resonance condition on the Kähler class (and a density condition for Enriques), the kernel of the natural map from the homotopy groups of the symplectomorphism group to those of the identity component of the diffeomorphism group is infinitely generated, in a range of even degrees controlled by n. The proof extends a family Seiberg-Witten invariant to higher homotopy groups and uses it to build surjective homomorphisms onto infinite direct sums of Z/2. If the paper is right, these kernels contain an infinite independent family of classes in every even degree in the stated range, so the symplectomorphism groups themselves have infinitely generated higher homotopy.","feed_headline":"Blowups of K3 and tori reveal infinite symplectic homotopy kernels","feed_subtitle":"A gauge-theoretic count produces independent Z/2 classes in symplectomorphism homotopy for each k=1,...,n.","key_machinery":"The load-bearing object is a blowup transversality formula (Theorem 3.9) for isolated regular solutions of the family Seiberg-Witten equation on Kähler families. The formula says that if a Kähler family over a disk has a unique regular solution represented by a holomorphic curve $C$, then blowing up the family along a holomorphic disk transverse to $C$ yields a new family whose unique regular solution is represented by the proper transform of $C$. This reduces the transversality check on the $n$-point blowup to the one-point base case, making the $q$-invariant evaluations inductive. The $q$-invariant itself is a homomorphism to $\\mathbb{Z}/2$ defined by taking the difference of two counts of solutions of the family Seiberg-Witten equation, and it is evaluated on spherical families $S^\\delta_\\kappa(i_1,\\dots,i_k)$ obtained as the boundaries of normal disk fibers to subvarieties of the parameter space of polarized complex structures.","core_discovery":"The paper's central claim, stated as Theorem A, is the following. Let $(M,\\omega)$ be a Kähler surface diffeomorphic to $T^4$, a K3 surface, or an Enriques surface, and let $(X_n,\\tilde{\\omega}_n)$ be its $n$-point Kähler blowup with equal exceptional sizes whose total is small; assume $\\kappa=[\\omega]$ is non-resonant, meaning $\\langle\\kappa,\\delta\\rangle\\neq 0$ for every nonzero integral class $\\delta$, with an additional density condition in the Enriques case. Then the kernel of $\\iota_*:\\pi_{2k}(\\mathrm{Symp}_s(X_n,\\tilde{\\omega}_n))\\to\\pi_{2k}(\\mathrm{Diff}_0(X_n))$ is infinitely generated for $k=1,\\dots,n$, and for $T^4$ the same holds in degree $2k-2$. The proof constructs homomorphisms $q_{k,n}$ from the relevant homotopy group of the space of symplectic forms to a direct sum of $\\mathbb{Z}/2$'s indexed by pairs $(\\delta,i_1,\\dots,i_k)$ with $\\delta$ in the infinite set $\\Delta_k$, and proves $q_{k,n}$ is surjective. Because $\\ker(\\iota_*)$ is the cokernel of the boundary map in the fibration, surjectivity of these homomorphisms forces the kernel to be infinitely generated.","pith_inferences":["The paper leaves implicit that the blowup formula should evaluate family Seiberg-Witten invariants on any iterated blowup of a Kähler surface along families of negative curves, not only on the tori, K3, and Enriques cases treated here.","A natural stress test is to weaken non-resonance to a generic condition on $\\kappa$: the proof only needs $\\Delta_k$ to be infinite and the chosen classes to stay away from the $(1,1)$ locus outside the central fiber.","All detected classes are $\\mathbb{Z}/2$-valued, so the argument establishes 2-torsion in these kernels; an extension to integer or higher-order invariants on the same spherical families could reveal non-torsion classes.","In the Enriques case the proof works by lifting to a K3 family with a fixed-point-free involution; a direct invariant on the quotient could independently test the density-condition conclusion."],"forward_implications":["For any non-resonant Kähler class on a K3 surface, the kernel of $\\iota_*$ in degree $2k$ is infinitely generated for every $k=1,\\dots,n$.","For a 4-torus, the same conclusion holds in degree $2k-2$, so the $n$-point blowup of a torus has infinitely many independent higher homotopy classes in its symplectomorphism group.","For Enriques surfaces satisfying the density condition, the result gives an example of a symplectic 4-manifold with $b^+=1$ whose symplectomorphism group has infinitely generated higher homotopy.","Because $q_{k,n}$ vanishes on families induced by diffeomorphisms, the detected classes are genuine obstructions to lifting spheres in the symplectomorphism group to the diffeomorphism group.","Since each $\\Delta_k$ is infinite, the proof gives not just one exotic class but an infinite independent family in the kernel."],"supporting_citations":[{"why":"Defines the family Seiberg-Witten invariant and the boundary-value homomorphism from homotopy groups of the space of symplectic forms to Z/2 that the paper extends.","marker":"[20]"},{"why":"Introduces the Z/2-valued q-invariant for blowups of T^4 and its two key properties: vanishing on diffeomorphism-induced families and surjectivity over Δ1.","marker":"[34]"},{"why":"Provides the base-case transversality result and the index set of (-2)-classes on K3 surfaces that starts the induction in the K3 case.","marker":"[33]"},{"why":"Establishes the correspondence between Seiberg-Witten solutions on Kähler surfaces and effective divisors, used to identify the unique central-fiber solution.","marker":"[36]"},{"why":"Supplies the criterion that a curve represents a regular family Seiberg-Witten solution exactly when the infinitesimal deformation map followed by restriction to the normal bundle is surjective.","marker":"[13]"},{"why":"Provides the family Seiberg-Witten invariant and the chamber/wall formalism that makes the family counts well-defined.","marker":"[24]"},{"why":"Gives the invariance of log-tangent cohomology under blowup, which Lemma 3.10 uses to prove the blowup transversality formula.","marker":"[12]"},{"why":"Establishes negative inflation, the operation used to define the Q^- term and to ensure the relevant curve classes have non-positive area.","marker":"[29]"}],"fun_headline_variants":["Blowups of K3 and tori give infinite symplectic homotopy kernels","Infinite homotopy from blowups: symplectomorphism groups of Kahler surfaces","Family Seiberg-Witten shows infinite symplectic homotopy on blowups","K3 and tori blowups: infinitely generated symplectic homotopy groups","Gauge theory on Kahler blowups yields infinite homotopy kernels"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on a blowup formula that says a unique regular family Seiberg-Witten solution stays unique and regular after blowing up the family along a transverse holomorphic disk; if that formula fails, the q-invariant evaluations have no basis.","fun_headline_variants_meta":{"raw":{"variants":["Blowups of K3 and tori give infinite symplectic homotopy kernels","Infinite homotopy from blowups: symplectomorphism groups of Kahler surfaces","Family Seiberg-Witten shows infinite symplectic homotopy on blowups","K3 and tori blowups: infinitely generated symplectic homotopy groups","Gauge theory on Kahler blowups yields infinite homotopy kernels"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000522,"raw_usage":{"total_tokens":2563,"prompt_tokens":1020,"completion_tokens":1543,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":636,"completion_tokens_details":{"reasoning_tokens":1433}},"tokens_in":636,"tokens_out":1543,"duration_ms":10867,"temperature":1.0,"reasoning_tokens":1433,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:42:00.335017+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check would be to take a K3 surface with a smooth rational curve $C$ of self-intersection $-2$, blow up a family of such surfaces along a disk transverse to $C$, and compute the log-tangent cohomology before and after the blowup; if the equality $h^2(\\tilde X,\\Theta_{\\tilde X}(-\\log \\tilde C)) = h^2(\\tilde X,\\Theta_{\\tilde X})$ used in Lemma 3.10 fails, Theorem 3.9 collapses. A second quantitative check is whether the index set $\\Delta_k$ is infinite for the chosen non-resonant $\\kappa$; if some $\\Delta_k$ were finite, the direct-sum image would have finite rank and infinite generation would not follow.","supporting_citations":[{"cited_title":"Some non-trivial families of symplectic structures.(1997)","cited_arxiv_id":null,"evidence_quote":"Defines the family Seiberg-Witten invariant and the boundary-value homomorphism from homotopy groups of the space of symplectic forms to Z/2 that the paper extends."},{"cited_title":"Symplectic mapping class groups of blowups of tori.Journal of Topology, 16(3):877–898, 2023","cited_arxiv_id":null,"evidence_quote":"Introduces the Z/2-valued q-invariant for blowups of T^4 and its two key properties: vanishing on diffeomorphism-induced families and surjectivity over Δ1."},{"cited_title":"Symplectic mapping class groups of K3 surfaces and seiberg–witten invariants.Geo- metric and Functional Analysis , 32(2):280–301, 2022","cited_arxiv_id":null,"evidence_quote":"Provides the base-case transversality result and the index set of (-2)-classes on K3 surfaces that starts the induction in the K3 case."},{"cited_title":"More constraints on symplectic forms from seiberg-witten invariants.Mathe- matical Research Letters, 2(1):9–13, 1995","cited_arxiv_id":null,"evidence_quote":"Establishes the correspondence between Seiberg-Witten solutions on Kähler surfaces and effective divisors, used to identify the unique central-fiber solution."},{"cited_title":"Obstruction bundles, semiregularity, and seiberg–witten in- variants","cited_arxiv_id":null,"evidence_quote":"Supplies the criterion that a curve represents a regular family Seiberg-Witten solution exactly when the infinitesimal deformation map followed by restriction to the normal bundle is surjective."},{"cited_title":"Family Seiberg-Witten invariants and wall crossing formulas","cited_arxiv_id":"math/0107211","evidence_quote":"Provides the family Seiberg-Witten invariant and the chamber/wall formalism that makes the family counts well-defined."},{"cited_title":"Q-acyclic surfaces and their deformations.Contemporary Mathematics, 162:143–143, 1994","cited_arxiv_id":null,"evidence_quote":"Gives the invariance of log-tangent cohomology under blowup, which Lemma 3.10 uses to prove the blowup transversality formula."}],"review_version":1}