{"id":"779eca47-6ac4-431c-902a-f9288e3294a6","arxiv_id":"2505.13120","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In high degree, fixed-domain Gromov-Witten invariants of positive symplectic manifolds equal signed counts of pseudo-holomorphic curves for a generic almost complex structure.","lead":"Fixed-domain Gromov-Witten invariants of positive symplectic manifolds are shown, in sufficiently high degree, to be signed counts of pseudo-holomorphic curves for a generic almost complex structure. This proves a symplectic analogue of a conjecture that fails in the complex algebraic setting.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's verdict of ACCEPT with moderate confidence is reasonable. The paper proves a substantial theorem by a long, intricate induction, and the central claim is internally consistent. The weakest hypothesis is indeed the quantitative regime inequality (2.2), which limits the theorem to codimensions that are small relative to the number of marked points; this is a stated restriction rather than a flaw. The proof's most delicate quantitative estimate, Lemma 4.4, is somewhat under-explained, but the intended tangent count appears correct. Since no load-bearing objection emerged, the reader's ACCEPT verdict should stand.","tokens_in":25646,"tokens_out":44494,"duration_ms":520873,"concrete_test":"Independently re-derive the differential dγ in Lemma 4.4, imposing the tangent constraint that ps=φ(p) as sets, and verify rank(dγ)≥max{0,m−b−3}. As a model case, take an unramified double cover of a genus-2 curve with m=20 target markings and both preimages of each marking included; compute dim H and rank(dγ), and check that dim H−rank(dγ)≤4d0=8 and that the claimed lower bound holds. If the rank bound survives this check, the induction in Proposition 4.3 is safe.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After reviewing the central induction and the quantitative codimension estimates, I find no internal inconsistency that would invalidate Theorem 1.1. The regime inequality (2.2) is genuinely restrictive, but it is consistent with Theorem 1.1 because the constant d(r) can be chosen so that codim(ev)≤dn implies k≤(r−1−ρ)n/5 for a fixed ρ>0. The most delicate technical step is the rank bound for γ in Lemma 4.4: the proof that rank(dγ)≥max{0,m−b−3} is tersely justified, and the wording about the kernel being “at most three” could easily be misread when several marked source points lie in the same fiber. However, the intended tangent count uses the constraint that all preimages of a given target marking move coherently, so the bound appears sound. I do not see a place where the argument breaks, only places where a reader must supply routine tangent-space bookkeeping.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for compact positive symplectic manifolds of real dimension at least six, the fixed-domain Gromov–Witten invariants can be represented, in sufficiently high degree and under a bound on the virtual codimension, by signed counts of unperturbed pseudo-holomorphic maps from a fixed marked curve. The main theorem (Theorem 1.1) is deduced from a general codimension statement (Theorem 2.16) for strata of stable maps modelled on newly introduced augmented graphs, using a simplification process that preserves the weighted homology class and a self-contained transversality theorem for simple maps (Section 6). The paper also answers, in this high-degree regime, a question of Ruan–Tian on whether the Gromov–Witten pseudocycle can be constructed without inhomogeneous perturbations.","tokens_in":25830,"tokens_out":36100,"duration_ms":401820,"significance":"If the result stands, it gives the first general enumerative interpretation of fixed-domain Gromov–Witten invariants in the symplectic category without domain-dependent or inhomogeneous perturbations, resolving the Ruan–Tian question in a positive direction. The technical machinery of augmented graphs and the refined simplification process is interesting in its own right and likely applicable to other enumerative problems. The paper is careful about its hypotheses: the high-degree condition and the codimension bound (2.2) are explicit, and the constants are existential rather than fitted to data. The proofs are unusually detailed for the field, including a complete transversality proof for simple maps and an appendix on Fredholm maps, which substantially increases confidence in the claims.","major_comments":[],"minor_comments":[{"comment":"In the paragraph after the definition of the augmented graph eΓ, the paper refers to \"condition (vi)\" of Definition 2.5, but that definition has only items (i)–(iv); this should read \"condition (iv)\".","section":"Section 5"},{"comment":"The statement says that the image \"has codimension 2k+2\", but the intended assertion is \"has codimension at least 2k+2\"; the inequality is important because the conclusion is a lower bound on codimension.","section":"Corollary 2.17"},{"comment":"The transition from the universal statements of Theorem 1.5 and Corollary 2.17 (images in X^n×J) to the fixed-generic-J statement of Theorem 1.1 is not written out. A standard Sard–Smale slicing argument for the Fredholm projection to J gives the fixed-J codimension bound, but the authors should state this step explicitly, since it is load-bearing for the pseudocycle claim.","section":"Section 5 / Theorem 1.1"},{"comment":"The rank estimate for γ says that the kernel of the restriction is \"at most three dimensional\". This wording is misleading when several marked source points lie in the same fiber of φ; the intended argument is to choose one unramified preimage for each target marking and then use the at-most-three-dimensional kernel of the map from marked-point tangent spaces to the moduli space of marked curves. The estimate is sound, but the proof should be rephrased for clarity.","section":"Lemma 4.4"},{"comment":"The proof of Theorem 6.1 explicitly assumes n=0 and does not explain how the case n>0 is obtained. The index formula (6.1) is stated for general n, so the proof should mention that including the n marked points adds n to the Teichmüller slice dimension and n evaluation conditions, leaving the stated index unchanged.","section":"Section 6"},{"comment":"There are several typographical slips: \"the the image\" in the paragraph after Theorem 1.5, \"an he thanks\" in the acknowledgments, and \"Proposition Proposition 4.5\" and \"Theorem Theorem\" in Section 4.3. These should be corrected.","section":"Throughout"},{"comment":"The assertion \"In particular, |E(Γ)|≥2ℓ\" is not immediate from the preceding sentence unless each of the ℓ special vertices has two edges to the main vertex, which is consistent with val(α_i)=3. A short explanatory sentence would help the reader verify the fixed-domain constraint in examples.","section":"Definition 2.5(iv)"}],"recommendation":"minor_revision","confidential_remarks":"This is a strong paper and the reader's accept recommendation is reasonable. The technical core appears sound; my main request is to make explicit the standard slicing argument that passes from the universal codimension bounds to the fixed generic almost complex structure, and to fix the small presentation issues listed. These are local and should not require a further round of technical refereeing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline result is real: for positive symplectic manifolds of dimension at least six, the fixed-domain Gromov-Witten pseudocycle can be built from unperturbed pseudo-holomorphic curves, provided the degree is high and the virtual codimension is not too large. That answers Ruan-Tian's question in a substantial regime and gives a literal signed-count interpretation to fixed-domain GW invariants. The algebraic analogue has counterexamples, so this is a new result in symplectic geometry, not a repackaging.\n\nWhat is genuinely good: the augmented graph stratification and the refined simplification process that preserves the weighted homology class. These are real technical ideas, and they are used honestly. The paper also includes a self-contained transversality theorem for simple maps and an appendix on Fredholm maps, which makes the proof much more checkable than is typical for this literature. The authorship is careful: the constants c and d are existential, not fitted to examples; the regime inequality (2.2) is stated as a hypothesis; and the authors explicitly flag that the earlier version of the paper omitted the Ruan-Tian and McDuff-Salamon discussion. I do not see a circular step or a hidden parameter.\n\nWhere are the soft spots? The proof is long and analytically dense. Section 4 relies on several numerical choices, including epsilon = 0.9 and delta = 0.2 in the case analysis, and the estimates are not machine-checked or independently verified. I would not call that a flaw; it is a reason to read carefully. The one spot that gave me real pause is Lemma 4.4: the rank bound for the differential of gamma is tersely justified, and the wording about the kernel being 'at most three' can be misread when marked points share a fiber. The stress-test note argues the intended tangent count is sound, and on reading the lemma that seems right, but the paper would benefit from a few more sentences of bookkeeping there. This is a minor presentational issue, not a mathematical gap. The qualitative restriction codim(ev) ≤ d n is genuine and means the theorem is not the full Ruan-Tian conjecture; the paper does not oversell it.\n\nWho is this for? Symplectic enumerative geometers working on Gromov-Witten pseudocycles and anyone tracking the Lian-Pandharipande conjecture. The reader's ACCEPT verdict is fair; the confidence at MODERATE is appropriate without formal verification. I would send it to a serious referee, expecting heavy scrutiny of Section 4, and I would cite it.","headline":"A substantial, carefully argued symplectic answer to Ruan-Tian in the high-degree fixed-domain regime; the main theorem deserves a serious referee, even though the analytic estimates are long and not machine-checked.","tokens_in":26316,"tokens_out":938,"would_cite":true,"duration_ms":12320,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D45","14N35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any compact positive symplectic manifold of real dimension at least six, fixed-domain Gromov-Witten invariants of sufficiently high degree are signed counts of pseudo-holomorphic curves.","keywords":["fixed-domain Gromov-Witten invariants","pseudo-holomorphic curves","pseudocycle","positive symplectic manifolds","augmented graphs","enumerative geometry","Tevelev degrees","transversality"],"falsifier":"Exhibit a compact positive symplectic manifold of real dimension \\(2r\\ge 6\\), a generic \\(J\\), and arbitrarily large classes \\(A\\) with \\(\\operatorname{codim}(\\operatorname{ev})\\le dn\\) but for which a non-simple stratum — for instance maps with a multiply covered rational tail on a fixed domain — has image closure of real codimension strictly less than \\(2k+2\\) in \\(X^n\\). Such a stratum would make \\(\\operatorname{ev}\\) fail the pseudocycle bound; a dimension calculation for \\(\\mathbb{CP}^3\\) with the standard structure, genus zero, and marked-point data tuned so that \\(k/n\\) approaches \\((r-1)/5\\) would settle whether the bound is sharp or false.","tokens_in":25471,"feed_emoji":"📐","tokens_out":13554,"duration_ms":130725,"temperature":0.7,"pith_summary":"This paper establishes a symplectic counterpart of the algebraic conjecture that fixed-domain Gromov-Witten invariants agree with geometric counts of curves in sufficiently high degree. The setting is a compact positive symplectic manifold — one whose first Chern class is positive on every symplectic sphere class, with smooth Fano varieties as the main examples. The algebraic version of the conjecture is known to fail in explicit examples, but the authors show that a fixed-domain version holds once complex structures are replaced by generic almost complex structures: for large \\langle c_1(X),A\\rangle and bounded relative codimension, the evaluation map from the moduli space of simple pseudo-holomorphic maps is a pseudocycle, so the invariant is a signed count of actual curves. In particular, the count needs no inhomogeneous or domain-dependent perturbations, resolving a question left open by the original pseudocycle construction. The proof works by controlling the image of all non-simple strata through a refined combinatorial stratification of the moduli space.","feed_headline":"Fixed-domain G-W counts are signed curve counts in high degree","feed_subtitle":"For positive symplectic manifolds, generic almost complex structures make fixed-domain invariants signed counts of curves.","key_machinery":"The workhorse is the moduli space stratified by augmented graphs. An augmented graph is a dual graph of the domain curve decorated with four extra pieces of data: a weight \\(m_\\$\\alpha$\\), a covering degree \\(d_\\$\\alpha$\\), the homology class \\(A_\\$\\alpha$\\) of the underlying simple map on each component, and a function \\(h\\) recording which two components have the same image in \\(X\\). These decorations are arranged so that every stable map modelled on an augmented graph \\(\\tilde\\Gamma\\) has a well-defined weighted homology class \\([\\tilde\\Gamma,m]\\), which is preserved under the paper's simplification process. The simplification step repeatedly replaces a multiply covered component by its underlying simple map, identifies two components with identical image, or collapses a contracted main component, while recording exactly which points are identified. The bookkeeping is finer than the older genus-zero simplification used in [21], and it is what lets the authors compute, by induction, that every non-simple stratum has image of real codimension at least \\(2k+2\\) under the evaluation map. That codimension bound is precisely what turns the evaluation map into a pseudocycle.","core_discovery":"The central claim is Theorem 1.1: if \\((X,\\omega)\\) is a compact positive symplectic manifold of real dimension \\(2r\\ge 6\\), there are constants \\(c=c(r,g)\\) and \\(d=d(r)\\) such that, for a generic almost complex structure \\(J\\) and a generic marked curve \\((C,p)\\in \\mathcal M_{g,n}\\), the evaluation map \\(\\operatorname{ev}: \\mathcal M^*(C,p;X,A,J)\\to X^n\\) is a pseudocycle — a map whose image has compact closure and whose limit set has real codimension at least two — whenever \\(\\langle c_1(X),A\\rangle\\ge c\\) and \\(\\operatorname{codim}(\\operatorname{ev})\\le dn\\). Corollary 1.3 then says that, for any homology class \\(\\gamma\\) of the appropriate degree, the corresponding fixed-domain Gromov-Witten invariant equals the signed count of intersection points of this evaluation map with any pseudocycle Poincaré dual to \\(\\gamma\\). Equivalently, the invariant constructed from virtual fundamental classes is, in this regime, an honest signed count of pseudo-holomorphic curves through the imposed point conditions, with no perturbed Cauchy-Riemann term \\(\\nu\\) required.","pith_inferences":["The same augmented-graph stratification is not tied to fixed domains, so a testable extension is to use it to detect enumerativity of ordinary, non-fixed-domain Gromov-Witten invariants of positive symplectic manifolds in high-degree regimes.","The quantitative bound \\(k\\le ((r-1-\\rho)/5)n\\) suggests a threshold phenomenon: enumerativity should appear when the number of marked points is large compared with the virtual codimension of the evaluation map, and probing whether the constant \\(1/5\\) can be improved would directly test the optimality of the method.","If the stratification is applied to non-positive targets, the positivity assumption is what forces contracted components off the main component to have bounded complexity; testing the same construction on a Calabi-Yau or symplectically aspherical target would show where the argument breaks."],"forward_implications":["When \\(\\langle c_1(X),A\\rangle\\ge c\\) and \\(\\operatorname{codim}(\\operatorname{ev})\\le dn\\), the fixed-domain Gromov-Witten invariant of a positive symplectic manifold can be computed as a signed count of \\(J\\)-holomorphic curves, without introducing any inhomogeneous term.","The pseudocycle is independent of the choice of generic almost complex structure and generic domain curve, so the signed count is a well-defined invariant rather than an artifact of a particular perturbation.","The disproof of the algebraic enumerativity conjecture does not contradict the symplectic result: replacing complex structures by generic almost complex structures is the corrective step that restores enumerativity.","The main theorem is a special case of Theorem 2.16, which makes the constants explicit in principle, and the paper leaves open whether the codimension bound \\(\\operatorname{codim}(\\operatorname{ev})\\le dn\\) is sharp.","The result requires real dimension at least six; the four-dimensional case is not covered and needs separate dimension counts."],"supporting_citations":[{"why":"introduces the perturbed-equation pseudocycle for fixed-domain invariants and asks whether the inhomogeneous perturbation can be dropped","marker":"[22]"},{"why":"supplies the genus-zero simplification procedure and transversality statements that the paper refines for higher genus","marker":"[21]"},{"why":"gives the counterexamples to the algebraic enumerativity conjecture that motivate the symplectic reformulation","marker":"[2]"},{"why":"formulates the original algebraic conjecture and defines the geometric Tevelev degrees being compared","marker":"[18]"},{"why":"provides the higher-genus transversality result with inhomogeneous perturbations that the present paper avoids by a self-contained proof","marker":"[23]"},{"why":"computes virtual Tevelev degrees via quantum cohomology and documents cases where virtual and geometric degrees differ","marker":"[5]"},{"why":"establishes that the pseudocycle class agrees with the pushforward of the virtual fundamental class, which justifies the signed-count identification","marker":"[14]"},{"why":"gives the Teichmüller slice construction used in the transversality proof for simple maps on arbitrary graphs","marker":"[26]"}],"fun_headline_variants":["Fixed-domain G-W invariants are honest curve counts without perturbations","In high degree, fixed-domain symplectic invariants count curves without perturbations","Positive symplectic manifolds: G-W invariants are signed counts in high degree","Generic almost complex structure makes fixed-domain invariants signed curve counts","Fixed-domain G-W counts become signed curve counts with generic J"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the quantitative regime inequality \\(k\\le \\frac{r-1-\\rho}{5}n\\): the virtual codimension of the evaluation map must be at most a fixed fraction of the number of marked points, and the target must have real dimension at least six; outside that regime the dimension counts for non-simple strata are not controlled.","fun_headline_variants_meta":{"raw":{"variants":["Fixed-domain G-W invariants are honest curve counts without perturbations","In high degree, fixed-domain symplectic invariants count curves without perturbations","Positive symplectic manifolds: G-W invariants are signed counts in high degree","Generic almost complex structure makes fixed-domain invariants signed curve counts","Fixed-domain G-W counts become signed curve counts with generic J"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2851,"prompt_tokens":908,"completion_tokens":1943,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":1849}},"tokens_in":524,"tokens_out":1943,"duration_ms":13076,"temperature":1.0,"reasoning_tokens":1849,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:19:36.182533+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a compact positive symplectic manifold of real dimension \\(2r\\ge 6\\), a generic \\(J\\), and arbitrarily large classes \\(A\\) with \\(\\operatorname{codim}(\\operatorname{ev})\\le dn\\) but for which a non-simple stratum — for instance maps with a multiply covered rational tail on a fixed domain — has image closure of real codimension strictly less than \\(2k+2\\) in \\(X^n\\). Such a stratum would make \\(\\operatorname{ev}\\) fail the pseudocycle bound; a dimension calculation for \\(\\mathbb{CP}^3\\) with the standard structure, genus zero, and marked-point data tuned so that \\(k/n\\) approaches \\((r-1)/5\\) would settle whether the bound is sharp or false.","supporting_citations":[{"cited_title":"Ruan and G","cited_arxiv_id":null,"evidence_quote":"introduces the perturbed-equation pseudocycle for fixed-domain invariants and asks whether the inhomogeneous perturbation can be dropped"},{"cited_title":"McDuff and D","cited_arxiv_id":null,"evidence_quote":"supplies the genus-zero simplification procedure and transversality statements that the paper refines for higher genus"},{"cited_title":"Beheshti, B","cited_arxiv_id":null,"evidence_quote":"gives the counterexamples to the algebraic enumerativity conjecture that motivate the symplectic reformulation"},{"cited_title":"Lian and R","cited_arxiv_id":null,"evidence_quote":"formulates the original algebraic conjecture and defines the geometric Tevelev degrees being compared"},{"cited_title":"Ruan and G","cited_arxiv_id":null,"evidence_quote":"provides the higher-genus transversality result with inhomogeneous perturbations that the present paper avoids by a self-contained proof"}],"review_version":1}