{"id":"fea00844-bddd-461e-b5e3-fbcdb7589901","arxiv_id":"1908.04936","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The logarithmic Gromov-Witten invariants of a product V×W equal the logarithmic Gysin pullback of the product of the invariants of V and W.","lead":"This paper proves a formula saying that logarithmic Gromov-Witten invariants of a product of two log smooth varieties are determined by the invariants of each factor separately. It is worth reading because it removes a technical restriction from an earlier product theorem by building logarithmic versions of normal cones and virtual fundamental classes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Everything downstream of Definition 1.7 depends on the unproved identification N^ℓ_{X/Y}=N_{X/LY}, deferred to the forthcoming [WH]; if it fails, the log virtual classes in Theorem 5.6 are not grounded.","rationale":"The reader's weakest assumption is the same deferred identification, and I concur that it is the load-bearing point. I also considered the application of the Log Costello Formula to the diagram in Situation 5.5: that is a real dependency, but it is downstream of the normal-sheaf identification, and its additional reliance on [Cos06] and [AHW] is a second-order issue. The primary foundation is whether Definition 1.7's functor of points is representable by the ordinary normal sheaf of X→LY. Without that, Definition 2.2, Definition 3.1, Construction 5.2, Theorem 4.1, and Theorem 5.6 all float on an unproved equivalence. The paper is honest about the dependence—it says 'See [WH] for details' and 'We show that this definition agrees'—so this is not a hidden flaw but an explicit incomplete proof. The concrete check is feasible without the forthcoming paper: compare both functors of points and, in particular, the torsor condition in Definition 1.7 with the square-zero lifting condition for X→LY. If the check succeeds, conditional acceptance remains appropriate; if it fails, the log normal cone and all subsequent virtual classes must be redefined before Theorem 5.6 can be evaluated.","tokens_in":22206,"tokens_out":26871,"duration_ms":291865,"concrete_test":"Independently derive the isomorphism in Definition 1.7 from Lemma 1.10(6) and the universal property of Olsson's L^2Y. Concretely, on a T-point, the data of a square-zero extension (A,M_A) of (O_{X|T},M_{X|T}) with M_A→M_{X|T} a torsor under 1+O_T must be shown to be canonically equivalent to a lift of T→X→LY to L^2Y, i.e. to a T-point of N_{X/LY}. A minimal case to compute is T=Spec k with A=k[ε]/(ε^2); both functors must give canonically bijective sets. If the torsor condition in Definition 1.7 is not exactly the lifting condition, then the log normal sheaf is misdefined and Theorem 5.6's virtual classes are not the ones claimed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central object is the log normal sheaf N^ℓ_{X/Y}, defined in Definition 1.7 through deformations of log structures. The only support for the identification N^ℓ_{X/Y}=N_{X/LY} is the sentence 'We show that this definition agrees with Definition 1.5 in [WH]', with details deferred to a forthcoming paper. The later machinery is built directly on this equality: Definition 2.2 defines C^ℓ_{X/Y}=C_{X/LY}, so the containment C^ℓ_{X/Y}⊆N^ℓ_{X/Y} required for a Log Perfect Obstruction Theory (Definition 3.1) and for the Log Gysin construction is only meaningful with the identification. Propositions 2.5 and 2.11 and Lemmas 2.15–2.16 transfer exact sequences of normal sheaves from the map X→LY; Construction 5.2 and Theorem 4.1 then produce the log virtual fundamental classes and the Log Costello formula used in the proof of Theorem 5.6. If the identification fails, the class h_*[M^ℓ_{g,n}(V×W),E(V×W)]^{ℓvir} in Theorem 5.6 is not the pushforward of the class defined by the stated obstruction theory, so the product formula is ungrounded. The manuscript neither proves the identification nor fully defines the square-zero closed-immersion notion used in Definition 1.7, citing [WH] for details. This is an acknowledged, load-bearing dependency rather than a matter of convention.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops logarithmic analogues of the intrinsic normal cone, perfect obstruction theory, Gysin pullback, and Costello's pushforward formula, and uses them to prove a product formula for logarithmic Gromov-Witten invariants of V × W in terms of those of V and W for log smooth quasiprojective schemes. The main theorem, Theorem 5.6, identifies the pushforward of the log virtual fundamental class of the moduli stack of stable log maps to V × W with the log Gysin pullback of the product of the two factor classes. The proof follows Behrend's strategy via a cartesian diagram of moduli stacks, replacing the ordinary normal cone and Gysin machinery with the newly introduced log versions. The manuscript is explicit about known failures of naive pushforward-Gysin compatibility (Remarks 2.14, 3.8, 3.9) and acknowledges several dependencies on forthcoming papers [WH] and [AHW].","tokens_in":22550,"tokens_out":14966,"duration_ms":145036,"significance":"If the technical foundations hold, the paper proves a natural and expected generalization of the product formula of Lee–Qu and Behrend to the logarithmic setting, removing the earlier restriction that one factor have trivial log structure. The paper also contributes a framework of log normal cones, log perfect obstruction theories, and log Gysin maps that is likely to be useful beyond the product formula. The author deserves credit for stating counterexamples to naive log analogues (Remark 2.14, Remarks 3.8 and 3.9) and for clearly separating what is proved here from what is deferred to [WH] and [AHW]. The central mathematical idea is coherent and the strategy of reducing log statements to strict, ordinary cases is attractive.","major_comments":[{"comment":"The equality N^ℓ_{X/Y} = N_{X/LY} is asserted with proof deferred to [WH], a forthcoming paper. This identification is load-bearing: Definition 2.2 defines the log normal cone as C_{X/LY}, Definition 3.1 requires the inclusion C^ℓ_{X/Y} ⊆ N^ℓ_{X/Y}, and the log virtual fundamental classes in Theorem 5.6 are constructed from this package. Without a proof or a published reference, the main theorem is conditional on an unverified premise. The paper should either prove this identification or replace the reference to [WH] with a published and accessible argument.","section":"Section 1, Definition 1.7 (also Definition 2.2)"},{"comment":"The notion of a square-zero closed immersion of log structures is not defined in the manuscript (the text says 'See [WH] for details'). Since the functor of points of N^ℓ is defined in terms of this notion, and Definition 1.7 claims agreement with N_{X/LY}, the reader cannot verify the central object of the paper. A precise definition and a proof of the claimed agreement are needed before the log normal sheaf can be used.","section":"Section 1, Definitions 1.5 and 1.7"},{"comment":"The proof of Theorem 4.1 asserts that after the reductions of Construction 1.1, 'the proof of Costello's Formula' applies to the strict closed immersions X ⊆ X^θ and X' ⊆ X'^θ, and that the exact sequences of Proposition 2.5 then show the cone map s is pure degree d. The first assertion is plausible only because the cones in question are then ordinary normal cones, but this is not stated, and the second assertion about preservation of pure degree under quotient by the tangent bundle stack is not justified. Since Theorem 4.1 is the mechanism producing the pushforward identity used in Theorem 5.6, this step should be written out in detail.","section":"Section 4, Theorem 4.1 (and Section 3, Theorem 3.10)"},{"comment":"The displayed chain of equalities in the proof applies φ^! to M_{g,n} × M_{g,n}, but Definition 3.3 permits φ^! to be applied only to a log-smooth equidimensional stack over Q' (the target of φ), and M_{g,n} × M_{g,n} is not such an input. The intended use of Corollary 3.15 must be written out with the correct Gysin pullbacks (the bottom horizontal map of the square, rather than φ, in the middle term). As printed, this is a gap in the proof of the main theorem.","section":"Section 5, proof of Theorem 5.6"}],"minor_comments":[{"comment":"The parenthetical '(equiv. N^ℓ_{X/Y} ⊆ E)' is not precise: N^ℓ is a sheaf, and the equivalence with a closed immersion of the cone C^ℓ_{X/Y} into a vector bundle stack requires explanation.","section":"Definition 3.1"},{"comment":"The expression C^ℓ_{C^ℓ_g|X / C^ℓ_g} uses a cone stack as the base of another cone stack, which is not defined in the text; please introduce notation for this construction.","section":"Theorem 3.12, Eq. (5)"},{"comment":"The paper relies on the forthcoming works [WH] and [AHW] for several supporting statements; for a journal submission these dependencies should either be removed or the statements proved in the paper.","section":"References and external dependencies"},{"comment":"There are frequent notational collisions between the prestable and stable curve stacks (both often rendered as 'M_{g,n}' in the text); the overline notation should be restored consistently to avoid ambiguity in the diagrams and the proof of Theorem 5.6.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the deferred identification in Definition 1.7: if the author cannot supply a proof or a published reference in the revision, the main theorem remains conditional. The type error in the displayed chain of the proof of Theorem 5.6 should be corrected. The explicit discussion of failures in Remarks 2.14, 3.8, 3.9 is a strength and should be kept."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a genuine extension of LQ18, not a repackaging. The paper proves the log Gromov-Witten product formula for arbitrary log smooth V and W, where LQ18 needed one factor to have trivial log structure. To do that it builds a log normal cone, log VFCs, a log Gysin map, and a log Costello formula. That is real work, and the author is unusually candid about what fails: log normal cones aren't invariant under log étale base change (Remark 2.14), and pushforward doesn't commute with log Gysin pullback (Remarks 3.8–3.9). Those admissions are not decorations; they shape the proofs.\n\nThe main theorem, Theorem 5.6, follows the Behrend/LQ18 strategy: compute the log normal cone of Q→Q' in two ways. The proof is short once Section 3's machinery is in place. Lemma 5.10 (compatibility of the obstruction theories) is a clean adaptation of Behrend's Proposition 6 and looks correct on a close read.\n\nNow the soft spots, in proportion. The load-bearing premise is the identification N^ℓ_{X/Y}=N_{X/LY}, stated after Definition 1.7 and deferred to [WH]. Everything downstream—Definition 2.2 of the log normal cone, the exact sequences in Section 2, the log POT in Definition 3.1, and hence the virtual classes in Theorem 5.6—depends on it. The stress-test note is right: if that identification fails, the product formula is ungrounded. This is a real dependency, not a matter of convention. The author knows; the sentence 'We show that this definition agrees...' is doing a lot of work. For a referee report I would want either the proof included or a precise statement with hypotheses in this paper.\n\nSecond dependency: the Log Costello Formula (Theorem 4.1) is proved by 'applying the proof of Costello's formula' after reducing to the ordinary case, with additional details left to [AHW]. That's thinner than the rest of the paper. The reduction to the strict case is sketched, and the argument is plausible, but it is the step that bridges from cone-level degree to Chow-level pushforward. It deserves a careful check.\n\nThe citation pattern is fine. The author cites Olsson, Manolache, Behrend-Fantechi, and the LQ18 paper it extends, plus the companion works [WH] and [AHW]. Relying on one's own forthcoming papers is mildly frustrating but not a flaw when the results are needed and the dependence is flagged.\n\nBottom line: this is a serious paper with a plausible central theorem and an honest accounting of its limitations. It is not a finished, self-contained reference—two key pieces are promised elsewhere. But it deserves a real referee, not desk rejection. I would send it to someone who knows Olsson's stack of log structures and ask them to verify the N^ℓ=N_{X/LY} identification and the Costello step. If those hold, the theorem stands.","headline":"The log product formula is likely correct, but the proof rests on a normal-sheaf identification deferred to a companion paper—worth a serious referee, not a desk reject.","tokens_in":23058,"tokens_out":2894,"would_cite":true,"duration_ms":27970,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14C17","14A20","14C15","14H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the Log Gromov-Witten Product Formula: the log virtual fundamental class of stable log maps to $V\\times W$ equals the log Gysin pullback of the product of the factor classes.","keywords":["log Gromov-Witten invariants","log normal cone","log virtual fundamental class","log Gysin map","product formula","log smooth schemes","fs log structures","intersection theory"],"falsifier":"Check Definition 1.7 directly on the log blowup example of Example 2.13: compute the square-zero deformations of log structures along a non-strict map such as the inclusion of a log point into the exceptional divisor, and compare the resulting sheaf with the ordinary normal sheaf of the map into the universal stack of log structures; a mismatch would invalidate the log virtual fundamental classes used in Theorem 5.6.","tokens_in":21949,"feed_emoji":"🧮","tokens_out":10746,"duration_ms":104191,"temperature":0.7,"pith_summary":"This paper proves a product formula for logarithmic Gromov-Witten invariants, the curve counts attached to stable maps from marked curves into log smooth targets: when the target is a product $V\\times W$ of log smooth quasiprojective schemes, the log virtual fundamental class of stable log maps to the product is determined by the corresponding classes for $V$ and $W$. The equality is stated as a log Gysin pullback along the diagonal after pushing forward from the space of stable log maps to $V\\times W$ to the fiber product $Q$. Earlier work had the same formula but only when one factor carried the trivial log structure; the present paper removes that restriction. A sympathetic reader should care because the result makes curve counting on product targets reducible to curve counting on the factors, and because the proof constructs the log normal cone, log virtual fundamental class, and log Gysin map that logarithmic intersection theory needs. Those constructions are the paper's main new tools and are reusable beyond this particular formula.","feed_headline":"Log curve counts in a product split into the factors","feed_subtitle":"A logarithmic intersection theory makes the product formula hold for all log smooth targets.","key_machinery":"The central object is the log normal cone $C^\\ell_{X/Y}=C_{X/LY}$, defined by sending a log map $X\\to Y$ to the ordinary normal cone of the strict map $X\\to LY$, where $LY$ is the universal stack of log structures over $Y$; for strict maps this cone agrees with the ordinary normal cone, and the definition is chosen to be invariant under log \\'etale base change. Its companion is the log normal sheaf $N^\\ell_{X/Y}$, whose points are square-zero deformations of log structures. A log perfect obstruction theory is an embedding $C^\\ell_{X/Y}\\subseteq E$ into a vector bundle stack, from which the log virtual fundamental class and the log Gysin map $f^!$ are extracted. The proof's load-bearing diagram is an fs pullback square involving the stack of partial stabilizations; the log Costello formula (Theorem 4.1) and the commutativity theorem for log Gysin maps (Theorem 3.12) are the two inputs that make the two computations agree.","core_discovery":"The paper's central claim is Theorem 5.6: for log smooth quasiprojective schemes $V$ and $W$, the identity\n$$h_*[M^\\ell_{g,n}(V\\times W), E(V\\times W)]^{\\ell\\mathrm{vir}} = \\$\\Delta$^!([M^\\ell_{g,n}(V), E(V)]^{\\ell\\mathrm{vir}}\\times [M^\\ell_{g,n}(W), E(W)]^{\\ell\\mathrm{vir}})$$\nholds in the Chow group of $Q$. Here $E(\\cdot)$ is the natural log perfect obstruction theory for stable log maps, $h$ is the comparison map to the fiber product $Q$, and $\\Delta^!$ is the log Gysin pullback. The proof follows a classical two-way computation: it forms a fine-and-saturated (fs) pullback square involving the stack of partial stabilizations, and computes the log normal cone of the comparison map $Q\\to Q'$ in two ways. One computation uses the log analogue of Costello's pushforward formula, the other uses commutativity of log Gysin maps; the two must agree, which is exactly the displayed formula.","pith_inferences":["Beyond the paper: the same two-way log normal cone computation should prove product formulas for other log moduli problems, such as stable log maps with contact-order conditions or relative expansions, since the machinery is not tied to the specific stack of stable maps.","Beyond the paper: the paper's counterexamples show ordinary pushforward loses logarithmic information, suggesting that a logarithmic Chow group is the right home for these classes; with such a theory, the equality $p_*[\\hat X]^{\\ell\\mathrm{vir}}=[X]^{\\ell\\mathrm{vir}}$ may hold without the log smoothness hypothesis.","Beyond the paper: because the log normal cone can differ from the ordinary normal cone even when the underlying schemes agree, the formula implies that ordinary product-formula computations can miss logarithmic contributions in mixed-degree cases; comparing the two outputs on a product of nontrivial log points would make this visible."],"forward_implications":["The log Gromov-Witten invariants of $V\\times W$ are determined by those of $V$ and $W$, so curve counting on product targets reduces to smaller problems.","The log Gysin map and log virtual fundamental class introduced here become available for any DM-type log map equipped with a log perfect obstruction theory, not only for stable log maps.","The log Costello formula gives a valid pushforward formula along log blowups of log smooth targets, even though pushforward compatibility with log Gysin maps fails in general.","The earlier product formula, proved only when one factor has trivial log structure, is included as the special case of the new theorem."],"supporting_citations":[{"why":"Formulated the product formula for log Gromov-Witten invariants and proved it when one factor has trivial log structure; the present paper removes that restriction.","marker":"[LQ18]"},{"why":"Established the ordinary product formula for Gromov-Witten invariants, whose two-way cone computation is the template for Theorem 5.6.","marker":"[Beh97]"},{"why":"Supplies the pushforward formula whose log analogue (Theorem 4.1) is one of the two main inputs.","marker":"[Cos06]"},{"why":"Provides the virtual pullback framework that the log Gysin map and log virtual classes adapt.","marker":"[Man08]"},{"why":"Defines the intrinsic normal cone and obstruction theory used to build the log normal cone and log classes.","marker":"[BF96]"},{"why":"Constructs log Gromov-Witten invariants and the natural log perfect obstruction theory for stable log maps to a log smooth target.","marker":"[GS11]"},{"why":"Gives the log cotangent complex and properties of stacks of log structures used for the log normal sheaf and exact sequences.","marker":"[Ols05]"},{"why":"Supplies the obstruction-theory formalism, embeddings into vector bundle stacks, used in the definitions.","marker":"[Wis11]"},{"why":"Defines log algebraic stacks and the universal stack of log structures on which the log normal cone is based.","marker":"[Ols03]"}],"fun_headline_variants":["Log Gromov-Witten counts split across product factors","Product formula for log curve invariants","Log invariants factor: V times W equals product of counts","Splitting log curve counts in a product"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the log normal sheaf defined in this paper coincides with the ordinary normal sheaf of the map $X\\to LY$ to the universal stack of log structures; the identification is asserted and deferred to another paper, yet Section 2's exact sequences, the obstruction-theory definitions, and the log Gysin machinery all depend on it.","fun_headline_variants_meta":{"raw":{"variants":["Log Gromov-Witten counts split across product factors","Product formula for log curve invariants","Log invariants factor: V times W equals product of counts","Splitting log curve counts in a product"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000491,"raw_usage":{"total_tokens":2359,"prompt_tokens":838,"completion_tokens":1521,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":1459}},"tokens_in":454,"tokens_out":1521,"duration_ms":12274,"temperature":1.0,"reasoning_tokens":1459,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:29:19.644842+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check Definition 1.7 directly on the log blowup example of Example 2.13: compute the square-zero deformations of log structures along a non-strict map such as the inclusion of a log point into the exceptional divisor, and compare the resulting sheaf with the ordinary normal sheaf of the map into the universal stack of log structures; a mismatch would invalidate the log virtual fundamental classes used in Theorem 5.6.","supporting_citations":[],"review_version":1}