{"id":"c5b6df1f-af58-4fb6-9e8d-e9548fbac799","arxiv_id":"2505.09535","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors define and study quasimodular forms for O(2,n), prove the constant-term isomorphism and a weight-depth criterion for theta lifts, and conjecture modularity for Enriques and bielliptic surface Gromov-Witten potentials.","lead":"This paper builds a theory of quasimodular forms for orthogonal groups of type (2,n), proves structural theorems about them and about the Borcherds theta lift, and gives explicit Fourier expansions. It then conjectures that the Gromov-Witten potentials of Enriques and bielliptic surfaces are such forms, and verifies the conjecture in important cases.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.12's only-if direction depends on an unproved assertion that the depth-d part L^d(F) has a negative-index Fourier coefficient when k<2d; without a proof or citation the 'if and only if' claim is not fully established.","rationale":"The reader's weakest-assumption pinpoints exactly the same statement: the unproved negative-coefficient lemma in the converse of Theorem 5.12. I agree that this is the most load-bearing unresolved point in the paper's central claim. The rest of the central theory appears sound: the constant-term isomorphism in Theorem 6.17 has two independent proof strategies, the omitted commutator computation in Proposition 4.17 is explicitly described as a direct calculation, and the theta-lift expansion in Section 7 is derived in detail. The coefficient lemma is standard and likely follows from the vanishing theorem for holomorphic vector-valued modular forms of negative weight, so the concern is about rigor rather than an actual mathematical falsehood. The paper also contains an overstatement in the abstract about K3/abelian fibrations, but the authors themselves acknowledge in Section 12.2 that they cannot rule out another almost-holomorphic completion; this is a wording issue, not a flaw in the central theorems. Because the missing lemma is fillable and the reader's conditional verdict already asks for this gap to be closed, I do not adjust the verdict.","tokens_in":80208,"tokens_out":27765,"duration_ms":253507,"concrete_test":"Check that every nonzero weakly holomorphic vector-valued modular form of negative weight for the Weil representation of a lattice M=U+direct sum+L, holomorphic on the upper half-plane, has at least one Fourier coefficient with n<0. Concretely, apply [26, Prop.4.4] to L^d(F) in the setting of Theorem 5.12; if L^d(F) is nonzero and holomorphic at infinity, the negative-index coefficient exists and the only-if direction is valid. A more direct verification is to take a concrete example, e.g., M=U+direct sum+U and F of depth 1 with k=1, expand L(F) in q and confirm a nonzero coefficient at a negative index.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 5.12 (Section 5.6), the converse direction is justified by the sentence: 'If k<2d, then L^d(F) is of negative weight. Therefore, F has nonzero Fourier coefficients c(d)(n,gamma) with n<0.' This is the only argument excluding almost-holomorphicity of Lift(F) when the weight-depth inequality fails, so the 'if and only if' in Theorem 5.12 and the abstract's necessary-and-sufficient claim rest on it. The inference is not immediate for vector-valued almost-holomorphic forms with a pole at infinity; it requires a vanishing theorem for holomorphic vector-valued modular forms of negative weight applied to L^d(F). The gap is probably fillable using [26, Prop.4.4] together with the cusp-condition discussion in Section 6.8, but the paper supplies no proof or reference at that point. If the lemma were false, there could exist almost-holomorphic lifts even when k<2d, invalidating the converse and the negative results for K3/abelian fibrations in Section 12.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of almost-holomorphic and quasimodular forms for the orthogonal group of a signature (2,n) lattice, using orthogonal lowering and raising operators. The main structural results are that the constant-term map from almost-holomorphic to quasimodular forms is an isomorphism (Theorem 6.17), that orthogonal quasimodular forms admit descriptions in terms of vector-valued modular forms, and that the theta lift interacts with the lowering and raising operators through explicit equivariance formulas. A central result is Theorem 5.12, which gives a necessary and sufficient weight-depth inequality k ≥ 2d for the theta lift of an almost-holomorphic modular form of depth d to be almost-holomorphic, under the assumption that the lattice splits a hyperbolic plane. The paper also gives a completely explicit Fourier expansion of the theta lift (Theorem 7.1) and proves that Fourier-Jacobi coefficients of orthogonal quasimodular forms are quasi-Jacobi forms. In the second part, the authors conjecture that Gromov-Witten potentials of Enriques and bielliptic surfaces are components of vector-valued logarithmic orthogonal quasimodular forms satisfying holomorphic anomaly equations, and they provide evidence from Hodge integrals and Fourier-Jacobi expansions. They further show that certain natural completions of fiber Gromov-Witten potentials of K3 and abelian surface fibrations are not almost-holomorphic.","tokens_in":80396,"tokens_out":5271,"duration_ms":53571,"significance":"If the central results hold, this paper gives the first systematic quasimodular formalism for type IV domains that is a genuine analogue of the classical SL2 theory, including an explicit and uniformly stated theta-lift expansion. The constant-term isomorphism, the equivariance of the theta lift with respect to lowering and raising operators, and the Fourier-Jacobi theorem are substantial and likely influential. The geometric applications to Enriques surfaces are also significant: the conjecture that GW potentials are vector-valued logarithmic orthogonal quasimodular forms is well-motivated and supported by concrete evidence, in particular the identification of Hodge integrals as theta lifts and the verification at the level of Fourier-Jacobi expansions. The paper is careful to separate theorems from conjectures and to attribute prior work, and it includes several explicit computations that can be checked independently. The main reservations concern two load-bearing points where supporting arguments are either omitted or only sketched: the converse direction of Theorem 5.12 and the proof of the commutator formula in Proposition 4.17.","major_comments":[{"comment":"The only-if direction of Theorem 5.12 depends on the unproved assertion that when k < 2d, the depth-d part L^d(F) of an almost-holomorphic modular form of negative weight has a nonzero Fourier coefficient c^{(d)}(n,γ) with n < 0. The proof states this without proof or reference. This assertion is load-bearing: without it, the conclusion that Lift(F) fails to be almost-holomorphic does not follow, and the same gap affects the negative claims about K3 and abelian fibrations in Section 12. Please supply a proof or a precise citation, for example a vanishing theorem for vector-valued holomorphic modular forms of negative weight applied to the top-depth component, together with the cusp-condition discussion in Section 6.8.","section":"Section 5.6, proof of Theorem 5.12"},{"comment":"The abstract and Section 12 claim that parallel statements for arbitrary K3 or abelian-surface fibrations 'do not hold', but the body only proves that the natural completion obtained from the theta lift is not almost-holomorphic. Footnote 14 explicitly concedes that the existence of some other almost-holomorphic completion is not ruled out. Please qualify the abstract and the section statement accordingly, or prove the stronger non-existence claim. Similarly, the assertion in Section 12.3 that one can 'show directly' that the series (12.1) is not quasimodular is made without proof; either provide the argument or mark it as a claim.","section":"Section 12.2 and footnote 14"},{"comment":"The proof of the commutator formula [L,R]F = (k/2)F⊗g + (1/2)Σ(...) is omitted, with the text saying the computation is 'lengthy but straightforward and omitted here'. This identity is used repeatedly in later sections, including the proof that raising preserves almost-holomorphicity (Proposition 4.23), the derivation of the Zemel operator identities (Proposition 5.10), and the commutator statement in the tube domain (Lemma 6.6). Since this is a foundational formula, the computation should be included in full or in an appendix, or at least reduced to a clearly verifiable symbolic identity.","section":"Section 4.17, Proposition 4.17"}],"minor_comments":[{"comment":"The label 'Seond Proof of Theorem 6.17' contains a typo; it should read 'Second Proof'.","section":"Section 6.5"},{"comment":"The second proof of Theorem 6.17 is only a sketch: it cites restriction results from [26, Chapter 4], invokes 'the discussion in Section 6.9 below', and then refers to the classical case. Since the first proof is complete, this sketch does not affect the theorem, but the text should be clarified so that it is clear that the second proof is supplementary.","section":"Section 6.5, second proof of Theorem 6.17"},{"comment":"The definition of the positive/negative cones via 'µ > 0 means that ⟨µ,v⟩ > 0 for all v sufficiently close to the boundary point (0,1)⊕0' is somewhat informal; a precise description of the domain of convergence of the displayed series would improve readability.","section":"Theorem 7.1"},{"comment":"In Definition 4.27 the phrase 'Its immediate to check' should read 'It is immediate to check'.","section":"Section 4.6"},{"comment":"Theorem 5.12 is stated before the cusp condition for n ≤ 2 is introduced in Section 6.8. Please state explicitly whether the theorem is intended for all n ≥ 2 in the sense of the later definition, or restrict to n ≥ 3.","section":"Section 5.6, Theorem 5.12"}],"recommendation":"major_revision","confidential_remarks":"This is a substantial and largely convincing paper, but the load-bearing gap in the only-if direction of Theorem 5.12 and the omitted proof of the commutator formula in Proposition 4.17 should be addressed before publication. The Section 12 negative claims also need to be qualified in line with footnote 14. I recommend major revision rather than rejection because the gaps appear fillable and the main structural results are well supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The core of Part 1 is genuinely new and likely correct: the vector-valued lowering/raising operators for O(2,n), the constant-term isomorphism AHMod → QMod, the weight-depth criterion for theta lifts, and the explicit Fourier expansion together give a working analogue of the classical SL2 quasimodular theory. Second, the paper is not yet airtight: two load-bearing verifications are missing, and the abstract overclaims in one spot.\n\nThe strong parts deserve credit. Theorem 6.17 gets two independent proofs, one through a transformation formula and one by restriction to classical SL2 cusps. The explicit theta-lift expansion (Cor 7.2) is a clean statement and recovers Borcherds' classical formula verbatim as the constant term. The Fourier-Jacobi theory in Section 8, especially the compatibility between the orthogonal lowering operator and the Jacobi lowering operators (Prop 8.10), is exactly what the geometric application needs. The geometric half is honest: Conjectures B and C are labeled as conjectures, and the evidence is real — the Hodge integrals of [31] are shown in Prop 10.7 to be constant terms of theta lifts, and the Fourier-Jacobi checks line up. Nothing circular there.\n\nSoft spots, in proportion. (1) The only-if direction of Theorem 5.12 asserts without proof that when k < 2d, the depth-d part L^d(F) has Fourier coefficients with n < 0. That step carries the converse and the negative results of Section 12. It is probably a quick fix — L^d(F) is holomorphic, and [26, Prop.4.4] plus the cusp discussion in Section 6.8 should rule out the nonnegative-index case — but as written it is an unproved lemma inside a theorem proof. (2) The commutator formula in Prop 4.17 is left as \"lengthy but straightforward.\" It drives the whole depth formalism; it should be written out once, even in an appendix. (3) The abstract says parallel statements for arbitrary K3 or abelian fibrations \"do not hold,\" but Section 12 only rules out the natural completion, with a footnote admitting another completion is not excluded. Soften the abstract.\n\nThe citation pattern is fine, and the discussion of how this relates to Shimura, Zemel, and Ma is honest. My own read is more optimistic than the soundness score in the report: I would be surprised if the theorems are false, but the gaps must be filled. This paper is for modular-form and enumerative-geometry people, it will be used, and it deserves a serious referee. Conditional accept: send it out, ask for the lemma and the commutator computation, soften the abstract.","headline":"Genuinely new and likely correct quasimodular formalism for O(2,n) with Enriques GW conjectures; two fillable verification gaps and an overstated abstract claim keep it from being airtight.","tokens_in":80929,"tokens_out":5606,"would_cite":true,"duration_ms":54366,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F55","11F50","11F37","14N35","14J28"],"pacs":[],"model":"deepseek-v4-flash","headline":"An almost-holomorphic theta lift stays almost-holomorphic precisely when k≥2d, and constant terms give an isomorphism to quasimodular forms.","keywords":["quasimodular forms","orthogonal groups","almost-holomorphic modular forms","theta lifts","Enriques surfaces","Gromov-Witten invariants","Fourier-Jacobi expansions","vector-valued modular forms"],"falsifier":"For a lattice M=U⊕L, construct an almost-holomorphic vector-valued form of weight κ=k+1−n/2 and depth d>k/2 whose depth-d layer has only nonnegative Fourier indices. Compute its theta lift; if the lift is almost-holomorphic, the claimed \"only if\" is false.","tokens_in":1874,"feed_emoji":"🧮","tokens_out":5852,"duration_ms":139116,"temperature":0.7,"pith_summary":"This paper builds a complete quasimodular theory for orthogonal groups of signature (2,n), mirroring the classical SL2 story. It proves that the constant-term map from almost-holomorphic modular forms to their holomorphic parts is an isomorphism, so every quasimodular form has a unique almost-holomorphic completion. It also proves a sharp criterion: for lattices splitting a hyperbolic plane, the regularized theta lift of an almost-holomorphic vector-valued form is almost-holomorphic exactly when the weight k is at least twice the depth d. The proof uses a lowering operator that intertwines the orthogonal and SL2 structures, and yields explicit Fourier expansions of theta lifts. Geometrically, the paper conjectures that Enriques and bielliptic Gromov-Witten potentials are orthogonal quasimodular forms satisfying holomorphic anomaly equations.","feed_headline":"Theta-lift depth bound decides almost-holomorphy for O(2,n)","feed_subtitle":"Enriques Gromov-Witten potentials may be orthogonal quasimodular forms, with a complete analogy to the classical SL2 theory.","key_machinery":"The vehicle is a pair of vector-valued lowering and raising operators L and R acting on smooth sections of L^k⊗$E^{{⊗s}}$ over the type IV domain, defined via the invariant Kähler metric and the identification E≅Ω_D⊗L^∨. In the tube-domain model for M=U⊕L, the non-holomorphic variables ν_j=∂/∂z_j log(Im z)^2 measure depth, so almost-holomorphic forms are exactly polynomials in the ν_j and the constant term sets ν=0. The $\\theta$ kernel satisfies an equivariance relation linking these operators with the classical Maaß operators, and the paper uses that relation to prove the depth-weight criterion and to compute the full Fourier expansion of the lift.","core_discovery":"The central theorem is the isomorphism ct: AHMod_{k,s}(Γ) → QMod_{k,s}(Γ), F ↦ F|_{ν=0}, for finite-index Γ in O⁺(M) when M has signature (2,n). Equivalently, every quasimodular form has a unique non-holomorphic completion that is annihilated by a power of the lowering operator L. For the $\\theta$ lift, the paper shows that if F is an almost-holomorphic vector-valued modular form of weight κ=k+1−n/2 and depth d>0, then Lift(F) is a logarithmic almost-holomorphic modular form of weight k and depth 2d when k≥2d; and when M=U⊕L, the lift is almost-holomorphic if and only if k≥2d. The argument hinges on the equivariance identity L_k[Lift(F)] = −(1/2π) R_{k−2}[Lift($L^{{Maaß}}$F)] and on an explicit series expansion of the lift, whose constant term reproduces the Borcherds formula from the constant term of the input.","pith_inferences":["The explicit lift expansion should make it possible to test the Enriques conjecture numerically before a proof is available.","The constant-term isomorphism suggests an sl2-action on the graded ring of orthogonal quasimodular forms, potentially giving a full structure theory analogous to the classical description via the second Eisenstein series.","The lowering and raising calculus may apply to other type IV period domains, where one might find logarithmic quasimodular objects rather than plain quasimodular forms.","A degeneration proof for the bielliptic case would provide the most direct geometric verification of the framework."],"forward_implications":["Every orthogonal quasimodular form has a unique almost-holomorphic completion, so the lowering and raising operators act on quasimodular forms with the same rigidity as in the classical SL2 theory.","For lattices splitting a hyperbolic plane, the theta lift is almost-holomorphic exactly when k≥2d; below that bound logarithmic singularities appear.","Fourier-Jacobi coefficients of orthogonal quasimodular forms are quasi-Jacobi forms, providing a bridge to Jacobi-form modularity.","If the Enriques conjecture holds, the full Gromov-Witten potential of an Enriques surface is a vector-valued logarithmic quasimodular form and the holomorphic anomaly equation is the lowering operator.","The same modularity fails for arbitrary K3 or abelian fibrations: their natural completions are not almost-holomorphic."],"supporting_citations":[{"why":"Supplies the regularized theta integral and the singularity analysis on hyperplanes that underpin the lift theorems.","marker":"[5]"},{"why":"Introduced quasimodular forms for SL2, the classical model that this paper generalizes.","marker":"[20]"},{"why":"Provides the theory of vector-valued orthogonal modular forms, including cusp conditions and the tangent-bundle description of E.","marker":"[26]"},{"why":"Computes the Enriques Hodge integrals whose theta-lift form is the main geometric evidence.","marker":"[31]"},{"why":"Gives the elliptic-fibration modularity and holomorphic anomaly framework used for bielliptic and structural arguments.","marker":"[33]"},{"why":"Provides the Katz-Klemm-Vafa formula for K3 fiber classes that underlies the negative results for K3 fibrations.","marker":"[36]"},{"why":"Sets out nearly-holomorphic modular forms on Hermitian symmetric spaces, the general notion here specialized to type IV.","marker":"[44]"},{"why":"Defines earlier quadratic raising and lowering operators and theta-lift equivariance identities that are recovered and extended by the vector-valued operators.","marker":"[51]"}],"fun_headline_variants":["Isomorphism revealed: nearly-holomorphic meets quasimodular for O(2,n)","Theta lift equivariance yields exact depth bound for almost-holomorphy","Orthogonal quasimodular forms: Enriques potentials may satisfy anomaly equations","Lowering operators classify theta lifts of almost-holomorphic forms","Depth inequality determines holomorphy of theta lifts for signature (2,n)"],"cache_read_input_tokens":83072,"weakest_assumption_plain":"The converse direction of the lift criterion assumes that when k<2d the deepest non-holomorphic layer of the input form has a nonzero Fourier coefficient at a negative index; if that assertion fails, the necessity of the inequality does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Isomorphism revealed: nearly-holomorphic meets quasimodular for O(2,n)","Theta lift equivariance yields exact depth bound for almost-holomorphy","Orthogonal quasimodular forms: Enriques potentials may satisfy anomaly equations","Lowering operators classify theta lifts of almost-holomorphic forms","Depth inequality determines holomorphy of theta lifts for signature (2,n)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000913,"raw_usage":{"total_tokens":3967,"prompt_tokens":1038,"completion_tokens":2929,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":2830}},"tokens_in":654,"tokens_out":2929,"duration_ms":19681,"temperature":1.0,"reasoning_tokens":2830,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:28:52.281789+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a lattice M=U⊕L, construct an almost-holomorphic vector-valued form of weight κ=k+1−n/2 and depth d>k/2 whose depth-d layer has only nonnegative Fourier indices. Compute its theta lift; if the lift is almost-holomorphic, the claimed \"only if\" is false.","supporting_citations":[{"cited_title":"Borcherds","cited_arxiv_id":null,"evidence_quote":"Supplies the regularized theta integral and the singularity analysis on hyperplanes that underpin the lift theorems."},{"cited_title":"A generalized Jacobi theta function and quasimodular forms","cited_arxiv_id":null,"evidence_quote":"Introduced quasimodular forms for SL2, the classical model that this paper generalizes."},{"cited_title":"Vector-valued orthogonal modular forms","cited_arxiv_id":"2209.10135","evidence_quote":"Provides the theory of vector-valued orthogonal modular forms, including cusp conditions and the tangent-bundle description of E."},{"cited_title":"Holomorphic anomaly equations and the Igusa cusp form conjecture","cited_arxiv_id":null,"evidence_quote":"Gives the elliptic-fibration modularity and holomorphic anomaly framework used for bielliptic and structural arguments."},{"cited_title":"Pandharipande and R","cited_arxiv_id":null,"evidence_quote":"Provides the Katz-Klemm-Vafa formula for K3 fiber classes that underlies the negative results for K3 fibrations."},{"cited_title":"Arithmeticity in the theory of automorphic forms , volume 82 of Mathematical Surveys and Mono- graphs","cited_arxiv_id":null,"evidence_quote":"Sets out nearly-holomorphic modular forms on Hermitian symmetric spaces, the general notion here specialized to type IV."},{"cited_title":"Weight changing operators for automorphic forms on Grassmannians and differential properties of certain theta lifts","cited_arxiv_id":null,"evidence_quote":"Defines earlier quadratic raising and lowering operators and theta-lift equivariance identities that are recovered and extended by the vector-valued operators."}],"review_version":1}