{"id":"eff92bf1-15d8-4ff7-ae2f-bf67425e365b","arxiv_id":"2505.03731","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Holomorphic mapping functors to projective space are polynomial in the Weiss calculus, with computable derivative spectra for continuous maps and a new proof of the Cohen-Cohen-Mann-Milgram stable splitting.","lead":"This mathematics paper studies the stable homotopy type of spaces of holomorphic maps into complex projective space, viewing them as functors of a vector space and analyzing their Weiss towers. It proves these functors are polynomial, computes the full Weiss tower for continuous maps, and derives a new proof of a known stable splitting for rational maps.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.10, the Picard-family proof of N-polynomiality, is explicitly only sketched; in particular the relative evaluation map must be an embedding over every point of Pic^α(X), which the stated hypotheses do not prove. This is the load-bearing step for Theorem A.","rationale":"The reader's weakest_assumption was the self-cited connectivity range from [Aum24]. That is a real external dependency, but it is less exposed than the internal gap in Theorem 3.10: the connectivity bound is stated consistently and a homology-range statement is sufficient for Proposition 2.3, since vanishing of the integral homology of the cofibre below a range forces its stable homotopy groups to vanish there by the Atiyah–Hirzebruch spectral sequence. The more immediate problem is that the proof of the polynomiality theorem itself is explicitly only a sketch once the Picard parameter enters. The text says so in Section 3.3.2, and the missing relative-embedding statement is a concrete point where the reduction from Hol_α to a family of linear maps could fail. Because Theorem A's first assertion is N-polynomiality and the derivative formulas depend on it, this is a load-bearing concern. It does not, however, move the verdict: the gap is fillable in principle, and the reader's CONDITIONAL verdict already reflects this level of uncertainty. I therefore recommend leaving the verdict unchanged.","tokens_in":30425,"tokens_out":24173,"duration_ms":250402,"concrete_test":"Write out the missing portion of Theorem 3.10 in full. A decisive first check is to verify that the relative evaluation map ϵ: Pic^α(X)×X → P((p_*P)^∨) is an embedding for every [L]∈Pic^α(X) under the stated assumptions. Concretely, take a smooth projective X with H^1(X,O_X)≠0 and a class α satisfying the hypotheses, and compute the base locus of |L_0⊗η| for every η in the connected component Pic^0(X); if some such line bundle is not very ample, Theorem 3.10 as stated is false, while if it is always very ample, the remaining parameterized versions of Lemmas 3.4–3.6 still need to be supplied before the N-polynomiality conclusion is established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is the N-polynomiality of V ↦ Σ^∞_+ Hol_α(X, P(C^{n+1}⊕V)). That is proved as Theorem 3.10, not as a formal consequence of the fixed-line-bundle case. But Section 3.3.2 is not a complete proof: after introducing the Poincaré bundle P over Pic^α(X), the text says 'we will be content with only indicating the most salient modifications.' The rank-stratification induction of §3.2 is asserted to adapt, but the parameterized versions of Lemmas 3.3–3.7 are never stated, and the homotopy pushout squares for the strata of Ψ(V) are not verified over the non-discrete base Pic^α(X). A concrete hidden assumption is that the relative evaluation map ϵ: Pic^α(X)×X → P((p_*P)^∨) is an embedding over every point [L]∈Pic^α(X), so that Hol_α is identified with the family of linear maps whose projectivized kernels avoid ϵ({[L]}×X). The hypotheses only supply one very ample line bundle L with c1(L)=α; nothing in the text proves that every line bundle in Pic^α(X) is very ample, or that the family evaluation map is an embedding. If some [L] in the same component is not very ample, this identification and the subsequent stratification fail. Since Theorem A's explicit derivative formulas are deduced only after Theorem 3.10, this gap is load-bearing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a unitary- (Weiss-) calculus approach to the stable homotopy type of spaces of holomorphic maps to projective space. For a connected smooth projective complex variety X and a class α admitting a very ample line bundle L with c1(L)=α and c1(L⊗K_X^∨) ample, Theorem A asserts that the functor V ↦ Σ^∞_+ Hol_α(X, P(C^{n+1}⊕V)) is N-polynomial with N = dim H^0(X,L), and that for 1 ≤ k ≤ d(α) the Weiss derivatives are given by explicit Thom spectra over labelled configuration spaces. Theorem B computes the complete Weiss tower for the analogous continuous mapping space. The paper also gives a new proof of the Cohen–Cohen–Mann–Milgram stable splitting for rational maps and computes several examples, including a formula for the top derivative.","tokens_in":30722,"tokens_out":18343,"duration_ms":201744,"significance":"If the proof gaps are closed, this is a valuable and genuinely new organizing framework for the unstable part of the homology of holomorphic mapping spaces. The continuous case (Theorem 4.16) is proved in detail through Malkiewich's and Goodwillie's calculi, and the passage from Goodwillie to Weiss towers is carefully set out. The explicit derivative formulas in Theorem A are concrete, parameter-free, and checkable, and the paper is honest in indicating where it sketches rather than proves. The potential pay-off is high: the unstable range is reframed as a finite extension problem in a Weiss tower.","major_comments":[{"comment":"The N-polynomiality of V ↦ Σ^∞_+ Hol_α(X, P(C^{n+1}⊕V)) is the first half of Theorem A, but the proof of Theorem 3.10 is explicitly a sketch: after introducing Ψ(V), the text says 'we will be content with only indicating the most salient modifications.' The parameterized versions of Lemmas 3.4–3.6 are never stated, and the homotopy pushout squares S(ν_r(V)) → D(ν_r(V)) ≃ Ψ_r(V), Ψ_{<r}(V) → Ψ_{≤r}(V) are not verified over the non-discrete base Pic^α(X). In particular, the identification of Hol_α(X,P(C^{n+1}⊕V)) with the C^×-quotient of Ψ(V) relies on the relative evaluation map ϵ: Pic^α(X)×X → P((p_*P)^∨) being an embedding over every [L]; the hypotheses only provide one very ample representative L in the class α, and very ampleness is not a numerical property, so this is not established. Since the explicit formulas for Θ_k in Theorem A are deduced only after Theorem 3.10, this gap is load-bearing and must be repaired.","section":"Section 3.3.2, proof of Theorem 3.10"},{"comment":"The formula for the top derivative Θ_N is advertised in the introduction, but its proof contains two unproved identifications. The sentence 'By inspection of our constructions, the morphisms between the derivatives are all induced by the natural projections and inclusions' identifies the maps in the Čech diagram (16), and the subsequent 'observe that this colimit is exactly the geometric realisation of the poset Gr(X,L)' identifies the colimit with |Gr(X,L)|; neither is demonstrated. The final passage from the section-space functor (14) to the holomorphic-map functor (20) is also left to the reader. Please provide complete arguments or explicitly mark these statements as conditional.","section":"Section 6.3, Proposition 6.8"},{"comment":"The proof of Lemma 5.1 states that 'the argument in the proof of Theorem 3.2 will show' d-polynomiality of the pointed rational-map functor, but the rank-filtration argument is not carried out for the Veronese embedding ν_d: P^1 → P^d. The pointed condition changes the linear-algebra model, so this is not a purely formal consequence of Theorem 3.2. Since Theorem 5.3 and its alternative proof via Theorem 5.6 rely on Lemma 5.1, this lemma needs a complete proof.","section":"Section 5, proof of Lemma 5.1"}],"minor_comments":[{"comment":"The phrase 'for 1≤1≤d(α)' should read 'for 1≤k≤d(α)'.","section":"Introduction, after Theorem A"},{"comment":"The text contains a duplicated word: 'we are are able' should read 'we are able'.","section":"Introduction"},{"comment":"The notation colim_{[K]∈P^3−Q} is applied to terms that do not depend on [K] in two corners of the diagram; please clarify whether these are constant diagrams and how the pushout maps are induced.","section":"Section 6.2, equation (12)"},{"comment":"The paper relies on [Aum24] for the connectivity range that identifies holomorphic and continuous derivatives, but the precise theorem used is not stated. Please state the connectivity result in the form needed here, since the equality Θ_k ≃ eΘ_k for k ≤ d(α) is a direct consequence of it.","section":"Section 4.7"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the dependence on the author's prior paper [Aum24] for the connectivity range; any error there propagates into Theorem A's explicit derivative formulas. The more immediate issue is that Theorem 3.10 is explicitly a sketch, and it is the core polynomiality claim. Proposition 6.8, while not needed for Theorem A's main statement, is advertised as a result and is also proved only by inspection. The manuscript fits the journal's scope, but the advertised central theorem will not be established until these proofs are completed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is one of the few papers that actually uses Weiss calculus to say something about holomorphic mapping spaces, and the continuous-side Theorem B looks like a complete, well-documented result. Second, the proof of Theorem A has a real gap: the Picard-parameter step in Theorem 3.10 is explicitly a sketch, and the stress-test concern lands. The hypotheses give one very ample L, not very ampleness of every line bundle in Pic^α(X). Nothing in the text proves the relative evaluation map is an embedding over every point of the Picard variety, and if some [L] is not very ample, the identification with Ψ(V) and the subsequent stratification do not follow. That is load-bearing for the polynomiality claim, not a cosmetic omission.\n\nWhat is genuinely good: the continuous-case Theorem B is proven carefully via Malkiewich's tower and Goodwillie calculus, with an explicit formula for all layers. The rank-stratification idea is attractive and yields concrete computations in Section 6. The new proof of the CCMM splitting is also a nice application, and it does not depend on the problematic part of Theorem 3.10. The paper is honest about its own sketches, which is to its credit, and the self-citation [Aum24] is an external connectivity input, not a circular restatement.\n\nSoft spots in proportion: Theorem 3.10 is the main one, and it is serious. The proof says 'we will be content with only indicating the most salient modifications' and then handwaves the key point that the family evaluation map is an embedding. The author should either prove that, or restrict the statement to the fixed-line-bundle case (Theorem 3.2), or add an extra hypothesis like every L in Pic^α(X) being very ample. The second soft spot is Proposition 6.8, where the colimit identification is justified 'by inspection'; that is not a proof, though it is a less central part of the paper.\n\nWho this is for: homotopy theorists working on calculus and spaces of holomorphic maps. The paper deserves a serious referee: the continuous-side theorem is likely correct and useful, and the framework is promising. But a referee should press hard on Theorem 3.10 before taking Theorem A as established.","headline":"A promising framework and a solid continuous-side theorem, but the main holomorphic polynomiality claim rests on a sketch that does not justify the Picard-family embedding.","tokens_in":31259,"tokens_out":6157,"would_cite":true,"duration_ms":61528,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P65","55R80"],"pacs":[],"model":"deepseek-v4-flash","headline":"A finite Weiss tower, whose early layers are explicit configuration spectra, governs the stable homotopy type of holomorphic maps to projective space.","keywords":["Weiss calculus","unitary calculus","holomorphic maps","projective space","Weiss tower","configuration spaces","stable splitting","rational maps"],"falsifier":"Evaluate both sides of the identification $\\Theta_k\\simeq \\tilde{\\Theta}_k$ for $k=d(\\alpha)$ on a known variety, such as $X=\\mathbb{P}^1$ with degree $d$, and compare their integral homology in degrees below $2(d(\\alpha)+1)\\dim V+d(\\alpha)-2$; the smallest degree where the two spectra differ must lie above that bound, so any disagreement below it would disprove the identification.","tokens_in":30200,"feed_emoji":"🧮","tokens_out":9874,"duration_ms":88315,"temperature":0.7,"pith_summary":"The paper claims that the stable homotopy type of the space of holomorphic maps from a smooth projective variety $X$ to projective space is governed by a finite Weiss tower in unitary calculus. Concretely, it proves that the functor $V \\mapsto \\Sigma^\\infty_+ \\mathrm{Hol}_\\alpha(X, \\mathbb{P}(\\mathbb{C}^{n+1}\\oplus V))$ is $N$-polynomial, with $N=\\dim H^0(X,L)$ for a very ample line bundle with $c_1(L)=\\alpha$, and it computes the first $d(\\alpha)$ layers of the tower explicitly as spectra made from configuration spaces of points on $X$. It also computes the complete Weiss tower for the continuous mapping-space functor. If the paper is right, the unstable homology of holomorphic mapping spaces is reduced to finitely many explicitly known spectra and extension problems between them.","feed_headline":"Holomorphic map spaces get a computable Weiss tower","feed_subtitle":"A finite tower of configuration spectra governs their unstable topology, and reproves a classical stable splitting.","key_machinery":"The load-bearing object is the Weiss tower of a unitary functor $F\\colon \\mathcal{J}\\to \\mathrm{Sp}$, whose $k$th layer is a homotopy quotient $(\\Theta_kF\\otimes S^{\\mathbb{C}^k\\otimes V})_{hU(k)}$. For holomorphic sections, the argument identifies sections of $L\\otimes V$ with linear maps $H^0(X,L)^\\vee\\to V$ and stratifies by the dimension of the kernel, requiring the projectivised kernel to miss the embedded variety $X$. Each stratum is shown polynomial using a stable splitting of Stiefel manifolds; the same stratification, with the Picard variety as parameter, gives polynomiality of holomorphic maps. For continuous sections, the restriction $V\\mapsto S(L\\otimes(\\mathbb{C}^M\\oplus V))$ turns the section-space tower into the Weiss tower, producing the explicit layer spectra. A homology connectivity theorem then identifies the holomorphic and continuous layers for all $k\\le d(\\alpha)$.","core_discovery":"The central discovery, stated on the paper's own terms, is that the holomorphic unitary functor is $N$-polynomial and its Weiss tower has layers $(\\Theta_k \\otimes S^{\\mathbb{C}^k\\otimes V})_{hU(k)} \\to T_k(V)\\to T_{k-1}(V)$, with $\\Theta_k$ for $1\\le k\\le d(\\alpha)$ given by $[\\mathrm{Ind}_{U(1)\\wr\\Sigma_k}^{U(k)} \\Sigma^{k(2n+1)} \\mathrm{Conf}_k(X,S(L))^{-kT_X}] \\times \\mathrm{Map}(X,U(1))$. For continuous maps the analogous tower converges and the same layer formula holds for every $k\\ge 1$. The mechanism is a rank filtration on the linear maps representing holomorphic sections, a stable splitting of Stiefel manifolds, and a comparison between the section-space tower and Weiss's tower; the two towers agree in the range where holomorphic and continuous mapping spaces have the same homology.","pith_inferences":["The explicit bottom layers suggest a practical route to unstable homology: compute a spectral sequence from the layers $\\Theta_1,\\dots,\\Theta_N$ and a few extension problems; the paper does not carry this out, and it is a natural next step.","For $X=\\mathbb{P}^n$ with $n\\ge 2$, $N=\\binom{n+d}{d}$ grows much faster than $d(\\alpha)=d$, so the first uncomputed layer $\\Theta_{d(\\alpha)+1}$ is likely where most of the unstable complexity concentrates; computing it is a concrete open problem suggested by the paper.","The same rank-filtration mechanism should extend to holomorphic maps into other homogeneous varieties, such as Grassmannians, where kernel and span conditions define analogous strata and the same pushout arguments apply.","The appearance of the geometric realisation of the poset of projective subspaces avoiding $X$ in the top derivative points to a direct connection between unstable holomorphic-map topology and the topology of spaces of linear subspaces disjoint from a subvariety, which could be studied independently."],"forward_implications":["For $1\\le k\\le d(\\alpha)$, the unstable homology of $\\mathrm{Hol}_\\alpha(X,\\mathbb{P}(\\mathbb{C}^{n+1}\\oplus V))$ is governed by the explicit configuration spectra $\\mathrm{Conf}_k(X,S(L))^{-kT_X}$ with $U(1)\\wr\\Sigma_k$ and $\\mathrm{Map}(X,U(1))$ actions.","The continuous functor $V\\mapsto \\Sigma^\\infty_+\\mathrm{Map}_\\alpha(X,\\mathbb{P}(\\mathbb{C}^M\\oplus V))$ has a converging Weiss tower, so its stable homotopy type is completely determined by the same layer formula for every $k\\ge 1$.","The top derivative $\\Theta_N$ is expressed as $\\mathrm{ad}_N\\otimes \\Sigma^\\infty_+PU(N)\\otimes \\Sigma^\\infty\\Sigma^{\\mathrm{un}}|\\mathrm{Gr}(X,L)|$, giving the first general description of the top layer.","For pointed degree-$d$ maps $\\mathbb{P}^1\\to \\mathbb{P}(\\mathbb{C}^2\\oplus V)$, the functor is $d$-polynomial and its Weiss tower splits, which yields a new proof of the classical stable splitting of spaces of rational maps.","The polynomiality degree $N=\\dim H^0(X,L)$ is independent of the chosen very ample $L$ under the ampleness hypothesis."],"supporting_citations":[{"why":"Introduces orthogonal/Weiss calculus, the tower and derivatives used throughout.","marker":"[Wei95]"},{"why":"Supplies the homology connectivity bound identifying holomorphic and continuous layers up to d(α).","marker":"[Aum24]"},{"why":"Provides the stable splitting of Stiefel manifolds used to prove polynomiality of rank strata.","marker":"[Mil85]"},{"why":"Builds the section-space tower for continuous maps that is converted into the Weiss tower.","marker":"[Mal15]"},{"why":"Gives the classical connectivity result for rational maps that motivates and calibrates the stable range.","marker":"[Seg79]"},{"why":"The target stable splitting of rational-map spaces reproved in Section 5.","marker":"[CCMM91]"},{"why":"Explains how to pass from homotopy-functor calculus to orthogonal/Weiss calculus.","marker":"[BE16]"},{"why":"Connects Miller's splitting to homogeneous derivatives, giving the explicit derivative formulas.","marker":"[Aro00]"}],"fun_headline_variants":["A computable Weiss tower for holomorphic maps","Weiss tower for holomorphic maps fully computed","Explicit Weiss layers for holomorphic maps","Weiss towers reprove a classical stable splitting","Polynomial Weiss tower for holomorphic map spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The explicit formulas for the holomorphic layers depend on a connectivity bound, quoted from the author's earlier paper, claiming that holomorphic and continuous maps agree in homology below a stated degree range.","fun_headline_variants_meta":{"raw":{"variants":["A computable Weiss tower for holomorphic maps","Weiss tower for holomorphic maps fully computed","Explicit Weiss layers for holomorphic maps","Weiss towers reprove a classical stable splitting","Polynomial Weiss tower for holomorphic map spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001525,"raw_usage":{"total_tokens":6034,"prompt_tokens":801,"completion_tokens":5233,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":417,"completion_tokens_details":{"reasoning_tokens":5163}},"tokens_in":417,"tokens_out":5233,"duration_ms":34442,"temperature":1.0,"reasoning_tokens":5163,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:44:06.481065+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate both sides of the identification $\\Theta_k\\simeq \\tilde{\\Theta}_k$ for $k=d(\\alpha)$ on a known variety, such as $X=\\mathbb{P}^1$ with degree $d$, and compare their integral homology in degrees below $2(d(\\alpha)+1)\\dim V+d(\\alpha)-2$; the smallest degree where the two spectra differ must lie above that bound, so any disagreement below it would disprove the identification.","supporting_citations":[],"review_version":1}