{"id":"3a7edba9-c61d-4a38-8d7c-b3a6305aeaa5","arxiv_id":"1908.05583","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"Complete almost scalar-flat Kähler metrics exist on X\\D when K_X^{-l}⊗L_X^m is very ample and m/l is sufficiently small relative to the average scalar curvature of D.","lead":"This paper proves that, under a positivity and smallness condition, the complement of a smooth hypersurface in a compact algebraic manifold admits complete Kähler metrics whose scalar curvature is flat on prescribed compact sets and tiny elsewhere. It is a step toward scalar-flat Kähler metrics on affine algebraic manifolds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.2 hinges on Theorem 1.1's derivative estimates with exponent a(n), but §3.3 never actually derives those estimates; the Schauder step is a black box and condition (1.2) is therefore unverified.","rationale":"The paper's central construction, gluing three Kähler potentials and using the regularized maximum, is plausible, but Theorem 1.2 is only as strong as the quantitative control on φ provided by Theorem 1.1. The manuscript itself signals the fragility: §3.3 is compressed, the Schauder constant s(n) is not specified, and the exponent a(n) is assigned in a remark rather than derived. Since condition (1.2) is an inequality involving this unproved integer, the stated theorem is not yet verified. I do not see an internal contradiction in the overall strategy, so the appropriate status remains conditional rather than a rejection. The reader's weakest assumption identifies the same load-bearing point, and my concern does not move the verdict.","tokens_in":18627,"tokens_out":17571,"duration_ms":174886,"concrete_test":"Re-derive §3.3 for the model equation det(u_{i\\bar j})=|w_F|^{-2/l}|w_D|^{2m/l} in the unit bidisk, tracking the Schauder constant through the difference-quotient argument with balls of radius comparable to min(|w_D|,|w_F|). Verify whether the pointwise third- and fourth-derivative exponents match Theorem 1.1 with a(n)=O(n^2), or acquire extra factors from f_h and ball shrinking. If they differ, recompute condition (1.2) with the corrected a(n) and check whether Theorem 1.2's smallness condition can still be satisfied.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2 is gated by condition (1.2), which contains the integer a(n) from Theorem 1.1. The proof of Theorem 1.1 does not establish the needed quantitative bound. Proposition 3.10 is only a C^{2,ε} estimate on a fixed relatively compact domain Ω in X\\(D∪F), so it cannot control pointwise derivatives as wD,wF→0. The subsequent argument applies a Schauder estimate (Proposition 3.12) to a difference quotient u_h, with a constant C_S=O((Λ/λ)^{s(n)}) and no specification of s(n); the source f_h and the shrinking of admissible balls near D∩F are not tracked. The proof of Theorem 1.1 then states third- and fourth-order bounds with exponent a(n), and Remark 3.14 asserts a(n)=O(n^2) only by 'examining the proof' of [8]. If the true constants give a larger a(n), or if the derivative bounds contain extra powers from the localization, the inequality (1.2) may no longer be sufficient, and Claims 2–4 in Section 4 do not go through. The central existence statement is therefore conditional on an unproved quantitative estimate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies an n-dimensional polarized manifold (X, L_X) with a smooth divisor D in |L_X| and considers the complete Kähler metric ω0 on X \\ D introduced by Bando and Kobayashi. The main result, Theorem 1.2, asserts that if there exist integers l > n and m such that K_X^{-l} ⊗ L_X^m is very ample and a(n)m/(2l) < Ŝ_D/[n(n-1)], then for any relatively compact Y ⋐ X \\ (D ∪ F) there is a complete Kähler metric ω_F on X \\ D whose scalar curvature vanishes on Y and is arbitrarily small on the complement. The proof solves a degenerate complex Monge-Ampère equation with right-hand side |σ_F|^{-2/l}|σ_D|^{2m/l}, estimates higher derivatives of the solution near D ∩ F (Theorem 1.1), and glues three Kähler potentials using the regularized maximum. The central issue is that the quantitative derivative estimates in Theorem 1.1 are not actually derived, and the gluing estimates depend on an unstated lemma from the author's companion paper.","tokens_in":18894,"tokens_out":12356,"duration_ms":111698,"significance":"If the main theorem is correct, it would be a notable contribution to the construction of almost scalar-flat complete Kähler metrics on affine algebraic manifolds, complementing the author's earlier weighted-analysis approach with a gluing method. The paper has a clear strategy and the use of the regularized maximum to combine potentials with controlled scalar curvature is attractive. However, the proof of the quantitative higher-order derivative estimates is not self-contained, and the scalar-curvature estimates in Section 4 rely on an unproved lemma from a companion preprint. The result is therefore conditional on missing technical verification rather than established by the manuscript.","major_comments":[{"comment":"The eigenvalue estimates Λ = O(||σ_F||^{-2/l}) and λ^{-1} = O(||σ_D||^{-2m/l}) are not justified. From (3.1) one obtains the upper bound for the largest eigenvalue, while (3.2) fixes the product of the eigenvalues as |σ_F|^{-2/l}|σ_D|^{2m/l} up to bounded factors. Combining these gives λ ≥ c |σ_D|^{2m/l} |σ_F|^{2(n-2)/l}, hence λ^{-1} = O(|σ_D|^{-2m/l} |σ_F|^{-2(n-2)/l}) for n ≥ 3. The additional |σ_F| factor is absent from the paper's stated estimate, so the ratio Λ/λ is underestimated. This ratio controls the constants in Lemma 3.4 and Proposition 3.10 and propagates into the higher-order estimates of Section 3.3, so the claimed bounds in Theorem 1.1 are not supported.","section":"§3.1, Eq. (3.1)–(3.2)"},{"comment":"The passage from the C^{2,ε} estimate on a fixed domain to the third- and fourth-order bounds is not proved. Proposition 3.12 states the Schauder constant as C_S = O((Λ/λ)^{s(n)}) without specifying s(n) or tracking the dependence of the constant on the distance to the boundary of the balls, which shrink as one approaches D ∩ F. The source term f_h and its C^{0,ε} norm in the difference-quotient equation are not estimated. Proposition 3.13 then asserts the third-order bounds with exponents -4m/l and -4/l, and the fourth-order bounds in Theorem 1.1 are said to follow after 'differentiating the equation' without controlling ˙a^{p,q}_h or ˙f_h. Remark 3.14 asserts a(n) = O(n^2) by referring to the proof of [8], but this is not a derivation. Since condition (1.2) involves a(n), the main theorem is conditional on this unproved quantitative estimate.","section":"§3.3, Propositions 3.12–3.13 and Theorem 1.1"},{"comment":"The scalar-curvature estimates depend essentially on 'Lemma 3.4 in [1]' at several points, in particular the displayed inverse-metric formula in Claim 1 and the trace estimate in Claim 2. This lemma is not stated, proved, or paraphrased in the present paper, and reference [1] is the author's own preprint. The reader therefore cannot check the key step in which the Ricci tensor is controlled after tracing with ω_{c,v,η}. A self-contained proof of Theorem 1.2 must either state and prove the needed lemma or give a precise reference with the full statement.","section":"§4, Claims 1–4"},{"comment":"The comparison that makes the derivatives of φ negligible is not derived. The proof asserts that by taking κ close to 1 'we may assume' an inequality involving ||σ_F|| and ||σ_D||, and that this follows from (1.2); no derivation is provided. The parameter choices (κ, a_i, β, v) are never fully specified, and the final conclusion S(ω_{c,v,η}) = O(c^{-2}) is stated to hold in each of the four regions, with Claim 4 closed by 'similarly'. This leaves a gap in the proof of Theorem 1.2 even assuming Theorem 1.1.","section":"§4, Claim 2 and proof of Theorem 1.2"}],"minor_comments":[{"comment":"The name 'Kołodziej' is corrupted as 'Ko/suppress lodziej' in the text; the references and acknowledgements should be corrected.","section":"Throughout"},{"comment":"The phrase 'scalar curvature is flat' should be replaced by 'scalar curvature is zero' or 'scalar-flat' for clarity.","section":"Abstract and Introduction"},{"comment":"The notation ∂^2/(∂z_i∂z_j∂^α φ) is ambiguous; it should be written as ∂^{2+|α|}φ/(∂z_i∂z_j∂z^α) or similar.","section":"Theorem 1.1"},{"comment":"'The third and the forth order estimates' contains a typo: 'forth' should be 'fourth'.","section":"§3.3 heading"},{"comment":"The displayed matrix for the inverse metric uses notation g^{i,j} without defining it as the inverse of g_{i,j}; the notation should be clarified.","section":"§4, Claim 1"},{"comment":"The statement that 'second and last terms above are zero on F' is not fully demonstrated; a short computation is needed.","section":"Lemma 2.4"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper offers a credible route to complete almost scalar-flat Kähler metrics on X\\D, but the proof as written is conditional on unverified quantitative estimates. I would not treat Theorem 1.2 as proven yet, but I would send it to a referee who can check the estimates.\n\nWhat is genuinely new: the gluing construction via the regularized maximum of three potentials—ω0 near D, a new potential γ_v^β near F, and the degenerate Monge-Ampère solution t+φ—is concrete, and the smallness condition (1.2) is explicit and checkable. The author also correctly identifies that losing asymptotically conical geometry near D∩F is the main remaining obstacle and defers that case to a sequel. That is honest.\n\nWhat the paper does well: the setup with l>n and m such that K_X^{-l}⊗L_X^m is very ample is sensible, and the use of Yau's theorem for degenerate Monge-Ampère plus Kołodziej's L∞ estimate is standard and appropriate. The division of the gluing region into four cases is careful, and the scalar-curvature estimates on each region, assuming Theorem 1.1, are plausible.\n\nWhere it is soft: Theorem 1.1 is the load-bearing piece and it is not actually proved. The text jumps from a C^{2,ε} estimate on a fixed domain to third- and fourth-order bounds with exponents involving a(n). The Schauder step (Proposition 3.12) has constant C_S = O((Λ/λ)^{s(n)}) with s(n) unspecified, and the source f_h together with the shrinking of admissible balls near D∩F is not tracked. Remark 3.14 asserts a(n)=O(n^2) by “examining the proof” of Gilbarg–Trudinger, which is not a derivation. Condition (1.2) depends on the actual integer a(n), not just its order. If a(n) is larger than the claimed size, Claims 2–4 may fail. The dependence on Lemma 3.4 from the companion paper [1] for the inverse metric estimate is also external; the reader cannot verify it from this text.\n\nMinor: the text has OCR-like corruption in names (“Ko/suppress lodziej”), which is cosmetic but should be fixed.\n\nOverall: the strategy is credible and the result would be a useful step, but the proof is incomplete in the place that matters most. I would send it to peer review with a specific request to check the Schauder constants and the derivation of Theorem 1.1. Not desk reject. I would not cite it yet.","headline":"A plausible gluing construction for almost scalar-flat Kähler metrics, but Theorem 1.2 rests on unproved derivative estimates with an unspecified constant a(n); worth a serious referee, not a desk reject.","tokens_in":19382,"tokens_out":2711,"would_cite":false,"duration_ms":25354,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C25","32Q15","53C21"],"pacs":[],"model":"deepseek-v4-flash","headline":"On a polarized manifold cut along an ample divisor, complete Kähler metrics exist with scalar curvature zero on any compact set and arbitrarily small outside, whenever a certain line bundle is very ample and the ratio m/l is small.","keywords":["constant scalar curvature Kähler metrics","complex Monge-Ampère equations","plurisubharmonic functions","complete Kähler metrics","affine algebraic manifolds","scalar curvature","regularized maximum"],"falsifier":"On an explicit polarized pair, take D and F as coordinate hypersurfaces in a toric surface, solve the degenerate Monge-Ampère equation (2.4), and measure the growth of the third and fourth derivatives of $\\phi$ as $w_D,w_F\\to0$; observing any growth faster than the paper's predicted rates $O(|w_D|^{-2a(n)m/l}|w_F|^{-2a(n)/l})$ for third derivatives, and one extra inverse power of the relevant coordinate for fourth derivatives, would disprove Theorem 1.1 and the sufficiency of the inequality $a(n)m/(2l)<\\hat S_D/[n(n-1)]$.","tokens_in":18421,"feed_emoji":"📐","tokens_out":15166,"duration_ms":129347,"temperature":0.7,"pith_summary":"The paper tries to establish that complements of smooth ample hypersurfaces in polarized manifolds carry complete Kähler metrics with nearly vanishing scalar curvature, under a numerical condition on a pair of positive integers. This matters because affine algebraic manifolds are noncompact and standard compact existence theorems do not apply, so the result offers a route to constructing scalar-flat or almost scalar-flat complete metrics on them. The construction solves a degenerate Monge-Ampère equation whose solution has prescribed blow-up near the intersection of two divisors, glues three plurisubharmonic potentials, and then shows that the glued metric has controlled scalar curvature. The main theorem states that the ratio must satisfy $a(n)m/(2l)<\\hat S_D/[n(n-1)]$, where $a(n)$ is a dimension-dependent constant.","feed_headline":"Kähler metric's scalar curvature can be made arbitrarily small","feed_subtitle":"The paper proves a numerical ratio m/l controls the curvature on complements of a divisor.","key_machinery":"The load-bearing machine is the solution $\\phi$ of the degenerate (meromorphic) Monge-Ampère equation $(\\theta_X+\\sqrt{-1}\\partial\\bar\\partial\\phi)^n=\\xi^{-1/l}\\wedge\\overline{\\xi^{-1/l}}$, with $\\xi=\\sigma_F\\otimes\\sigma_D^{-m}$, together with the explicit derivative bounds of Theorem 1.1. Those bounds state that near $D\\cap F$, third derivatives of $\\phi$ are $O(|w_D|^{-2a(n)m/l}|w_F|^{-2a(n)/l})$, and fourth derivatives have one additional inverse power of the normal coordinate in the differentiated direction. These estimates are what allow the Ricci-tensor terms of the glued metric, which contain derivatives of $\\phi$, to be controlled after tracing against the complete metric. The integer $a(n)$ encodes the dimension dependence of the Schauder-estimate constants and enters the main smallness condition; Remark 3.14 only bounds it by $O(n^2)$.","core_discovery":"The central claim is Theorem 1.2: if $(X,L_X)$ is an $n$-dimensional polarized manifold, $D\\in|L_X|$ is smooth, $\\hat S_D>0$, and there exist integers $l>n$ and $m$ such that $K_X^{-l}\\otimes L_X^m$ is very ample and $a(n)m/(2l)<\\hat S_D/[n(n-1)]$, then for any relatively compact domain $Y\\Subset X\\setminus(D\\cup F)$ there is a complete Kähler metric $\\omega_F$ on $X\\setminus D$ whose scalar curvature is zero on $Y$ and arbitrarily small on the complement of $Y$. Moreover $\\omega_F=\\omega_0$ in a neighborhood of $D\\setminus(D\\cap F)$, where $\\omega_0$ is the standard complete conical metric defined from the defining section of $D$. The metric is formed by gluing three strictly plurisubharmonic potentials — the conical potential $\\Theta(t)$ near $D$, the potential $\\tilde G_v^\\beta(b)$ concentrated near $F$, and the Ricci-flat potential $t+\\phi+c$ away from $D\\cup F$ — using the regularized maximum $M_\\eta$; the proof that the glued scalar curvature is small rests on Theorem 1.1, which bounds the third and fourth derivatives of the degenerate Monge-Ampère solution $\\phi$ near $D\\cap F$.","pith_inferences":["The same gluing scheme could plausibly be adapted to produce metrics with prescribed small Ricci curvature, not just small scalar curvature, by matching the Ricci forms of the three potentials; the paper does not pursue this.","The derivative bounds identify $D\\cap F$ as the only place where the glued metric fails to have the conical asymptotics needed for a global scalar-flat theorem, which the author states as the next step.","One testable sharpening would be to compute the optimal exponent $a(n)$ on toric models; if it is smaller than the paper's $O(n^2)$ bound, the condition on $m/l$ could be relaxed."],"forward_implications":["Whenever the hypotheses hold, $X\\setminus D$ admits complete Kähler metrics with scalar curvature smaller than any prescribed $\\varepsilon>0$, not merely on compact sets.","If $K_X^{-1}$ is nef, the numerical hypothesis is automatic, so the theorem supplies nearly scalar-flat complete Kähler metrics on the complement of a smooth divisor in every Fano manifold.","The scalar curvature can be made exactly zero on any fixed compact piece while the metric remains complete and conical near $D$ away from $D\\cap F$.","The explicit dependence of the admissible ratio on the dimension, through $a(n)=O(n^2)$, gives an effective range of $m/l$ for which the conclusion holds."],"supporting_citations":[{"why":"Supplies the existence theorem for the degenerate Monge-Ampère equation on compact Kähler manifolds.","marker":"[14]"},{"why":"Gives the a priori estimate that keeps the solution $\\phi$ bounded on $X$.","marker":"[11]"},{"why":"Provides the $C^2$-estimate controlling the ellipticity of the Kähler metric defined by $\\phi$.","marker":"[12]"},{"why":"Provides the $C^{2,\\varepsilon}$-estimate and Schauder-estimate framework used to prove Theorem 1.1.","marker":"[9]"},{"why":"Supplies the weak Harnack and Schauder estimates whose constants determine the exponent $a(n)$.","marker":"[8]"},{"why":"Constructs the complete conical metric $\\omega_0$ and its scalar curvature decay near $D$.","marker":"[3]"},{"why":"Defines the regularized maximum function used to glue the three Kähler potentials.","marker":"[6]"},{"why":"Sets up the asymptotically conical scalar-flat problem that motivates the comparison with $\\hat S_D$.","marker":"[1]"}],"fun_headline_variants":["Almost scalar-flat Kähler metrics on affine varieties","Kähler metric with arbitrarily small scalar curvature","Complete Kähler metric nearly scalar-flat on divisor complement","Scalar curvature arbitrarily small: a Kähler construction","Kähler metrics with near-zero scalar curvature on affine manifolds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the third- and fourth-order derivatives of the Monge-Ampère solution near the intersection of the two divisors grow no faster than fixed powers of the distances to the divisors, with an exponent a(n) that is at most O($n^{2}$).","fun_headline_variants_meta":{"raw":{"variants":["Almost scalar-flat Kähler metrics on affine varieties","Kähler metric with arbitrarily small scalar curvature","Complete Kähler metric nearly scalar-flat on divisor complement","Scalar curvature arbitrarily small: a Kähler construction","Kähler metrics with near-zero scalar curvature on affine manifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000415,"raw_usage":{"total_tokens":2139,"prompt_tokens":934,"completion_tokens":1205,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":1123}},"tokens_in":550,"tokens_out":1205,"duration_ms":11214,"temperature":1.0,"reasoning_tokens":1123,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:10:44.734298+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On an explicit polarized pair, take D and F as coordinate hypersurfaces in a toric surface, solve the degenerate Monge-Ampère equation (2.4), and measure the growth of the third and fourth derivatives of $\\phi$ as $w_D,w_F\\to0$; observing any growth faster than the paper's predicted rates $O(|w_D|^{-2a(n)m/l}|w_F|^{-2a(n)/l})$ for third derivatives, and one extra inverse power of the relevant coordinate for fourth derivatives, would disprove Theorem 1.1 and the sufficiency of the inequality $a(n)m/(2l)<\\hat S_D/[n(n-1)]$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the existence theorem for the degenerate Monge-Ampère equation on compact Kähler manifolds."},{"cited_title":"Ko/suppress lodziej, The complex Monge-Amp` ere equation, Acta Math","cited_arxiv_id":null,"evidence_quote":"Gives the a priori estimate that keeps the solution $\\phi$ bounded on $X$."},{"cited_title":"Pˇ aun, Regularity properties of the degenerate Monge-Amp` ere equations on com- pact K¨ ahler manifolds, Chin","cited_arxiv_id":null,"evidence_quote":"Provides the $C^2$-estimate controlling the ellipticity of the Kähler metric defined by $\\phi$."},{"cited_title":"Guedj and A","cited_arxiv_id":null,"evidence_quote":"Provides the $C^{2,\\varepsilon}$-estimate and Schauder-estimate framework used to prove Theorem 1.1."},{"cited_title":"Gilbarg and N","cited_arxiv_id":null,"evidence_quote":"Supplies the weak Harnack and Schauder estimates whose constants determine the exponent $a(n)$."},{"cited_title":"Bando and R","cited_arxiv_id":null,"evidence_quote":"Constructs the complete conical metric $\\omega_0$ and its scalar curvature decay near $D$."},{"cited_title":"Demailly","cited_arxiv_id":null,"evidence_quote":"Defines the regularized maximum function used to glue the three Kähler potentials."},{"cited_title":"Toward a construction of scalar-flat K\\\"{a}hler metrics on affine algebraic manifolds","cited_arxiv_id":"1907.09780","evidence_quote":"Sets up the asymptotically conical scalar-flat problem that motivates the comparison with $\\hat S_D$."}],"review_version":1}