{"id":"3163d383-5672-428d-8896-03092a1e0fae","arxiv_id":"2608.03572","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A sufficiently positive Lefschetz pencil on any smooth projective variety has a base-locus blowup that admits cscK metrics in an explicit adiabatic polarization.","lead":"This paper proves that every complex projective variety can be blown up to a smooth manifold carrying a constant scalar curvature Kähler metric, even when the original variety is unstable. It settles a long-open point-blowup conjecture for projective surfaces and gives explicit polarizations for the resulting cscK metrics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lefschetz fiber lct computation in Prop. 2.8 is wrong for n≥3; B_f≠0, so the Hattori-adiabatic entropy bound in Lemma 4.8 is not justified as written.","rationale":"I checked the slope-localization part (Proposition 3.1 and the inequality in its proof) and the algebra in Proposition 4.6; those appear consistent. The serious problem is the explicit false lct computation in Proposition 2.8, which is used to set B_f=0. The reader's concern about Hattori's theorem is well placed, and this is the precise spot where the manuscript's application is not justified. The final theorem may still be true by a corrected argument, so the verdict should remain conditional pending a fix.","tokens_in":1062,"tokens_out":940,"duration_ms":239516,"concrete_test":"Check the lct of the local model z_1^2+...+z_n^2=0 in C^n for n=3 by the standard resolution: blowing up the origin, the exceptional divisor E has discrepancy n-1-2c; log canonicity requires n-1-2c ≥ -1, giving c≤n/2, so lct=3/2 rather than 1. If the citation [Mus12, Example 1.8] instead supports n/2, this confirms the error in Proposition 2.8. Then inspect the hypotheses of [Hat25, Theorem 4.12] to determine whether B_f is required to be an effective divisor; if it is, Lemma 4.8 is invalid for dim Y≥3, and a modified proof must show lim δ = 2 despite B_f having negative coefficients.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The entropy half of the proof rests on Lemma 4.8, which concludes δ(Y,L_{N,ε})>1 for small ε by applying Theorem 2.9 with B_f=0. The vanishing B_f=0 is derived from Proposition 2.8, where it is claimed that every Lefschetz fiber Y_t satisfies lct(Y,Y_t)=1 because lct_0(z_1^2+...+z_n^2)=min(1,n/2)=1. This lct statement is false for n≥3: for the ordinary double point model Q={z_1^2+...+z_n^2=0}⊂C^n, blowing up the origin gives an exceptional divisor whose coefficient in the discrepancy formula is n-1-2c, so the pair (C^n,cQ) is log canonical exactly for c≤n/2. Thus lct=n/2 (e.g., 3/2 for n=3, 2 for n=4), not 1. Consequently, singular fibers of the Lefschetz fibration have lct>1, the discriminant boundary B_f is nonzero, and it has negative coefficients at the singular fibers. Corollary 2.10 and Lemma 4.8 therefore do not follow as written. The limiting value 2 might still hold because smooth fibers have lct=1, but only if Hattori's theorem remains valid for sub-pairs with non-effective B_f; the paper does not establish this, and standard log-delta definitions usually require an effective boundary. Since δ>1 is one of the two sufficient conditions for coercivity in Proposition 4.4, this gap is load-bearing for the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper claims that every complex projective variety is birational to a smooth projective variety admitting a constant scalar curvature Kähler (cscK) metric. For a smooth projective n-fold X and a sufficiently positive very ample line bundle H, the paper considers a general Lefschetz pencil in |H| and the blowup Y of its codimension-two base locus. The resulting fibration f:Y→P^1 gives an adiabatic class L_N,ε = F + εΘ, where F is the fiber class and Θ = c1(K_{Y/P^1}) is shown to be Kähler. The main theorem asserts that Y admits a cscK metric in L_N,ε for all sufficiently small positive rational ε. The proof splits the Mabuchi functional into energy and entropy thresholds: the energy threshold is computed exactly by a slope computation (Proposition 4.6), while the entropy threshold is controlled by Hattori's adiabatic limit for the delta invariant (Lemma 4.8). The paper also derives a surface point-blowup bound, a uniform Fano statement, and an explicit projective-space example.","tokens_in":18401,"tokens_out":19764,"duration_ms":176161,"significance":"If the proof is fully correct, the result is highly significant: it would show that cscK existence is birationally unobstructed and would solve the folklore surface point-blowup conjecture with an effective bound. The construction is explicit, and the exact slope computation in Proposition 4.6 is a real asset; the projective-space example provides a concrete and verifiable computation. The use of Hattori's adiabatic theorem is natural, and the paper does not appear to be circular: the delta invariant is quoted from independent external results rather than assumed. The main fragility is the entropy half of the proof, which depends on a log-canonical threshold computation and on the precise hypotheses of Hattori's theorem.","major_comments":[{"comment":"The assertion that every fiber Y_t satisfies lct(Y,Y_t)=1 is false for n≥3. In the local model {z_1^2+...+z_n^2=0}⊂C^n, the log canonical threshold at the origin is n/2, not min{1,n/2}=1. Consequently the discriminant boundary B_f in §2.4 has negative coefficients at the singular fibers when n≥3, and Corollary 2.10's conclusion B_f=0 is not established. Since Lemma 4.8 derives δ(Y,L_N,ε)>1 from Corollary 2.10, the entropy estimate is not justified as written. The limiting value 2 may still be correct because smooth fibers have lct=1, but the proof must be corrected.","section":"§2.3, Proposition 2.8"},{"comment":"The application of Hattori's Theorem 2.9 requires a log delta invariant δ_{(C,B_f)}(L_C) for the base pair. When n≥3, B_f is a non-effective sub-boundary with b_p=1-n/2<0 at the singular fibers, whereas Section 2.4 defines δ_{(Y,B)} only for klt pairs with an effective boundary. The manuscript does not state whether the quoted theorem covers sub-pairs with negative coefficients. This is load-bearing: δ(Y,L)>1 is one of the two sufficient conditions in Proposition 4.4. Please either quote a version of Hattori's theorem that explicitly allows non-effective B_f, or supply a direct proof of the needed convergence for the Lefschetz fibration.","section":"§2.4 and §4.5, Theorem 2.9 / Lemma 4.8"}],"minor_comments":[{"comment":"The word 'For exmaple' should be corrected to 'For example'.","section":"§1.1(c)"},{"comment":"After Proposition 4.6, the chain Γ_pp_{K_Y+L}(L)>0 and hence Γ_pp_{K_Y}(L)>-1 is correct, but it would be clearer to explicitly state that this uses Lemma 2.4 with a=1.","section":"§4.6, proof of Theorem 4.1"},{"comment":"The displayed slope formula would be less ambiguous with parentheses: E=(1+(n+1)^2 ε)/(n(n+2)+(n+1)^2 ε).","section":"§5.3"},{"comment":"The definition of the discriminant boundary B_f would benefit from a sentence clarifying that scheme-theoretic fibers f^*p are used throughout, since the log canonical threshold is computed on the scheme-theoretic fiber.","section":"§2.4"},{"comment":"The entries [JSD26] and [SSD26] are listed as 'in preparation'; if they are needed for the surface discussion, the dependence should be stated explicitly.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The lct error in Proposition 2.8 is clear-cut, but it is local and the numerical limit 2 in Lemma 4.8 likely survives because smooth fibers have lct=1. The decisive point to check is whether Hattori's Theorem 4.12 as quoted genuinely covers non-effective discriminant boundaries; if it does, the main theorem is probably correct and the paper will be a major contribution. The reliance on several 'in preparation' references is a secondary concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Dyrefelt paper. The headline is exactly what it claims: for every smooth projective X, blowing up the base locus of a sufficiently positive Lefschetz pencil yields a cscK metric in an adiabatic class. So every projective variety is birational to a cscK model, and the surface point-blowup conjecture follows. That is a major result.\n\nThe proof is short and mostly clean. The energy side is an exact slope computation: the nef threshold class is a multiple of the fiber class, so Proposition 3.1 localizes the destabilizers to vertical divisors, and Proposition 4.6 gives E_{K_Y+L}(L)>0. I checked the arithmetic; it works. The entropy side uses Hattori's adiabatic theorem, and the key input is that every Lefschetz fiber has lct=1. I also checked the local model: for f=z_1^2+...+z_n^2, the log canonical threshold of the pair (C^n,c{f=0}) is min(1,n/2)=1, because the strict transform's log discrepancy is 1-c. The stress-test note claiming lct=n/2 is wrong; it only looked at the exceptional divisor after blowing up the origin, ignoring the strict transform. So B_f=0 holds and Corollary 2.10 is fine.\n\nSoft spots are minor. The Section 5.3 display for the projective space slope is missing a parenthesis: it should be (1+(n+1)^2ε)/(n(n+2)+(n+1)^2ε), not 1+(n+1)^2ε/(...). Cosmetic. The proof leans on two big external theorems, [Che21] and [Hat25], as black boxes; a referee should verify the hypotheses, but the applications look right. The adiabatic parameter ε is non-effective, and the paper says so. There is some reliance on an in-preparation paper [JSD26], but only for remarks, not the main line.\n\nVerdict: this is a serious paper. I'd send it to a strong referee. The central argument holds as written (modulo the typo), and the theorem is important. For the reading group: yes.","headline":"Major, correct theorem: every projective variety is birational to a cscK model, with a clean proof; the stress-test's lct concern is a false alarm.","tokens_in":18947,"tokens_out":19892,"would_cite":true,"duration_ms":164593,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","14E05","32Q26","14J45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every complex projective variety is birational to a smooth projective variety carrying a constant scalar curvature Kähler metric, obtained by a Lefschetz blowup.","keywords":["cscK metric","constant scalar curvature Kähler metric","Lefschetz pencil","birational model","blowup","K-stability","delta invariant","Mabuchi functional"],"falsifier":"On the explicit $\\mathbb P^2$ example, the cscK classes are $(4+5\\epsilon)h-(1+\\epsilon)\\sum_{i=1}^{16}E_i$; computing the delta invariant for a decreasing sequence of small rational $\\epsilon$ and checking whether it stays above $1$ and limits to $2$ would test the entropy estimate directly. A second check is to inspect any singular fiber of a general Lefschetz pencil: if some fiber fails to have log canonical threshold one, the proof's identification of the base curve threshold is no longer valid.","tokens_in":17849,"feed_emoji":"📐","tokens_out":9436,"duration_ms":81469,"temperature":0.7,"pith_summary":"The paper proves that cscK metrics—Kähler metrics whose scalar curvature is constant—exist on a birational model of every complex projective variety, with no stability or metric assumption on the original variety. The model is explicit: choose a very ample line bundle $H$ with $K_X+H$ ample, take a general Lefschetz pencil, and blow up its smooth codimension-two base locus. On the resulting manifold $Y$, the polarization $\\Omega_\\epsilon=F+\\epsilon\\Theta$, with fiber class $F$ and relative canonical class $\\Theta$, is Kähler and carries cscK metrics for all sufficiently small rational $\\epsilon>0$. Because $H$ can be any sufficiently large multiple of an ample class, every smooth projective $X$ fits the construction, and resolution of singularities extends the conclusion to all complex projective varieties. The novelties are that the cscK class is given explicitly and that the proof does not propagate an existing cscK metric, so unstable varieties are allowed.","feed_headline":"One blowup gives every projective variety a cscK model","feed_subtitle":"Blowing up the base locus of a general Lefschetz pencil yields explicit metrics without any stability assumption.","key_machinery":"The load-bearing object is the Lefschetz pencil package: after blowing up the smooth codimension-two base locus $Z$ of a general pencil in $|H|$, the blowup $Y$ comes with a fibration $f:Y\\to\\mathbb P^1$ whose fiber class $F$ obeys $F=\\pi^*H-E$, and the relative canonical class $\\Theta=c_1(K_{Y/\\mathbb P^1})=K_Y+2F$ is Kähler because it restricts from a Kähler class on $X\\times\\mathbb P^1$. The polarization is $L_{N,\\epsilon}=F+\\epsilon\\Theta$. Its energy side is computed exactly: the nef threshold class is a pullback of a positive multiple of the fiber class, so the slope threshold is localized on vertical divisors and evaluates to a positive explicit quotient; positivity of that quotient is exactly the energy coercivity criterion. The entropy side is controlled by sending $\\epsilon\\to0$, when the delta invariant of $(Y,L_{N,\\epsilon})$ converges to the delta invariant of $(\\mathbb P^1,\\mathcal O(1))$, namely $2$, because every Lefschetz fiber has log canonical threshold one. Adding the two lower bounds makes the Mabuchi functional coercive, which by the existence theorem yields the cscK metric.","core_discovery":"On any smooth complex projective variety $X$ of dimension $n\\ge2$, take a very ample line bundle $H$ with $K_X+H$ ample and a general Lefschetz pencil $|W|\\subset|H|$. Let $\\pi:Y=\\mathrm{Bl}_Z X\\to X$ be the blowup of the smooth codimension-two base locus $Z$, with $f:Y\\to\\mathbb P^1$ the associated Lefschetz fibration. The paper proves that the relative canonical class $\\Theta=c_1(K_{Y/\\mathbb P^1})$ is Kähler, and that $Y$ carries constant scalar curvature Kähler metrics in the classes $\\Omega_\\epsilon=F+\\epsilon\\Theta$ for all sufficiently small rational $\\epsilon>0$, where $F=f^*c_1(\\mathcal O_{\\mathbb P^1}(1))$. Since $H$ can be taken as a sufficiently large multiple of any ample line bundle, every smooth projective variety admits such a birational cscK model; passing through a resolution of singularities extends the statement to every complex projective variety. The proof shows the Mabuchi energy is coercive, not merely modulo automorphisms.","pith_inferences":["Inference: the same two-threshold strategy should work for any fibration over a curve whose fibers all have log canonical threshold one and whose nef threshold class is pulled back from the base; Lefschetz pencils are one convenient source, not the only one.","Inference: in the projective-space example the cscK classes approach the fiber class boundary as $\\epsilon\\to0$; this suggests a collapse or adiabatic limit picture that could be studied independently.","Inference: an effective version of the adiabatic convergence theorem would turn the theorem into an explicit construction, and on surfaces would sharpen the coarse bound $\\mathfrak b_{\\mathrm{cscK}}(X)\\le16A^2$ toward the exact minimal number of blowup points.","Inference: because the construction applies to Fano K-moduli components with the CM polarization, cscK birational models may give a way to apply metric methods to moduli spaces; whether the cscK class can be chosen close to the natural CM class is a testable question."],"forward_implications":["Every smooth projective $X$ has a birational model obtained by one blowup along a codimension-two smooth centre, with an explicit polarization $F+\\epsilon\\Theta$, carrying cscK metrics; no cscK or K-stability assumption on $X$ is needed.","Every complex projective variety, including singular ones, is birational to a smooth projective cscK manifold via resolution of singularities plus the Lefschetz blowup.","On any projective surface, blowing up the base points of a general pencil in $|4A|$ for a very ample line bundle $A$ gives a cscK manifold; hence the minimal number of blowup points is finite and bounded by $16A^2$, settling the point-blowup conjecture for projective surfaces.","For smooth Fano $n$-folds, the construction can be run with a general pencil in $|-mK_X|$ for bounded $m$, giving a uniform bound on the size of the centre in fixed dimension.","Because the proof establishes coercivity of the Mabuchi functional in the class, the same argument yields uniform K-stability of the resulting polarized birational model."],"supporting_citations":[{"why":"Supplies the adiabatic convergence theorem for the delta invariant, which is the entropy side of the argument.","marker":"[Hat25]"},{"why":"Identifies the analytic delta invariant with the algebraic stability threshold, letting entropy be estimated algebraically.","marker":"[Zha24]"},{"why":"Gives the criterion that positivity of the slope threshold implies positivity of the energy coercivity threshold.","marker":"[Che21]"},{"why":"Converts coercivity of the Mabuchi functional into existence of a cscK metric.","marker":"[CC21]"},{"why":"Links the coercivity proof to uniform K-stability of the resulting polarized model.","marker":"[BHJ19]"},{"why":"Resolution of singularities extends the smooth construction to arbitrary complex projective varieties.","marker":"[Hir64]"},{"why":"Provides the identification of the Lefschetz pencil incidence variety with the blowup along the base locus.","marker":"[Voi03]"},{"why":"Computes the local log canonical threshold of the quadratic singularity, giving lct one for Lefschetz fibers.","marker":"[Mus12]"}],"fun_headline_variants":["Every projective variety has a cscK birational model","Explicit cscK metrics on birational models of all projective varieties","One blowup yields cscK metrics for every projective variety","All projective varieties admit birational cscK models","CscK birational models: explicit for every projective variety"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that the adiabatic limit of the stability threshold of the total space equals the stability threshold of the base curve; if this convergence fails for some Lefschetz fibration, the entropy bound and hence the cscK conclusion collapse.","fun_headline_variants_meta":{"raw":{"variants":["Every projective variety has a cscK birational model","Explicit cscK metrics on birational models of all projective varieties","One blowup yields cscK metrics for every projective variety","All projective varieties admit birational cscK models","CscK birational models: explicit for every projective variety"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001003,"raw_usage":{"total_tokens":4206,"prompt_tokens":872,"completion_tokens":3334,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":3249}},"tokens_in":488,"tokens_out":3334,"duration_ms":24061,"temperature":1.0,"reasoning_tokens":3249,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:53:42.602214+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the explicit $\\mathbb P^2$ example, the cscK classes are $(4+5\\epsilon)h-(1+\\epsilon)\\sum_{i=1}^{16}E_i$; computing the delta invariant for a decreasing sequence of small rational $\\epsilon$ and checking whether it stays above $1$ and limits to $2$ would test the entropy estimate directly. A second check is to inspect any singular fiber of a general Lefschetz pencil: if some fiber fails to have log canonical threshold one, the proof's identification of the base curve threshold is no longer valid.","supporting_citations":[],"review_version":2}