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CscK metrics on birational models of projective varieties

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Pith's one-line read Every complex projective variety is birational to a smooth projective variety carrying a constant scalar curvature Kähler metric, obtained by a Lefschetz blowup.

desk verdict 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. read the letter →

arxiv 2608.03572 v1 pith:ZSPNTO2P submitted 2026-08-04 math.DG math.AG

Zakarias Sjöström Dyrefelt This is my paper · ORCID
classification math.DGmath.AG MSC 53C5514E0532Q2614J45
keywords cscKmetricconstantscalarcurvatureKählerLefschetzpencilbirationalmodelblowupK-stabilitydeltainvariantMabuchifunctional
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [§2.3, Proposition 2.8] 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.
  2. [§2.4 and §4.5, Theorem 2.9 / Lemma 4.8] 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.
minor comments (5)
  1. [§1.1(c)] The word 'For exmaple' should be corrected to 'For example'.
  2. [§4.6, proof of Theorem 4.1] 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.
  3. [§5.3] The displayed slope formula would be less ambiguous with parentheses: E=(1+(n+1)^2 ε)/(n(n+2)+(n+1)^2 ε).
  4. [§2.4] 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.
  5. [References] The entries [JSD26] and [SSD26] are listed as 'in preparation'; if they are needed for the surface discussion, the dependence should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central proof computes the slope threshold from definitions and imports the entropy bound from Hattori's external theorem, with no fitted input or self-referential reduction.

full rationale

I walked the derivation chain of Theorems 1.2 and 4.1. The energy side is a direct algebraic computation: Proposition 4.6 derives the exact slope formula E_{K_Y+L}(L) = (B_N^n + εΘ_N^n)/(n H_N·B_N^{n-1} + εΘ_N^n) from the definitions of the slope threshold, using F^2=0, K_Y+F=π^*B_N, and the Lefschetz fiber identity; no parameter is fitted and no quantity is defined in terms of the desired cscK conclusion. The positivity of the energy threshold then follows from Gao Chen's independently stated J-stability theorem [Che21]. The entropy side is imported from Hattori's adiabatic theorem [Hat25, Theorem 4.12], an external result, combined with the log-canonical threshold computation in Proposition 2.8. Even if that lct computation were incorrect, as the skeptic argues, that would be a mathematical correctness problem in an input lemma, not a circularity: the paper does not define lct(Y,Y_t)=1 in order to force δ>1, nor does it fit δ to the desired existence result. The self-citations that appear are [SD20] for J-functional notation, [SD22] for an automorphism remark, [KSD24] and [KSD26] for motivational framing of destabilizing subvarieties, and [JSD26] for an optional surface remark; none of these supplies the load-bearing entropy or energy bounds. The cscK criterion in Proposition 4.4 combines two independent sufficient thresholds, and the final existence step invokes Chen–Cheng [CC21]. I find no step where a prediction is equivalent by construction to an input, no renamed fit, and no load-bearing self-citation chain.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claim rests on several external theorems but introduces no fitted constants and no new objects. The only free choices are the adiabatic epsilon and the positivity multiplier N, both existence-level parameters.

free parameters (2)
  • adiabatic parameter epsilon = rational, 0<epsilon<epsilon_0, epsilon_0 not effective
    The cscK polarization is F+epsilon Theta. The proof establishes existence of a sufficiently small positive rational epsilon via a limit argument but gives no bound, leaving the class only conditionally explicit.
  • positivity multiplier N = any sufficiently large integer
    One chooses N so that H_N=NA is very ample and K_X+H_N is ample. Such N exists elementarily; it is a construction choice, not fitted to data.
assumptions (7)
  • domain assumption Hattori adiabatic limit for delta invariants (Theorem 2.9, [Hat25])
    Used in Lemma 4.8 to conclude delta(Y,F+epsilon Theta_N) tends to 2, hence exceeds 1 for small epsilon. This is the entire entropy side of the proof.
  • domain assumption Chen's equivalence between slope threshold positivity and energy threshold positivity for Kähler beta (Theorem 2.6, [Che21])
    Converts the exact slope computation E_{K_Y+L}(L)>0 into Gamma^pp_{K_Y+L}(L)>0 in Proposition 4.6.
  • domain assumption Chen-Cheng coercivity implies existence of a cscK metric (Theorem 2.1, [CC21])
    Final step of Theorem 4.1: coercive Mabuchi energy yields a cscK metric.
  • domain assumption Zhang equality of analytic and algebraic delta invariants (Theorem 2.7, [Zha24])
    Justifies writing delta(Y,L) for the entropy threshold and using algebraic computations.
  • domain assumption Existence of general Lefschetz pencils and lct=1 for Morse fibers (SGA7, [Mus12])
    Provides the fibration and the vanishing discriminant boundary B_f=0 used in Corollary 2.10.
  • standard math Mori cone theorem length bound on surfaces (KM98)
    Used in Corollary 5.2 to show K_X+4A is ample for any very ample A.
  • standard math Hironaka resolution of singularities
    Extends the smooth main theorem to arbitrary projective varieties in Corollary 4.10.

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Pith. "Pith review of CscK metrics on birational models of projective varieties." pith.science (2026). https://pith.science/paper/ZSPNTO2P

@misc{pith2026260803572,
  author       = {Pith},
  title        = {Pith review of: CscK metrics on birational models of projective varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSPNTO2P}},
  note         = {Machine review of arXiv:2608.03572}
}
read the original abstract

We prove that every complex projective variety is birational to a smooth projective manifold admitting a constant scalar curvature K\"ahler (cscK) metric. For any smooth projective variety, a birational cscK model is obtained by resolving the codimension two base locus of a general Lefschetz pencil in a sufficiently positive linear system. The cscK polarization is given explicitly, producing a cscK model also when the initial variety is unstable. In dimension two this proves the folklore conjecture that the blowup of any complex projective surface in enough points admits cscK metrics.

Figures

Figures reproduced from arXiv: 2608.03572 by the authors.

Figure 1
Figure 1. The fibered geometry of the regularizing blowup. map f is called the Lefschetz fibration associated with the pencil. The fiber Yt := f −1 (t) is the strict transform of Dt , and π|Yt : Yt ≃−−→ Dt , since Z ⊂ Dt is a Cartier divisor. Let E ⊂ Y denote the exceptional divisor, and let F denote the divisor class of a fiber of f. The following summarizes the only properties we need about the Lefschetz fibration and its b… view at source ↗

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