A solution to the Yau-Tian-Donaldson Conjecture through Special Fujita Approximations

Antonio Trusiani

A solution to the Yau-Tian-Donaldson Conjecture through Special Fujita Approximations

T0 review · 0 major / 3 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read A polarized smooth projective variety admits a cscK metric if and only if it is Aut°(X,L)-uniformly K-stable.

desk verdict Trusiani claims a solution to the uniform YTD conjecture for general polarized varieties by proving a special Fujita approximation for big line bundles. read the letter →

arxiv 2605.30063 v1 pith:6ZMZYMG5 submitted 2026-05-28 math.AG math.CVmath.DG

Antonio Trusiani This is my paper · ORCID

classification math.AGmath.CVmath.DG

keywords Yau-Tian-DonaldsonconjecturecscKmetricsK-stabilityFujitaapproximationnon-Archimedeanentropypolarizedvarietiesconstantscalarcurvaturebiglinebundles

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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 every big line bundle on a smooth projective variety admits a special Fujita approximation, in which its volume and first Riemann-Roch coefficient are matched arbitrarily closely by those of ample rational line bundles on birational models. Combined with earlier results of Boucksom, Jonsson and Li, this settles the regularization conjecture for the non-Archimedean entropy functional. The main payoff is a complete solution of the uniform Yau-Tian-Donaldson conjecture: existence of a constant-scalar-curvature Kähler metric on (X,L) is equivalent to uniform K-stability with respect to the connected component of the automorphism group. The statement recovers and extends the known correspondence that previously held only for smooth Fano varieties.

What carries the argument

The special Fujita approximation: a sequence of birational models and ample Q-line bundles whose volumes and first Riemann-Roch coefficients converge to those of the given big line bundle.

What would settle it

An explicit big line bundle on a smooth projective variety whose volume or first Riemann-Roch coefficient cannot be approximated to arbitrary precision by ample Q-line bundles on birational models, or a polarized variety that is Aut°-uniformly K-stable yet carries no cscK metric.

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Extended reading notes

Core claim

Any big line bundle on a smooth projective variety admits a special Fujita approximation. Exploiting this fact together with prior work, the Boucksom-Jonsson regularization conjecture is solved, which in turn yields that a polarized smooth projective variety (X,L) admits a constant scalar curvature Kähler metric if and only if it is Aut°(X,L)-uniformly K-stable.

Load-bearing premise

Every big line bundle on a smooth projective variety admits a special Fujita approximation by ample Q-line bundles on higher models.

Editorial extensions

If this is right

The Boucksom-Jonsson regularization conjecture holds for the non-Archimedean entropy.
The uniform Yau-Tian-Donaldson conjecture is true for every polarized smooth projective variety.
Existence of cscK metrics is characterized by uniform K-stability outside the Fano case.
Algebraic approximation techniques now suffice to establish the analytic existence statement.

Reading between the lines

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

The same approximation may let researchers test uniform K-stability by direct computation on finite sequences of models rather than analytic limits.
Similar regularization arguments could be attempted for other non-Archimedean functionals arising in K-stability theory.
The result suggests that the passage from algebraic to analytic data is controlled once volume and linear terms are matched.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Trusiani claims to solve the uniform Yau-Tian-Donaldson conjecture for any polarized smooth projective variety. The key step is a new result that any big line bundle admits a special Fujita approximation, matching both volume and the first Riemann-Roch coefficient with ample Q-line bundles on higher models. This feeds into the Boucksom-Jonsson regularization of the non-Archimedean entropy and then applies earlier results to get the cscK versus uniform K-stability equivalence.

The approximation is the genuinely new piece. Prior work covered Fano varieties, so extending the correspondence requires this control over big bundles. The high-level chain runs cleanly from the new input through known regularization to the final statement.

The main soft spot is that the actual construction of the approximation is not visible here, so it is difficult to check whether the models deliver simultaneous approximation without extra assumptions that might affect the entropy step. Nothing in the outline points to circularity or self-reference.

This paper is for people already working on K-stability and non-Archimedean methods. A reader who knows the Boucksom-Jonsson-Li papers will follow the argument without much extra background. It deserves a serious referee because the claim is large and the approach builds directly on established lines rather than starting from scratch.

I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper proves that any big line bundle on a smooth projective variety admits a special Fujita approximation in which both the volume and the first Riemann-Roch coefficient are approximated by the corresponding quantities for ample Q-line bundles on higher models. Combining this new approximation with prior results of Boucksom–Jonsson–Li, the authors solve the Boucksom–Jonsson regularization conjecture for the non-Archimedean entropy functional. As the main application they obtain the uniform Yau–Tian–Donaldson conjecture: a polarized smooth projective variety (X,L) admits a cscK metric if and only if it is Aut°(X,L)-uniformly K-stable. The result extends the known correspondence that holds for smooth Fano varieties.

Significance. If the central technical step is correct, the manuscript supplies a complete solution to the uniform version of the YTD conjecture for arbitrary polarized smooth projective varieties, a long-standing open problem. The new ingredient—the special Fujita approximation for arbitrary big line bundles—directly enables the required regularization of the non-Archimedean entropy and thereby closes the argument via the cited earlier works.

minor comments (3)

[Introduction] §1, paragraph following Theorem A: the precise statement of which prior results of Boucksom–Jonsson–Li are invoked for the regularization step should be made explicit rather than summarized as “exploiting previous works.”
[Section 2] Definition 2.3 (special Fujita approximation): the notation for the sequence of models and the error terms in the volume and χ1 approximations could be introduced with a short displayed equation to improve readability.
[Section 4] Theorem 4.1 (regularization of entropy): the dependence on the special Fujita approximation is stated only in the proof; a one-sentence reference in the theorem statement itself would clarify the logical flow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained

full rationale

The paper's central new input is the special Fujita approximation theorem for arbitrary big line bundles on smooth projective varieties, which approximates volume and first Riemann-Roch coefficients by ample Q-line bundles on higher models. This is presented as an independent technical result. The argument then invokes prior independent results of Boucksom-Jonsson-Li on non-Archimedean entropy regularization to obtain the uniform YTD statement. No equation or step in the provided chain reduces the target YTD correspondence to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation. The derivation chain remains externally supported by the cited prior works and does not collapse by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the newly introduced special Fujita approximation together with domain assumptions from prior literature on non-Archimedean entropy.

assumptions (1)

domain assumption Prior results by Boucksom, Jonsson and Li on the Non-Archimedean entropy functional
Exploited to solve the Boucksom-Jonsson Regularization Conjecture after establishing the special Fujita approximation.

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Cite this review

Pith. "Pith review of A solution to the Yau-Tian-Donaldson Conjecture through Special Fujita Approximations." pith.science (2026). https://pith.science/paper/6ZMZYMG5

@misc{pith2026260530063,
  author       = {Pith},
  title        = {Pith review of: A solution to the Yau-Tian-Donaldson Conjecture through Special Fujita Approximations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ZMZYMG5}},
  note         = {Machine review of arXiv:2605.30063}
}

abstract

We show that any big line bundle on a smooth projective variety admits a special Fujita approximation: the volume and the first Riemann-Roch coefficient are both approximated by those of ample $\mathbb{Q}$-line bundles on higher models. Exploiting previous works by Boucksom, Jonsson and Li, we solve the Boucksom-Jonsson Regularization Conjecture on the Non-Archimedean entropy functional. As a main consequence, we obtain a solution to the (uniform version of the) Yau-Tian-Donaldson Conjecture: a polarized smooth projective variety $(X,L)$ admits a cscK metric if and only if it is $\mathrm{Aut}^\circ(X,L)$-uniformly $K$-stable. This extends the known Yau-Tian-Donaldson correspondence for smooth Fano varieties.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Poisson K-stability and the semiclassical Yau--Tian--Donaldson correspondence
math.DG 2026-07 accept novelty 7.5 of 10

Polystable Poisson structures on Kähler–Einstein Fano manifolds admit constant-scalar-curvature symplectic generalized Kähler metrics for small enough Poisson tensors, settling the semiclassical YTD conjecture on P².
On the geometry of non-collapsed polarized cscK surfaces
math.DG 2026-06 unverdicted novelty 4.0 of 10

Gromov-Hausdorff convergence of non-collapsed polarized cscK surfaces is realized as Hilbert scheme convergence, with Bergman kernel estimates enabling Zariski openness of cscK metrics in smooth polarized families.

Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages · cited by 2 Pith papers

[1]

4, 5 [72]Hashimoto, Y.Relative uniformK-stability over models implies existence of extremal metrics.Math. Z. 312, 4 (2026), Paper No. 109. 5 [73]Hattori, M.A decomposition formula for J-stability and its applications.arXiv preprint arXiv:2103.04603 (2021). 2 [74]He, W.On Calabi’s extremal metric and properness.Trans. Amer. Math. Soc. 372, 8 (2019), 5595–5...

work page arXiv 2026
[2]

K-stability of constant scalar curvature polarization

Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. 8, 18 [83]Lazarsfeld, R.Positivity in algebraic geometry. II, vol. 49 ofErgebnisse der Mathematik und ihrer Grenzge- biete. 3. ...

work page Pith review arXiv 2004
[3]

6 [120]Tian, G.K¨ ahler-Einstein metrics with positive scalar curvature.Invent. Math. 130, 1 (1997), 1–37. 1, 11 28 ANTONIO TRUSIANI [121]Tian, G.Canonical metrics in K¨ ahler geometry. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, 2000. Notes taken by Meike Akveld. 2 [122]Tian, G.K-stability and K¨ ahler-Einstein metrics.Comm. Pure Ap...

1997

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

4, 5 [72]Hashimoto, Y.Relative uniformK-stability over models implies existence of extremal metrics.Math. Z. 312, 4 (2026), Paper No. 109. 5 [73]Hattori, M.A decomposition formula for J-stability and its applications.arXiv preprint arXiv:2103.04603 (2021). 2 [74]He, W.On Calabi’s extremal metric and properness.Trans. Amer. Math. Soc. 372, 8 (2019), 5595–5...

work page arXiv 2026

[2] [2]

K-stability of constant scalar curvature polarization

Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. 8, 18 [83]Lazarsfeld, R.Positivity in algebraic geometry. II, vol. 49 ofErgebnisse der Mathematik und ihrer Grenzge- biete. 3. ...

work page Pith review arXiv 2004

[3] [3]

6 [120]Tian, G.K¨ ahler-Einstein metrics with positive scalar curvature.Invent. Math. 130, 1 (1997), 1–37. 1, 11 28 ANTONIO TRUSIANI [121]Tian, G.Canonical metrics in K¨ ahler geometry. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, 2000. Notes taken by Meike Akveld. 2 [122]Tian, G.K-stability and K¨ ahler-Einstein metrics.Comm. Pure Ap...

1997