From K\"ahler Ricci solitons to Calabi-Yau K\"ahler cones

Abdellah Lahdili; Eveline Legendre; Vestislav Apostolov

From K\"ahler Ricci solitons to Calabi-Yau K\"ahler cones

T0 review · 0 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A Fano manifold with a Kähler–Ricci soliton gives Calabi–Yau cones after adding a large projective-space factor.

desk verdict Asymptotic Mabuchi–Nakagawa is proved cleanly via Han–Li's weighted YTD criterion; the paper deserves a serious referee. read the letter →

arxiv 2412.02564 v2 pith:SXT2DGFZ submitted 2024-12-03 math.DG math.AG

Vestislav Apostolov , Abdellah Lahdili , Eveline Legendre This is my paper · ORCID

classification math.DGmath.AG MSC 53C2553C5532Q2014J45

keywords Kähler–RiccisolitonCalabi–YauconeSasaki–Einsteinmetricv-solitonweightedDingfunctionalK-stabilityFanomanifoldcoercivity

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 aims to show that soliton-type canonical metrics on a Fano manifold produce Ricci-flat cone metrics on a related space. More precisely, it claims that if the smooth Fano manifold $X$ admits a Kähler–Ricci soliton, then for every sufficiently large $k$ the canonical cone $K^\times_Z$ of $Z = X \times \mathbb{P}^k_{\mathbb{C}}$ carries a Ricci-flat Kähler cone metric, also called a Calabi–Yau cone. This is an asymptotic version of the conjecture that $k=0$ already works. The engine is Theorem 1.1: the set of weights $v$ for which the $v$-weighted Ding functional is coercive relative to the complexified torus is an open convex cone in the space of weights with vanishing $v$-Futaki invariant. The argument approximates the soliton weight $e^{\langle\tau,x\rangle}$ by explicit rational weights, obtains $v$-solitons on $X$ for all sufficiently large $N$, and converts the product soliton on $Z$ into a Calabi–Yau cone metric.

What carries the argument

The carrying object is the $v$-weighted Ding functional $D_v$ on the space of $T$-invariant Kähler potentials: its critical points are exactly the $v$-solitons, and its coercivity relative to the complex torus $T^{\mathbb{C}}$ is the analytic criterion for existence. Theorem 1.1 proves that the coercive locus $D(X)$ is open and convex in $F(X)$ by a comparison estimate: if normalized weights $\bar v_1,\bar v_2$ satisfy $\inf_{P_X}(\bar v_1-\bar v_2)=-\lambda_0$, then $D_{v_1}-D_{v_2} \geq -\lambda_0 J(\omega) + \text{constant}$, so coercivity of slope $\Lambda_0$ at one weight transfers to nearby weights with slope $\Lambda_0-\lambda_0$. Convexity comes from linearity of the weighted Aubin–Mabuchi functional in the weight. The approximation sequence $v_N$ arises from convex volume functionals and an implicit-function-theorem deformation of the Kähler–Ricci soliton weight.

What would settle it

Take a smooth Fano $X$ with a Kähler–Ricci soliton and compute the weighted $\beta$-invariant of a $T$-equivariant prime divisor for the approximating weights $v_N$; finding $\beta_{v_N} < 0$ for arbitrarily large $N$ would contradict the openness of $S(X)$ and invalidate Theorem 1.2. Alternatively, producing weights $v_N \in F(X)$ that converge in $C^0$ to an existing soliton weight but admit no $v_N$-soliton for infinitely many $N$ would falsify the paper's central mechanism.

Watch

Extended reading notes

Core claim

Let $X$ be a smooth Fano manifold and $T$ a maximal compact torus of automorphisms, with canonically normalized momentum polytope $P_X$. Let $F(X)$ be the cone of positive weight functions $v$ on $P_X$ whose $v$-Futaki invariant vanishes. Theorem 1.1 asserts that the subset $D(X)$ where the $v$-weighted Ding functional $D_v$ is $T^{\mathbb{C}}$-coercive is open and convex in $F(X)$. By the criterion of [37], a $v$-soliton exists exactly when $D_v$ is $T^{\mathbb{C}}$-coercive, so $S(X)=D(X)$. The main application, Theorem 1.2, constructs for a manifold $X$ admitting a Kähler–Ricci soliton a sequence of weights $v_N(x) = (1 - \langle x, \xi_N\rangle/N)^{-N}$ in $F(X)$ that converges uniformly to $e^{\langle\tau,x\rangle}$; for $N$ large these weights are in $S(X)$, the product of the resulting $v_N$-soliton with the Fubini–Study metric is a $v_N$-soliton on $Z = X \times \mathbb{P}^{N-n-2}_{\mathbb{C}}$ with weight of the form $\ell^{-(\dim Z+2)}$, and the correspondence of [3] turns this into a Ricci-flat Kähler cone metric on the canonical cone $K^\times_Z$.

Load-bearing premise

The load-bearing premise is that a $v$-soliton exists precisely when the $v$-weighted Ding functional is coercive; if that equivalence fails for some smooth positive weights, the openness theorem and the Calabi–Yau cone conclusion do not follow.

Editorial extensions

If this is right

If $X$ admits a Kähler–Ricci soliton, then for every $k \geq k_0$ the canonical cone of $X \times \mathbb{P}^k_{\mathbb{C}}$ admits a Ricci-flat Kähler cone metric; the proof gives no explicit bound on $k_0$.
The projective-space factor can be replaced by any $k$-dimensional Kähler–Einstein Fano manifold.
A transversal Kähler–Ricci soliton on a Sasaki structure persists under nearby Sasaki–Reeb polarizations, recovering a known deformation theorem as Theorem 1.3.
A Kähler–Ricci soliton on $X$ forces $\langle\tau,x\rangle < n$ on the canonical polytope $P_X$, with the analogous bound $\langle\tau_\xi,x\rangle < n(\langle\xi,x\rangle+1)$ for transversal Kähler–Ricci solitons.
For a normalized $v$-soliton weight, the first Chern number satisfies $c_1(X)^n \leq \left(\frac{n+1}{\inf_{P_X} v}\right)^n$, a weighted version of Fujita's volume bound.

Reading between the lines

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

The openness theorem implies that the soliton locus is convex in weight space, so convex combinations of soliton weights in $F(X)$ should again admit solitons; the paper does not isolate this as a separate statement.
If the coercivity slope could be estimated explicitly from the Kähler–Ricci soliton weight, the proof would yield a quantitative $k_0$; the paper only establishes existence of some such $k_0$.
The weighted Fujita bound can be read as an obstruction on the size of admissible weights: a normalized $v$-soliton weight cannot have its minimum too small relative to the anti-canonical volume.
The same deformation mechanism, run through the Appendix A argument, should give openness of transversal Kähler–Ricci solitons for Fano orbifolds, an extension the paper gestures at in Remark 4.5 but does not state as a theorem.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The headline: this paper proves an asymptotic version of Mabuchi–Nakagawa that actually holds up. If X is a smooth Fano with a Kähler–Ricci soliton, then for all sufficiently large k the canonical cone of X × P^k carries a Ricci-flat Kähler cone metric. The k0 is not effective, but the existence statement is solid.

What's genuinely new: Theorem 1.1, the set D(X) of weights whose v-weighted Ding functional is T^C-coercive is an open convex cone in F(X). Since Han–Li proved S(X)=D(X), this is effective openness of the soliton locus. The approximation lemma, writing the KRS weight e^{<τ,x>} as a limit of powers of affine-linear functions, is the clever bit that makes the product construction work. The paper also recovers Petrecca's theorem on deformations of transversal KRS and adds a weighted Fujita volume bound and a Lichnerowicz-type obstruction. These are useful side results, honestly derived.

Where the soft spots are: the main theorem leans entirely on Han–Li's weighted YTD theorem (Theorem 2.1) as a black box. That's not a flaw in itself — the theorem is published and independent — but it means the paper's contribution is conditional on that criterion being as general as stated. The non-quantitative k0 is admitted and is a true limitation. The maximum principle argument in Appendix B is terse; Corollary B.1 (positive transversal scalar curvature) is sketched rather than proved in detail, though it looks correct. The self-citation count is high, but the cited results are used as framework, not as the target, so I don't read it as circular.

I checked the stress-test note's chain: the weights v_N are smooth and positive, their v-Futaki vanishes by construction, they converge uniformly to e^{<τ,x>}, and then either Theorem 1.1 or Corollary A.1 gives v_N-solitons. The product with P^k and the conversion to a Calabi–Yau cone uses their earlier [3, Prop.2] exactly in the stated setting. That all lines up.

Who this is for: people working in Kähler–Ricci solitons, weighted K-stability, and Sasaki geometry. It's a clean, well-written paper that deserves a serious referee. I'd send it to review and expect acceptance after the usual checks.

Referee Report

0 major / 6 minor

Summary. The paper studies the space S(X) of weight functions v on the momentum polytope of a smooth Fano manifold for which a v-soliton exists. The main structural result, Theorem 1.1, states that the coercivity locus D(X) of the v-weighted Ding functional is an open convex cone inside the vanishing-v-Futaki space F(X); combined with Han-Li's criterion S(X)=D(X), this yields quantitative openness of S(X). The main application, Theorem 1.2, shows that if X admits a Kähler-Ricci soliton, then for all sufficiently large k the product Z=X×P^k admits an ℓ^{-(dim Z+2)}-soliton, hence the canonical cone of Z is a Calabi-Yau cone, giving an asymptotic version of the Mabuchi-Nakagawa conjecture. The paper also proves relative openness of transversal KRS Sasaki structures (Theorem 1.3), an upper bound on the soliton potential (Theorem 1.4), and a weighted Fujita volume bound (Theorem 1.5), with an appendix giving a LeBrun-Simanca style deformation argument.

Significance. If the results hold, the paper gives a clean new route to constructing Calabi-Yau cones by perturbing KRS weights, and establishes openness of the v-soliton locus in C^0 topology as a corollary of Han-Li's analytic criterion. The proof of Theorem 1.2 is a genuine consequence of external results applied in their valid regime: Han-Li's weighted YTD theorem (Theorem 2.1) identifies S(X)=D(X), and the authors' earlier correspondence [3, Prop.2] converts ℓ^{-(m+2)}-solitons into Ricci-flat cone metrics. The approximation of the KRS weight by v_N via the implicit function theorem is explicit and coherent, and the absence of a quantitative k0 is honestly stated as a limitation. The secondary results give useful complements: a new Lichnerowicz-type obstruction and a weighted Fujita bound.

minor comments (6)

[Introduction, organization paragraph] The organization paragraph says 'In Sect. 5, we prove Theorem 1.4' and then 'The final Sect. 5, we prove Theorem 1.5', but Theorem 1.5 is proved in Section 6; the numbering should be corrected.
[Abstract and throughout] There are several typos that should be fixed: 'caries' should be 'carries', 'Nikagawa' should be 'Nakagawa', 'Licherowicz' should be 'Lichnerowicz', 'Fujita'a' should be 'Fujita's', and 'Ricci-flal' should be 'Ricci-flat'.
[Equation (2.5)] The Duistermaat-Heckman measure is denoted dµDH in the text but the displayed equation (2.5) writes dµDM; the notation should be made uniform.
[Proof of Theorem 1.1] The proof of convexity is too terse: the sentence 'The convexity of D(X) follows from the fact that the subspace of normalized weight functions is linearly convex' does not by itself explain why D(X) is a cone, since a cone also requires dilation invariance; a one-sentence clarification would be helpful.
[Corollary 3.1 and proof of Theorem 1.1] In the comparison of normalized weights it is assumed that inf(˚v - ˚v0) = -λ0 with λ0 > 0; since both weights are normalized the case λ0 = 0 forces ˚v = ˚v0, but this reasoning is implicit and could be stated explicitly.
[Appendix B, Corollary B.1] The maximum principle argument for positivity of the transversal scalar curvature is very compressed: at a minimum of Scal the displayed identity (B.2) gives ΔScal ≥ 0 and |Ric|² - Scal ≥ 0, which does not by itself show Scal > 0; the sign conventions for the Laplacian and the precise maximum principle argument should be expanded.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.2 follows from the external Han–Li weighted YTD criterion and the independently established [3, Prop.2] correspondence, neither of which is equivalent to the target result.

full rationale

The claimed derivation chain is not circular. Theorem 1.2 is obtained by (i) constructing weights v_N in F(X) that converge uniformly to the KRS weight e^{<tau,x>}, with v_N's Futaki vanishing by construction; (ii) invoking the new openness Theorem 1.1, whose proof uses the external Han–Li theorem [37, Theorem 1.7] only to identify coercivity with existence of v-solitons; and (iii) converting the resulting product w_N-soliton on Z = X x P^{N-n-2} into a Ricci-flat Kähler cone metric via the earlier correspondence [3, Prop.2]. The Han–Li theorem is external to this paper and does not have the target result as an input. The same-author citations [3] and [41] supply supporting equivalences (v-solitons versus weighted cscK metrics, and the LeBrun–Simanca deformation argument) that are stated with their own assumptions and proved in prior work; they are used as tools, not as the conclusion. The weights v_N are not fitted to the output: their v-Futaki vanishing is the necessary normalization forced by membership in F(X), and convergence to the KRS weight is proved by the implicit function theorem. No equation in the paper defines a quantity in terms of the result it is used to prove, and the non-quantitative nature of k0 is explicitly acknowledged as a limitation. The central derivation is therefore self-contained against external benchmarks, and no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new geometric or physical entities. The central claim rests on imported theorems: Han-Li's weighted YTD criterion, the authors' earlier [3] Sasaki correspondence, and Fujita's volume inequality. These are legitimate published results, but they carry most of the analytic burden.

assumptions (4)

domain assumption Han-Li weighted YTD criterion: X admits a T-invariant v-soliton iff D_v is coercive relative to T^C, and S(X)=D(X).
Stated as Theorem 2.1 and [37, Theorem 1.7]; it is the main bridge between coercivity and existence and is used without proof throughout the paper.
domain assumption Existence of T^C-orbit minimizers of the Aubin functional J, as used in the proof of Theorem 1.1.
Imported from [37, Lemma 29]; needed to transfer coercivity from a representative to the infimum over the complex torus orbit.
domain assumption Apostolov-Jubert-Lahdili correspondence [3, Prop.2]: an l^{-(dim Z+2)}-soliton on a Fano Z is equivalent to a Calabi-Yau cone structure on K_Z^×.
This is the final step of Theorem 1.2, converting the approximate soliton on the product into the desired Ricci-flat cone metric.
domain assumption Fujita volume lower bound [29, Theorem 2.3]: Vol(b^*K_X - xD) ≥ Vol(-K_X) - x^n for prime divisors over a Fano manifold X.
Used without proof in Theorem 1.5; the theorem is imported as a black box and its hypotheses must cover the v-soliton case.

how reviews work

0 comments

Cite this review

Pith. "Pith review of From K\"ahler Ricci solitons to Calabi-Yau K\"ahler cones." pith.science (2026). https://pith.science/paper/SXT2DGFZ

@misc{pith2026241202564,
  author       = {Pith},
  title        = {Pith review of: From K\"ahler Ricci solitons to Calabi-Yau K\"ahler cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXT2DGFZ}},
  note         = {Machine review of arXiv:2412.02564}
}

abstract

We show that if $X$ is a smooth Fano manifold which caries a K\"ahler Ricci soliton, then the canonical cone of the product of $X$ with a complex projective space of sufficiently large dimension is a Calabi--Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions $v$ over the momentum polytope of a given smooth Fano manifold, for which a $v$-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a K\"ahler Ricci soliton and a Fujita type volume bound for the existence of a $v$-soliton.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

66 extracted references · 63 canonical work pages

[1]

Apostolov, D

V. Apostolov, D. J. Calderbank, The CR geometry of weighted extremal Kahler and Sasaki metrics , Math. Ann. 379 (2021), 1047–1088

work page 2021
[2]

Apostolov, D

V. Apostolov, D. J. Calderbank, E. Legendre, Weighted K-stability of polarized varieties and extremal- ity of Sasaki manifolds , Advances in Math. 391 (2021), Paper No 107969

work page 2021
[3]

Apostolov, S

V. Apostolov, S. Jubert and A. Lahdili, Weighted K-stability and coercivity with applications to ex- tremal K¨ ahler and Sasaki metrics, Geometry & Topology 8 (2023), 3229–3302

work page 2023
[4]

Atiyah, Convexity and commuting Hamiltonians , Bull

M.F. Atiyah, Convexity and commuting Hamiltonians , Bull. London Math. Soc. 14 (1982) 1—15

work page 1982
[5]

Aubin, R´ eduction du cas positif de l’´ equation de Monge-Amp` eere sur les vari´ et´ es k¨ ahl´ eriennes compactes ` a la d´ emonstration d’une in´ egalit´ e, J

Th. Aubin, R´ eduction du cas positif de l’´ equation de Monge-Amp` eere sur les vari´ et´ es k¨ ahl´ eriennes compactes ` a la d´ emonstration d’une in´ egalit´ e, J. Funct. Anal. 57 (1984), 143–153

work page 1984
[6]

Berline, M

M. Berline, M. Vergne, Z´ eros d’un champs de vecteur et classes charact´ eristiques ´ equivarientes, Duke Math. J. 50 (1983), 539–549

work page 1983
[7]

Berman, K-polystability of Q-Fano varieties admitting K¨ ahler-Einstein metrics, Invent

R. Berman, K-polystability of Q-Fano varieties admitting K¨ ahler-Einstein metrics, Invent. Math. 203 (2016), 973—1025

work page 2016
[8]

The projective space has maximal volume among all toric K\"ahler-Einstein manifolds

R. Berman and B. Berndtsson, The projective space has maximal volume among all toric K¨ ahler- Einstein manifolds , arXiv:1112.4445

work page Pith review arXiv

Show all 66 references

[9]

Berman and B

R. Berman and B. Berndtsson, The volume of K¨ ahler–Einstein Fano varieties and convex bodies , J. Reine Angew. Math., 723 (2017), 127–152

2017
[10]

Berman, S

R. Berman, S. Boucksom and M. Jonsson, A variational approach to the Yau-Tian-Donaldson conjec- ture, J. Amer. Math. Soc. 34 (2021), 605–652

2021
[11]

R. J. Berman and D. Witt-Nystr¨ om,Complex optimal transport and the pluripotential theory of K¨ ahler- Ricci solitons, arXiv:1401.8264. 27

arXiv
[12]

Boucksom, C

S. Boucksom, C. Favre and M. Jonsson, Differentiability of volumes of divisors and a problem of Teissier. J. Algebraic Geom. 18 (2009), 279—308

2009
[13]

Boyer, H.Huang, E

C. Boyer, H.Huang, E. Legendre, C. W. Tønnesen–Friedman,Existence and Non-Existence of Constant Scalar Curvature and Extremal Sasaki Metrics , Math. Z. 304 (2023), Article No 61

2023
[14]

C. P. Boyer and C. van Coevering, Relative K-stability and extremal Sasaki metrics , Math. Res. Lett. 28 (2018), 1–19

2018
[15]

Calabi, On K¨ ahler manifolds with vanishing canonical class, in Algebraic Geometry and Topology : A symposium in honour of Lefschetz, Princeton University Press (1955), 78–89

E. Calabi, On K¨ ahler manifolds with vanishing canonical class, in Algebraic Geometry and Topology : A symposium in honour of Lefschetz, Princeton University Press (1955), 78–89

1955
[16]

H.-D. Cao, G. Tian and X. Zhu, K¨ ahler–Ricci solitons on compact complex manifolds withc1(M ) > 0, Geom. Funct. Anal. 15 (2005), 697–719

2005
[17]

Donaldson and S

X.X.Chen, S. Donaldson and S. Sun, K¨ ahler–Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities . J. Amer. Math. Soc. 28 (2015), 183—197

2015
[18]

Donaldson and S

X.X.Chen, S. Donaldson and S. Sun, K¨ ahler–Einstein metrics on Fano manifolds, II: limits with cone angle less than 2π. J. Amer. Math. Soc. 28 (2015), 199—234

2015
[19]

Donaldson and S

X.X.Chen, S. Donaldson and S. Sun, K¨ ahler–Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof . J. Amer. Math. Soc. 28 (2015), 235—278

2015
[20]

X.X. Chen, S. Sun and B. Wang, K¨ ahler-Ricci flow, K¨ ahler-Einstein metric, and K-stability, Geometry & Topology 22 (2018), 3145–3173

2018
[21]

Collins and G

T. Collins and G. Sz´ ekelyhidi, K-semistability for irregular Sasaki manifolds , J. Differential Geom. 109 (2018), 81–109

2018
[22]

Collins and G

T. Collins and G. Sz´ ekelyhidi, Sasaki-Einstein metrics and K-stability , Geometry & Topology 23 (2019), 1339–1413

2019
[23]

Darvas and Y

T. Darvas and Y. Rubinstein, Tian ’s properness conjectures and Finsler geometry of the space of K¨ ahler metrics, J. Amer. Math. Soc. 30 (2017), 347–387

2017
[24]

Datar and G

V. Datar and G. Sz´ ekelyhidi, K¨ ahler-Einstein metrics along the smooth continuity method . Geom. Funct. Anal. 26 (2016), 975–1010

2016
[25]

Delcroix, Weight sensibility in K-stability of Fano varieties , arXiv:2411.07864

T. Delcroix, Weight sensibility in K-stability of Fano varieties , arXiv:2411.07864

arXiv
[26]

Dervan, E

R. Dervan, E. Legendre Valuative stability of polarized varieties , Math. Ann. 385 (2023), 357–391

2023
[27]

Donaldson, Scalar curvature and stability of toric varieties

S. Donaldson, Scalar curvature and stability of toric varieties . J. Differential Geom. 62 (2002), 289— 349

2002
[28]

Duistermaat and G

J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259—268

1982
[29]

Fujita, Optimal bounds for the volumes of K¨ ahler-Einstein Fano manifolds , American J

K. Fujita, Optimal bounds for the volumes of K¨ ahler-Einstein Fano manifolds , American J. Math., 140 (2018), 391–414

2018
[30]

Futaki, An obstruction to the existence of Einstein K¨ ahler metrics, Invent

A. Futaki, An obstruction to the existence of Einstein K¨ ahler metrics, Invent. Math. 73 (1983) 437— 443

1983
[31]

Mabuchi Bilinear forms and extremal K¨ ahler vector fields associated with K¨ ahler classes, Math

A.Futaki and T. Mabuchi Bilinear forms and extremal K¨ ahler vector fields associated with K¨ ahler classes, Math. Ann. 301 (1995) 199–210

1995
[32]

Futaki, H

A. Futaki, H. Ono, G. Wang, Transverse K¨ ahler geometry of Sasaki manifolds and toric Sasaki– Einstein manifolds J. Differential Geom. 83, (2009) 585–635

2009
[33]

Gauduchon Calabi’s extremal metrics: An elementary introduction , Lecture notes available upon request

P. Gauduchon Calabi’s extremal metrics: An elementary introduction , Lecture notes available upon request
[34]

Gauntlett, D

J.P. Gauntlett, D. Martelli, J. Sparks, S.-T. Yau Obstructions to the Existence of Sasaki–Einstein Metrics. Commun. Math. Phys. 273 (2007), 803–827

2007
[35]

Guillemin and S

V. Guillemin and S. Sternberg, Convexity properties of the moment mapping , Invent. Math. 67 (1982), 491—513

1982
[36]

Hallam, The Geometry and Stability of Fibrations , Oxford PhD thesis 2022, availabe at https://ora.ox.ac.uk/objects/uuid:941370e7-4267-40ce-ae45-8b009c168130

M. Hallam, The Geometry and Stability of Fibrations , Oxford PhD thesis 2022, availabe at https://ora.ox.ac.uk/objects/uuid:941370e7-4267-40ce-ae45-8b009c168130

2022
[37]

Han and C

J. Han and C. Li, On the Yau-Tian-Donaldson conjecture for generalized K¨ ahler-Ricci soliton equa- tions, Comm. Pure Appl. Math. 76 (2023), 1793–1867

2023
[38]

J. Han, C. Li, Algebraic uniqueness of K¨ ahler-Ricci flow limits and optimal degenerations of Fano varieties, Geometry & Topology 28 (2024) 539—592

2024
[39]

H¨ attig, J

D. H¨ attig, J. Hausen, H. S¨ uss, Log DelPezzo C∗-surfaces, K¨ ahler–Einstein metrics, K¨ ahler–Ricci solitons and sasaki–Einstein metrics , to appear in Michigan Math. J., arXiv:2306.03796v1

arXiv
[40]

He and S

W. He and S. Sun, Frankel conjecture and Sasaki geometry, Adv. Math. 291 (2016), 912—960

2016
[41]

Lahdili, K¨ ahler metrics with constant weighted scalar curvature and weighted K-stability , Proc

A. Lahdili, K¨ ahler metrics with constant weighted scalar curvature and weighted K-stability , Proc. London Math. Soc. 119 (2019), 1065–1114

2019
[42]

Lazarsfeld, Positivity in algebraic geometry I: Classical setting line bundles and linear series

R. Lazarsfeld, Positivity in algebraic geometry I: Classical setting line bundles and linear series. Ergeb- nisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics,
[43]

LeBrun and S

C.R. LeBrun and S. Simanca, Extremal K¨ ahler metrics and complex deformation theory, Geom. Funct. Anal. 4 (1994), 298–336

1994
[44]

K. L. Lee, J. Sturm and X. Wang, Moment map, convex function and extremal point, arXiv:2208.03724

arXiv
[45]

Li, Minimizing normalized volumes of valuations , Math

C. Li, Minimizing normalized volumes of valuations , Math. Z. 289 (2018), 491–513

2018
[46]

Li, G-uniform stability and K¨ ahler-Einstein metrics on singular Fano varieties, Invent

C. Li, G-uniform stability and K¨ ahler-Einstein metrics on singular Fano varieties, Invent. Math. 227 (2022), 661–744

2022
[47]

C. Li, Y. Liu, C. Xu A guided tour to normalized volume , Geometric Analysis, volume 333 of Progr. Math., pages 167-219. Birkauser/Springer, Cham, 2020

2020
[48]

xviii+387 p 28 V

Springer-Verlag, Berlin, 2004. xviii+387 p 28 V. APOSTOLOV, A. LAHDILI, AND E. LEGENDRE

2004
[49]

Lichnerowicz, G´ eom´ etrie des groupes de transformation, Travaux et Recherches Math´ ematiques 3, Dunod (1958)

A. Lichnerowicz, G´ eom´ etrie des groupes de transformation, Travaux et Recherches Math´ ematiques 3, Dunod (1958)

1958
[50]

Mabuchi, Multiplier Hermitian structures on K¨ ahler manifolds, Nagoya J

T. Mabuchi, Multiplier Hermitian structures on K¨ ahler manifolds, Nagoya J. Math. 170 (2003), 73– 115

2003
[51]

Mabuchi and Y

T. Mabuchi and Y. Nakagawa, New examples of Sasaki–Einstein manifolds , Tohoku Math. J. 65 (2013), 243-252

2013
[52]

Martelli, J

D. Martelli, J. Sparks, S.-T. Yau Sasaki-Einstein manifolds and volume minimisation , Comm. Math. Phys. 280 (2008), 611–673

2008
[53]

Matsushima, Sur la structure du groupe d’hom´ eomorphismes analytiques d’une certaine vari´ et´ e k¨ ahl´ erienne, Nagoya Math

Y. Matsushima, Sur la structure du groupe d’hom´ eomorphismes analytiques d’une certaine vari´ et´ e k¨ ahl´ erienne, Nagoya Math. J. 11 (1957), 145—150

1957
[54]

Jerison, D

D. Jerison, D. and J. M. Lee, Extremals for the Sobolev inequalitiy for the Heisenberg group and the CR Yamabe problem. J. Amer. Math. Soc. 1 (1988), 1—13

1988
[55]

Petersen and W

P. Petersen and W. Wylie, Rigidity of gradient Ricci solitons , Pacific J. Math. 241 (2009), 329–345

2009
[56]

Petrecca, On Sasaki–Ricci solitons and their deformations , Adv

D. Petrecca, On Sasaki–Ricci solitons and their deformations , Adv. Geom. 16 (2016), 57–70

2016
[57]

Ross and R.P

J. Ross and R.P. Thomas An obstruction to the existence of constant scalar curvature K¨ ahler metrics J. Differential Geom. 72 (2006), no. 3, 429-466

2006
[58]

Tian, K¨ ahler–Einstein metrics with positive scalar curvature

G. Tian, K¨ ahler–Einstein metrics with positive scalar curvature. Invent. Math. 130 (1997), 1–37

1997
[59]

Tian, K-stability and K¨ ahler–Einstein metrics

G. Tian, K-stability and K¨ ahler–Einstein metrics. Commun. Pure Appl. Math. 68 (2015), 1085–1156; Corrigendum, 68 (2015), 2082–2083

2015
[60]

Tian and X

G. Tian and X. Zhu, A new holomorphic invariant and uniqueness of K¨ ahler-Ricci solitons, Comment. Math. Helv. 77 (2002) 297–325

2002
[61]

Tian and X

G. Tian and X. Zhu, Uniqueness of K¨ ahler-Ricci solitons, Acta Math. 184 (2000), 271–305

2000
[62]

van Coevering, Examples of asymptotically conical Ricci-flat K¨ ahler manifolds, Math

C. van Coevering, Examples of asymptotically conical Ricci-flat K¨ ahler manifolds, Math. Z. 267 (2011), 465–496

2011
[63]

S. T. Yau, Open problems in Geometry . In: Differential Geometry: Partial Differential Equations on Manifolds. Proc. Sympos. Pure Math., vol. 54, pp. 1–28. AMS, Los Angeles, CA (1990)

1990
[64]

Wang A valuative criterion of K-polystability , arXiv:2406.06176

L. Wang A valuative criterion of K-polystability , arXiv:2406.06176
[65]

Woodward, Moment maps and geometric invariant theory , Les cours du CIRM, Tome 1 (2010) no

C. Woodward, Moment maps and geometric invariant theory , Les cours du CIRM, Tome 1 (2010) no. 1, pp. 55-98

2010
[66]

Zhang, A quantization proof of the uniform Yau–Tian–Donaldson conjecture

K. Zhang, A quantization proof of the uniform Yau–Tian–Donaldson conjecture . J. Eur. Math. Soc. (2023), published online, DOI 10.4171/JEMS/1373. Vestislav Apostolov, D´epartement de Math´ematiques, UQAM, C.P. 8888, Succursale Centre- ville, Montr´eal (Qu´ebec), H3C 3P8, Canad...

2023 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Apostolov, D

V. Apostolov, D. J. Calderbank, The CR geometry of weighted extremal Kahler and Sasaki metrics , Math. Ann. 379 (2021), 1047–1088

work page 2021

[2] [2]

Apostolov, D

V. Apostolov, D. J. Calderbank, E. Legendre, Weighted K-stability of polarized varieties and extremal- ity of Sasaki manifolds , Advances in Math. 391 (2021), Paper No 107969

work page 2021

[3] [3]

Apostolov, S

V. Apostolov, S. Jubert and A. Lahdili, Weighted K-stability and coercivity with applications to ex- tremal K¨ ahler and Sasaki metrics, Geometry & Topology 8 (2023), 3229–3302

work page 2023

[4] [4]

Atiyah, Convexity and commuting Hamiltonians , Bull

M.F. Atiyah, Convexity and commuting Hamiltonians , Bull. London Math. Soc. 14 (1982) 1—15

work page 1982

[5] [5]

Aubin, R´ eduction du cas positif de l’´ equation de Monge-Amp` eere sur les vari´ et´ es k¨ ahl´ eriennes compactes ` a la d´ emonstration d’une in´ egalit´ e, J

Th. Aubin, R´ eduction du cas positif de l’´ equation de Monge-Amp` eere sur les vari´ et´ es k¨ ahl´ eriennes compactes ` a la d´ emonstration d’une in´ egalit´ e, J. Funct. Anal. 57 (1984), 143–153

work page 1984

[6] [6]

Berline, M

M. Berline, M. Vergne, Z´ eros d’un champs de vecteur et classes charact´ eristiques ´ equivarientes, Duke Math. J. 50 (1983), 539–549

work page 1983

[7] [7]

Berman, K-polystability of Q-Fano varieties admitting K¨ ahler-Einstein metrics, Invent

R. Berman, K-polystability of Q-Fano varieties admitting K¨ ahler-Einstein metrics, Invent. Math. 203 (2016), 973—1025

work page 2016

[8] [8]

The projective space has maximal volume among all toric K\"ahler-Einstein manifolds

R. Berman and B. Berndtsson, The projective space has maximal volume among all toric K¨ ahler- Einstein manifolds , arXiv:1112.4445

work page Pith review arXiv