The constant scalar curvature K\"ahler condition is very general

Ruadha\'i Dervan

REVIEW 3 major objections 3 minor 2 cited by

The constant scalar curvature K\"ahler condition is very general

T0 review · 3 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read This paper argues that arc K-semistability is a very general property in flat families of polarized varieties.

desk verdict An ambitious and plausible-looking paper whose main theorem rests on a false lemma about incidence graph closures; the result is unproved and the paper should be rejected, though the strategy may be salvageable. read the letter →

arxiv 2504.15195 v3 pith:CII2Q443 submitted 2025-04-21 math.AG math.DG

Ruadhaí Dervan This is my paper · ORCID

classification math.AGmath.DG MSC 14L2414D2014J10

keywords arcK-semistabilityuniformK-stabilitystabilityofpairsZariskiopennessverygeneralpropertyconstantscalarcurvatureKählermetricsflatfamiliespolarizedvarietiesHilbertscheme

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 sets out to show that arc K-semistability, a stability condition that tests all formal one-parameter degenerations, behaves algebraically in families. The main theorem claims that in any flat family of polarized varieties the locus of arc K-semistable fibres is very general, meaning its complement is a countable union of Zariski-closed subsets, and the same holds for uniform arc K-stability. The route is to prove a general statement: semistability of a pair in the sense of Paul is Zariski open, and stability of pairs is very general. This matters because it gives the first algebraic very-generality result for these conditions in arbitrary polarized families, and it turns known constant scalar curvature Kähler examples into new uniformly arc K-stable varieties by generic deformation.

What carries the argument

The central machinery is stability of pairs: for a group $G$ acting on vector spaces $V$ and $W$, a point $[v:w]\in\mathbb{P}(V\oplus W)$ is semistable when the orbit closure of $[v:w]$ does not meet $\mathbb{P}(0\oplus W)$. The paper proves the semistable locus is Zariski open by considering the incidence graph of the group action and its closure in $Y\times Y$, whose fibres are asserted to be orbit closures; intersecting with $Y\times\widehat{W}$ and projecting makes the degeneracy locus closed. A numerical criterion via arcs, formal maps $\operatorname{Spec}\mathbb{C}((t))$ to $G$, identifies this condition with the Futaki inequalities defining arc K-semistability. On the Hilbert scheme, the condition is expressed through the CM line bundle written as $L_0 - L_1$, so the argument does not require the CM line bundle to be ample.

What would settle it

For the $\mathbb{C}^*$-action $t\cdot(x,y)=(tx,t^{-1}y)$ on $\mathbb{A}^2$, compute the fibre of the closure of the graph over the origin: it contains $(0,y)$ for every $y$, although the orbit of the origin is just the origin. That directly tests the equality $\pi^{-1}(y_1)=\overline{G\cdot y_1}$ used in Proposition 3.1.

Watch

Extended reading notes

Core claim

The central claim is that arc K-semistability is very general in flat families of polarized varieties, and uniform arc K-stability is as well. The proof first shows that semistability of a pair is Zariski open, then uses arc-based numerical criteria to translate pair-semistability at a fixed exponent into arc K-semistability at that exponent. This translation takes place on the Hilbert scheme of subvarieties of projective space, where the CM line bundle is expressed as a difference $L_0 - L_1$ of linearised line bundles, so no positivity of the CM line bundle is required. Combining with the known result that smooth polarized varieties with discrete automorphism group and a constant scalar curvature Kähler metric are uniformly arc K-stable, the paper concludes that the constant scalar curvature Kähler locus is very general in flat families, and that generic deformations of finite covers that break covering symmetry give the first examples of uniformly arc K-stable smooth polarized varieties not known to admit such metrics.

Load-bearing premise

The load-bearing premise is that the closure of the graph of the group action has exactly the orbit closures as fibres; if a fibre over a degenerate point contains extra points coming from smaller orbits, the Zariski-closedness conclusion can fail.

Editorial extensions

If this is right

In any flat family of polarized varieties, arc K-semistability is very general, and uniform arc K-stability behaves the same way.
Semistability of pairs is Zariski open and stability of pairs is very general, supplying the main algebraic ingredient toward Tian's proposed moduli spaces of pairs.
For smooth polarized varieties with discrete automorphism group, the constant scalar curvature Kähler locus is very general in any flat family.
Generic deformations of finite covers of constant scalar curvature Kähler manifolds that break the covering symmetry are uniformly arc K-stable, yielding the paper's advertised examples.
If a fibre admits an arc destabilization whose central fibre is arc K-semistable, then the original fibre must be arc K-semistable.

Reading between the lines

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

If the incidence-graph step is correct, the same pair-stability mechanism should give Zariski openness for any stability theory expressed as semistability of a pair, including the mixed Monge-Ampère and harmonic Chern-Weil situations the paper lists.
The Hilbert-scheme formulation covers singular fibres, so the result should feed into moduli constructions for singular polarized varieties once the remaining moduli ingredients such as boundedness and separatedness are supplied.
Because the argument decouples openness from positivity of the CM line bundle, analogous very-generality results should hold for twisted or weighted K-stability conditions.
A useful check is whether the identification of fibres with orbit closures in Proposition 3.1 survives when orbit dimension drops; the $\mathbb{C}^*$-action $t\cdot(x,y)=(tx,t^{-1}y)$ on $\mathbb{A}^2$ is the minimal case to inspect.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

The paper sets out to prove that arc K-semistability and uniform arc K-stability are very general in arbitrary flat families of polarized varieties, going well beyond the known Fano case. That is a genuinely important goal, and the idea of reducing it to Paul's stability of pairs is the right kind of move. The paper is also honest about what depends on prior work with Reboulet [DR24]; the numerical criteria are proved there, so the self-citation itself is not the problem.

The problem is the new geometry. Proposition 3.1, which claims that the fiber of the closure of the incidence graph over y is the orbit closure of y, is false. The proof implicitly assumes that if g_j·y1_j → y2 and y1_j → y1, then g_j·y1 → y2. That is wrong: the group elements can diverge while the moving points compensate. A concrete example is C* acting on P^1 by λ·[u:v]=[λu:λ^{-1}v]. Take y=[1:0] and x_j=[1:1/j], with λ_j=1/j. Then x_j→y, but λ_j·x_j=[1/j:j]→[0:1]. So the pair ([1:0],[0:1]) lies in the closure of the incidence graph, although [0:1] is not in the orbit closure of [1:0]. The equality of fibers is exactly what identifies the image of Z∩(Y×W) with the degenerate locus, so Proposition 3.1 collapses. Consequently Corollary 3.2 and Theorem 1.2, which assert Zariski openness of semistability of pairs, are not proved. In fact they are false: with C* acting on C^2⊕C^2 with weights 0,-1 on both factors, the point [1:0:1:0] is semistable, but every neighborhood contains points [1:ε:1:δ] whose orbit closure meets P(0⊕W). So the semistable locus is not open.

Because Theorem 1.1 is derived from these statements, the main conclusion is unsupported. There is also a mismatch between the abstract and the body: the abstract claims “first examples of cscK metrics whose existence only follows from the recent solution of Yau–Tian–Donaldson,” while the body says these are varieties not yet known to admit cscK metrics. That should be fixed regardless of the mathematical fate.

The paper deserves a serious referee because the claim is significant and the error is subtle and instructive. But as written, the central result is false, so my recommendation is reject. If the author can repair Lemma 3.1 or replace it with a different openness argument, the strategy may still yield something real.

Referee Report

3 major / 3 minor

Summary. The paper aims to prove that arc K-semistability and uniform arc K-stability are very general properties in flat families of polarized varieties (Theorem 1.1). The strategy is to prove that Paul's semistability of pairs is a Zariski open property (Theorem 1.2) via a geometric argument about orbit-closure incidence (Proposition 3.1), then to translate arc K-stability into pair stability using the numerical criteria of Dervan–Reboulet, and finally to apply this to smooth polarized varieties to obtain new examples relevant to constant scalar curvature Kähler metrics.

Significance. If the results were valid, Theorem 1.1 would be a substantial extension of the Blum–Liu–Xu openness results from the Fano case to arbitrary polarized varieties, and the paper would provide a new construction of uniformly arc K-stable varieties. The paper also usefully packages the arc-based numerical criteria from prior work. However, the central geometric lemma is false, and the main theorem is contradicted by an explicit example. Because the claimed Zariski openness of the semistable pair locus fails, the manuscript cannot stand in its present form.

major comments (3)

[§3.1, Proposition 3.1] The assertion that the fiber of the closure of the incidence graph equals the orbit closure is false. For G=C* acting on P^1 by λ·[u:v]=[λu:λ^{-1}v], take y0=[1:0], y_j=[1:1/j], and λ_j=1/j. Then λ_j·y_j=[1/j:j]→[0:1], so the pair ([1:0],[0:1]) lies in the closure of the incidence graph. But [0:1] is not in the orbit closure of [1:0], which is just {[1:0]}. The proof's step "Thus g_j(y1) converges to y2" is invalid because g_j acts on y1^j, not on y1. This invalidates the identification of the degenerating locus with the image of a closed set.
[Corollary 3.2 / Theorem 1.2] The claimed Zariski openness of semistable pairs is false. Let G=C* act on V⊕W=C^2⊕C^2 with weights 0,-1 on V and 0,-1 on W, and set Ŵ=P(0⊕W). The point p=[1:0:1:0] is fixed, hence semistable. For any ε,δ≠0, the point q=[1:ε:1:δ] satisfies lim_{λ→0} λ·q=[0:ε:0:δ]∈Ŵ, so q is unstable. Thus every neighborhood of the semistable point p contains unstable points, and the semistable locus is not Zariski open. This directly contradicts Theorem 1.2.
[Corollaries 3.5 and 3.6 (Theorem 1.1)] The main theorem depends essentially on the false Corollary 3.2 (and on Corollary 3.3, which uses the same openness assertion). Therefore Theorem 1.1, the very generality of arc K-semistability and uniform arc K-stability, is unsupported. Since the counterexample shows the proposed approach cannot work, these claims are not merely unproved but false as stated.

minor comments (3)

[Definition 2.4] The notation G·[v:w] in the definition of semistability of pairs should specify whether the orbit or its closure is meant. The proof of Proposition 3.1 uses the closure, and the ambiguity is consequential for the numerical criterion invoked from [DR24].
[Throughout] The supplied text contains numerous character-encoding artifacts (for example "C/llparenthesist/rrparenthesis" and similar corrupted sequences). The authors should ensure the final PDF renders all mathematical symbols correctly.
[Proof of Corollary 3.6] The sentence "so their complement is a countable union of Zariski closed subsets" followed by "the complement of a countable union of countable unions" is grammatically tangled and should be reworded for clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: central error is mathematical, not circular; self-citations to [DR24] carry independent prior proofs.

full rationale

Theorem 1.1 is derived from Theorem 1.2 (Zariski openness of semistability of pairs) together with the numerical criteria of [DR24]. The openness claim is not obtained by renaming a known fact or by fitting a parameter to the desired output: Proposition 3.1 attempts an independent geometric argument via the incidence graph, and the reduction from arc K-semistability to pair semistability uses the stated numerical criteria from [DR24]. Those criteria are prior results proved elsewhere, with stated assumptions not including the present conclusion, so the dependency is a substantive citation rather than a circular import. The paper's actual fatal weakness is the false assertion in the proof of Proposition 3.1 that π^{-1}(y1) equals the orbit closure G·y1; the supplied convergence argument incorrectly replaces y1^j by y1. That is a mathematical error, not a circularity: it does not assume its conclusion, it asserts a false identification. Similarly, Corollary 3.3 and Remark 3.4 inherit their validity from the definition of stability via semistability of an associated pair and from Corollary 3.2; these are logical consequences of the definitions rather than circular reductions. The self-citations to [DR24] are numerous and load-bearing, which warrants a small nonzero score, but they do not reduce the derivation to its own inputs. Consequently, the correct circularity finding is essentially negative, with the caution that the central mathematical claim is unsupported for a different reason.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No empirical constants or new physical entities appear. The paper rests on standard algebraic geometry plus several imported results from [DR24]; the one genuinely ad hoc input is the false orbit-closure fiber equality.

assumptions (5)

domain assumption Flat family of polarised varieties with fixed Hilbert polynomial and relatively ample Q-line bundle L; fibres may be singular.
Used throughout to form the Hilbert scheme and Knudsen-Mumford expansions (Section 3.2).
standard math Properness and separatedness of the Hilbert scheme allow extension of C((t))-points to C[[t]]-points (valuative criteria).
Invoked in Corollary 3.5 to turn an arc in GL(N+1) into a model.
standard math Knudsen-Mumford expansion det π_*(kL) = H0 k^{n+1} + H1 k^n + ... for flat families.
Used to define the Donaldson-Futaki invariant and the CM line bundle (Section 2.1).
domain assumption The numerical criteria of [DR24, Theorems 2.8 and 2.10] and the weight-degree comparison [DR24, Lemma 3.7] and norm comparison [DR24, Corollary 4.7] hold in the generality needed here, including for singular varieties.
Central reduction in Corollary 3.5 depends on these cited results from prior work with Reboulet; they are not proved in this paper.
ad hoc to paper The fibre of the closure of the incidence graph over y equals the closure of the orbit G·y.
This assertion in Proposition 3.1 is load-bearing and is false in general; it is the source of the counterexample.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The constant scalar curvature K\"ahler condition is very general." pith.science (2026). https://pith.science/paper/CII2Q443

@misc{pith2026250415195,
  author       = {Pith},
  title        = {Pith review of: The constant scalar curvature K\"ahler condition is very general},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CII2Q443}},
  note         = {Machine review of arXiv:2504.15195}
}

read the original abstract

Recent work of Trusiani implies that the existence of a constant scalar curvature K\"ahler metric on a smooth polarised variety with discrete automorphism group is equivalent to uniform arc K-stability. We prove that uniform arc K-stability is essentially algebraic in flat families of polarised varieties. When the polarised varieties are further smooth and have discrete automorphism group, combining these two results implies that the constant scalar curvature K\"ahler locus is very general. We use this result to give the first examples of constant scalar curvature K\"ahler metrics whose existence only follows from the recent solution of the Yau--Tian--Donaldson conjecture. Our technique is to prove a general result stating that stability of a pair in the sense of Paul is essentially an algebraic property in families, and to employ prior work with Reboulet relating uniform arc K-stability to stability of an associated pair.

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

CscK metrics on birational models of projective varieties
math.DG 2026-08 conditional novelty 8.0 of 10

A sufficiently positive Lefschetz pencil on any smooth projective variety has a base-locus blowup that admits cscK metrics in an explicit adiabatic polarization.
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

37 extracted references · 28 canonical work pages · cited by 2 Pith papers

[1]

Reductivity of the automorphism group of K -polystable Fano varieties

Jarod Alper, Harold Blum, Daniel Halpern-Leistner, and Chenyang Xu. Reductivity of the automorphism group of K -polystable Fano varieties. Invent. Math. , 222(3):995--1032, 2020

work page 2020
[2]

Constant scalar curvature K \"ahler metrics on ramified G alois coverings

Claudio Arezzo, Alberto Della Vedova, and Yalong Shi. Constant scalar curvature K \"ahler metrics on ramified G alois coverings. J. Reine Angew. Math. , 799:229--247, 2023

work page 2023
[3]

Uniform K -stability, D uistermaat- H eckman measures and singularities of pairs

S\'ebastien Boucksom, Tomoyuki Hisamoto, and Mattias Jonsson. Uniform K -stability, D uistermaat- H eckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble) , 67(2):743--841, 2017

work page 2017
[4]

The normalized volume of a singularity is lower semicontinuous

Harold Blum and Yuchen Liu. The normalized volume of a singularity is lower semicontinuous. J. Eur. Math. Soc. (JEMS) , 23(4):1225--1256, 2021

work page 2021
[5]

Openness of uniform K -stability in families of Q - F ano varieties

Harold Blum and Yuchen Liu. Openness of uniform K -stability in families of Q - F ano varieties. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , 55(1):1--41, 2022

work page 2022
[6]

Openness of K -semistability for Fano varieties

Harold Blum, Yuchen Liu, and Chenyang Xu. Openness of K -semistability for Fano varieties. Duke Math. J. , 171(13):2753--2797, 2022

work page 2022
[7]

Uniqueness of K -polystable degenerations of F ano varieties

Harold Blum and Chenyang Xu. Uniqueness of K -polystable degenerations of F ano varieties. Annals of Mathematics , 190(2):609--656, 2019

work page 2019
[8]

Positivity of the CM line bundle for families of K -stable klt F ano varieties

Giulio Codogni and Zsolt Patakfalvi. Positivity of the CM line bundle for families of K -stable klt F ano varieties. Invent. Math. , 223(3):811--894, 2021

2021

Show all 37 references

[9]

Infinitesimal deformations of double covers of smooth algebraic varieties

Slawomir Cynk and Duco van Straten. Infinitesimal deformations of double covers of smooth algebraic varieties. Math. Nachr. , 279(7):716--726, 2006

2006
[10]

Uniform stability of twisted constant scalar curvature K \"ahler metrics

Ruadha\'i Dervan. Uniform stability of twisted constant scalar curvature K \"ahler metrics. Int. Math. Res. Not. IMRN , (15):4728--4783, 2016

2016
[11]

Moduli of polarised manifolds via canonical K \"ahler metrics

Ruadha\'i Dervan and Philipp Naumann. Moduli of polarised manifolds via canonical K \"ahler metrics . Ann. Inst. Fourier (Grenoble) (to appear) , 2018

2018
[12]

Donaldson

Simon K. Donaldson. Scalar curvature and stability of toric varieties. J. Differential Geom. , 62(2):289--349, 2002

2002
[13]

Donaldson

Simon K. Donaldson. Stability, birational transformations and the K \"a hler- E instein problem. In Surveys in differential geometry. V ol. XVII , volume 17 of Surv. Differ. Geom. , pages 203--228. Int. Press, Boston, MA, 2012

2012
[14]

Algebraic families of constant scalar curvature K \"ahler metrics

Simon Donaldson. Algebraic families of constant scalar curvature K \"ahler metrics. In Surveys in differential geometry 2014. R egularity and evolution of nonlinear equations , volume 19 of Surv. Differ. Geom. , pages 111--137. Int. Press, Somerville, MA, 2015

2014
[15]

Arcs, stability of pairs and the Mabuchi functional

Ruadha\'i Dervan and R \'e mi Reboulet . Arcs, stability of pairs and the Mabuchi functional . arXiv e-prints , page arXiv:2409.13617, September 2024

2024 arXiv
[16]

Fine and J

J. Fine and J. Ross. A note on positivity of the CM line bundle. Int. Math. Res. Not. , pages Art. ID 95875, 14, 2006

2006
[17]

The moduli space of extremal compact K \"ahler manifolds and generalized W eil- P etersson metrics

Akira Fujiki and Georg Schumacher. The moduli space of extremal compact K \"ahler manifolds and generalized W eil- P etersson metrics. Publ. Res. Inst. Math. Sci. , 26(1):101--183, 1990

1990
[18]

Boundedness of Q - F ano varieties with degrees and alpha-invariants bounded from below

Chen Jiang. Boundedness of Q - F ano varieties with degrees and alpha-invariants bounded from below. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , 53(5):1235--1248, 2020

2020
[19]

The projectivity of the moduli space of stable curves I : preliminaries on det and D iv

Finn Knudsen and David Mumford. The projectivity of the moduli space of stable curves I : preliminaries on det and D iv. Mathematica Scandinavica , 39(1):19--55, 1976

1976
[20]

On the proper moduli spaces of smoothable K \"ahler- E instein F ano varieties

Chi Li, Xiaowei Wang, and Chenyang Xu. On the proper moduli spaces of smoothable K \"ahler- E instein F ano varieties. Duke Math. J. , 168(8):1387--1459, 2019

2019
[21]

Finite generation for valuations computing stability thresholds and applications to K -stability

Yuchen Liu, Chenyang Xu, and Ziquan Zhuang. Finite generation for valuations computing stability thresholds and applications to K -stability. Ann. Math. (2) , 196(2):507--566, 2022

2022
[22]

Mumford, J

D. Mumford, J. Fogarty, and F. Kirwan. Geometric invariant theory. , volume 34 of Ergeb. Math. Grenzgeb. Berlin: Springer-Verlag, 3rd enl. ed. edition, 1994

1994
[23]

On the moduli of K \"ahler-- E instein F ano manifolds

Yuji Odaka. On the moduli of K \"ahler-- E instein F ano manifolds. Proceeding of Kinosaki Algebraic Geometry Symposium , pages 112--126, 2013

2013
[24]

Moment maps and stability of holomorphic submersions

Annamaria Ortu . Moment maps and stability of holomorphic submersions . arXiv e-prints , page arXiv:2407.03246, July 2024

2024 arXiv
[25]

Hyperdiscriminant polytopes, C how polytopes, and M abuchi energy asymptotics

Sean Timothy Paul. Hyperdiscriminant polytopes, C how polytopes, and M abuchi energy asymptotics. Annals of Mathematics , pages 255--296, 2012

2012
[26]

On the generalized numerical criterion

Sean Timothy Paul, Song Sun, and Junsheng Zhang. On the generalized numerical criterion. Int. Math. Res. Not. IMRN , (17):12150--12160, 2024

2024
[27]

Paul and Gang Tian

Sean T. Paul and Gang Tian . CM Stability and the Generalized Futaki Invariant I . arXiv Mathematics e-prints , page math/0605278, May 2006

2006 arXiv
[28]

A Hilbert--Mumford criterion for generalised Monge--Amp \`e re equations

R \'e mi Reboulet . A Hilbert--Mumford criterion for generalised Monge--Amp \`e re equations . arXiv e-prints , page arXiv:2504.06612, April 2025

2025 arXiv
[29]

K\"ahler- E instein metrics with positive scalar curvature

Gang Tian. K\"ahler- E instein metrics with positive scalar curvature. Invent. Math. , 130(1):1--37, 1997

1997
[30]

Stability of pairs

Gang Tian. Stability of pairs. arXiv preprint arXiv:1310.5544 , 2013

2013 arXiv
[31]

K-stability implies CM -stability

Gang Tian. K-stability implies CM -stability. In Geometry, analysis and probability , volume 310 of Progr. Math. , pages 245--261. 2017

2017
[32]

Height and GIT weight

Xiaowei Wang. Height and GIT weight. Math. Res. Lett. , 19(4):909--926, 2012

2012
[33]

Discriminants and Higher K-energies on Polarized K \"a hler Manifolds

Quinton Westrich . Discriminants and Higher K-energies on Polarized K \"a hler Manifolds . arXiv e-prints , page arXiv:1507.01152, July 2015

2015 arXiv
[34]

A minimizing valuation is quasi-monomial

Chenyang Xu. A minimizing valuation is quasi-monomial. Ann. of Math. (2) , 191(3):1003--1030, 2020

2020
[35]

K-stability of F ano varieties: an algebro-geometric approach

Chenyang Xu. K-stability of F ano varieties: an algebro-geometric approach. EMS Surv. Math. Sci. , 8(1-2):265--354, 2021

2021
[36]

On positivity of the CM line bundle on K -moduli spaces

Chenyang Xu and Ziquan Zhuang. On positivity of the CM line bundle on K -moduli spaces. Ann. Math. (2) , 192(3):1005--1068, 2020

2020
[37]

Open problems in geometry

Shing-Tung Yau. Open problems in geometry. In Differential geometry: partial differential equations on manifolds ( L os A ngeles, CA , 1990) , volume 54 of Proc. Sympos. Pure Math. , pages 1--28. 1993

1990

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Reductivity of the automorphism group of K -polystable Fano varieties

Jarod Alper, Harold Blum, Daniel Halpern-Leistner, and Chenyang Xu. Reductivity of the automorphism group of K -polystable Fano varieties. Invent. Math. , 222(3):995--1032, 2020

work page 2020

[2] [2]

Constant scalar curvature K \"ahler metrics on ramified G alois coverings

Claudio Arezzo, Alberto Della Vedova, and Yalong Shi. Constant scalar curvature K \"ahler metrics on ramified G alois coverings. J. Reine Angew. Math. , 799:229--247, 2023

work page 2023

[3] [3]

Uniform K -stability, D uistermaat- H eckman measures and singularities of pairs

S\'ebastien Boucksom, Tomoyuki Hisamoto, and Mattias Jonsson. Uniform K -stability, D uistermaat- H eckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble) , 67(2):743--841, 2017

work page 2017

[4] [4]

The normalized volume of a singularity is lower semicontinuous

Harold Blum and Yuchen Liu. The normalized volume of a singularity is lower semicontinuous. J. Eur. Math. Soc. (JEMS) , 23(4):1225--1256, 2021

work page 2021

[5] [5]

Openness of uniform K -stability in families of Q - F ano varieties

Harold Blum and Yuchen Liu. Openness of uniform K -stability in families of Q - F ano varieties. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , 55(1):1--41, 2022

work page 2022

[6] [6]

Openness of K -semistability for Fano varieties

Harold Blum, Yuchen Liu, and Chenyang Xu. Openness of K -semistability for Fano varieties. Duke Math. J. , 171(13):2753--2797, 2022

work page 2022

[7] [7]

Uniqueness of K -polystable degenerations of F ano varieties

Harold Blum and Chenyang Xu. Uniqueness of K -polystable degenerations of F ano varieties. Annals of Mathematics , 190(2):609--656, 2019

work page 2019

[8] [8]

Positivity of the CM line bundle for families of K -stable klt F ano varieties

Giulio Codogni and Zsolt Patakfalvi. Positivity of the CM line bundle for families of K -stable klt F ano varieties. Invent. Math. , 223(3):811--894, 2021

2021