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On K-stability of Fano's last Fanos

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Pith's one-line read The paper proves an explicit equivariant criterion for K-stability of Fano's last Fanos, and classifies their automorphism groups.

desk verdict Explicit, checkable K-stability criterion for Fano's last Fanos, plus a full automorphism group classification; the proof is careful but leans on an imported lemma the referee must verify. read the letter →

arxiv 2507.08528 v2 pith:W3FM4B6P submitted 2025-07-11 math.AG

Ivan Cheltsov , DongSeon Hwang , Alex Küronya , Antonio Laface , Frédéric Mangolte , Alex Massarenti , Jihun Park , Junyan Zhao This is my paper · ORCID
classification math.AG MSC 14J4514J3014J5032Q2014L24
keywords K-stabilityFanothreefoldsKähler–EinsteinmetricsconicbundlesdiscriminantquarticdelPezzosurfacesautomorphismgroupsequivariant
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

Fano's last Fanos are smooth Fano threefolds of degree 22 and Picard rank 2 obtained by blowing up a complete intersection of two quadrics in $\mathbb{P}^5$ along a conic. This paper gives a checkable criterion for when one of them is K-stable, phrased in terms of the discriminant quartic $\Delta$ of the conic bundle $g\colon X\to\mathbb{P}^2$: if a finite group $G\subset\operatorname{Aut}(X)$ fixes no $k$-point of $\operatorname{Sing}(\Delta)$, then the geometric model $X_{\mathbb{C}}$ is K-stable and therefore carries a K\"ahler\textendash Einstein metric. The paper also shows that every such threefold has finite automorphism group and that the possible nontrivial groups are exactly 17 finite groups. A sympathetic reader should care because K-stability is the algebro-geometric condition controlling the existence of these metrics and the structure of K-moduli spaces.

What carries the argument

The carrying mechanism is the admissible-flag method for lower-bounding local stability thresholds. For each point $P$, the paper chooses a general surface $S$ in $|g^{*}\mathcal{O}_{\mathbb{P}^2}(1)|$ containing $P$; apart from one special case, $S$ is a smooth del Pezzo surface of degree $4$, with ${\left.-K_X\right|}_{S}=-K_S$, and the fiber $C$ of the conic bundle through $P$ falls into smooth, reducible, or non-reduced cases. The argument computes the Zariski decomposition of $-K_X-uS$, reduces local invariants to explicit integrals of volumes on $S$ and on curves inside it, and feeds in lower bounds for $\delta_P(S,D_t)$ on del Pezzo surfaces. The output is explicit constants such as $\delta_P(X)\ge 176/171$ at points lying over nonsingular points of $\Delta$, with the remaining special configurations treated separately.

What would settle it

Take an explicit Fano's last Fano satisfying the Main Theorem's hypothesis, for instance the $A_4$-symmetric example from [7, Section 5.6] over $k=\mathbb{Q}$ with $G=A_4$, and compute $\delta(X_{\mathbb{C}})$ or check the $\beta$-criterion. If any $G$-invariant divisor $E$ over $X$ has $\beta(E)\le 0$, or if a local threshold from Propositions 5.2.3, 5.2.4, or 5.2.6 drops below $1$ at a point lying over a nonsingular point of $\Delta$, the theorem fails. A cheaper check is to test inequalities (5.1) and (5.2) on every degree-$4$ del Pezzo surface $S\in |g^{*}\mathcal{O}_{\mathbb{P}^2}(1)|$; a single violation would invalidate Section 5.3.

Watch

Extended reading notes

Core claim

The central claim is the Main Theorem: if $X$ is a smooth Fano threefold in Family no. 2.16 defined over a subfield $k\subset\mathbb{C}$, $G$ is a finite subgroup of $\operatorname{Aut}(X)$, and $G$ fixes no $k$-point of $\operatorname{Sing}(\Delta)$, where $\Delta$ is the discriminant quartic of the conic bundle $g\colon X\to\mathbb{P}^2$, then the geometric model $X_{\mathbb{C}}$ is K-stable. The proof argues by contradiction: if $X_{\mathbb{C}}$ were not K-polystable, equivariant K-stability results produce a $G$-invariant geometrically irreducible divisor $E$ over $X$ with $\beta(E)\le 0$ whose center is not a surface. Projecting that center to $\mathbb{P}^2$ yields a $k$-point that is not singular on $\Delta$, and at such a point the local $\delta$-invariant is shown to be strictly larger than $1$, contradicting $\delta_{P}(X_{\mathbb{C}})\le 1$. Since $\operatorname{Aut}(X)$ is finite, K-polystability upgrades to K-stability.

Load-bearing premise

The proof leans on an imported lemma stating that on every degree-$4$ del Pezzo surface obtained here, the local stability threshold $\delta_P(S,D_t)$ stays above the explicit rational bounds (5.1) and (5.2); if that lemma fails for even one such surface, the inequalities $S(W^S_{\bullet,\bullet};F)\le \gamma A_S(F)$ can fail and the contradiction in Section 5.3 collapses.

Editorial extensions

If this is right

  • If the Main Theorem is correct, K-stability of an explicitly given Fano's last Fano is decided by a finite computation: form the discriminant quartic $\Delta$, inspect $\operatorname{Sing}(\Delta)$ over $k$, and apply the criterion.
  • A smooth discriminant quartic forces K-stability; so does absence of $k$-points on $X$, and so does the condition that the symmetry group fixes no $k$-point of $\Delta$ or of $X$.
  • Together with the established equivalence between K-polystability and existence of K\"ahler\textendash Einstein metrics, every member satisfying the hypothesis admits a K\"ahler\textendash Einstein metric.
  • The classification of automorphism groups into 17 finite groups makes the hypothesis checkable from explicit equations for the threefold and its invariant planes.
  • The openness of K-stability turns the criterion into a Zariski-open locus of K-stable members, making the earlier non-effective statement for general members effective.

Reading between the lines

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

  • The method stops at non-reduced fibers $C=2L$, where the surface-level computation gives $S(W^S_{\bullet,\bullet};L)=31/22>1$ and does not prove $\delta_P(X)>1$; extending the local computation to this case is the natural next step toward deciding K-stability for all Fano's last Fanos.
  • Because the hypothesis is phrased in terms of $k$-points of the discriminant quartic, the theorem creates a direct bridge between rational-point behavior of $\Delta$ and analytic geometry: fields that force rational points on $\Delta$ may obstruct K-stability, while pointless discriminants guarantee it.
  • If the conjectured identification of the K-moduli space with a GIT quotient is established, the explicit threshold criterion would describe which boundary points of the K-moduli stack occur, identifying the K-semistable locus inside the parameter space as a GIT-semistable locus.
  • The short list of 17 automorphism groups suggests that one can in principle enumerate all pairs $(X,G)$ and verify the Main Theorem's hypothesis group by group, turning K-stability for this family into a finite classification statement.
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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 / 4 minor

Summary. The paper studies smooth Fano threefolds of Picard rank 2 and degree 22 that arise as blow-ups of a smooth (2,2)-complete intersection in P^5 along a conic (Family №2.16). Its main theorem gives an explicit criterion for K-stability: if X is defined over a subfield k⊂C, G is a finite subgroup of Aut(X), and G fixes no k-point of Sing(Δ), where Δ is the discriminant quartic of the conic bundle g:X→P^2, then the geometric model X_C is K-stable. The proof combines Zhuang's equivariant K-stability theorem with the Abban–Zhuang method: for each point P with g(P) outside Sing(Δ), the local δ-invariant is bounded below by explicit constants >1 via Zariski decomposition computations on a degree-4 del Pezzo surface S∈|g^*O(1)|. The paper also classifies the possible nontrivial automorphism groups of these threefolds as 17 finite groups, using a mix of manual arguments and Magma computations.

Significance. If the main theorem is correct, it provides an effective, checkable K-stability criterion for a large class of Fano threefolds, going substantially beyond the non-effective openness argument in the existing literature. The paper contains detailed and explicit computations of local δ-invariants, with all Zariski decompositions spelled out, together with reproducible Magma code for the automorphism-group classification. The main risk is the unproved import of the local del Pezzo estimates (5.1) and (5.2) from [27]; these bounds are load-bearing for the main theorem. Apart from that risk, the central argument is coherent and the result is significant.

major comments (2)
  1. [§5.2, Propositions 5.2.3 and 5.2.4] The lower bounds for δ_P(X) are obtained from the inequalities (5.1) and (5.2), quoted from [27, Lemma 24] without proof and without a detailed verification that the hypotheses of that lemma hold for every degree-4 del Pezzo surface S that arises here, including the cases where C is reducible, where P∈E, and the weak del Pezzo case treated in Proposition 5.2.6. These inequalities feed directly into the displayed coefficients γ (for example 176/161, 176/169, 88/85, 88/89); if they fail, the inequalities S(W^S_{•,•};F) ≤ γ A_S(F) are unsupported and the contradiction in §5.3 collapses. Please either prove (5.1)–(5.2) in the present setting or state Lemma 24 in full and verify its hypotheses case by case.
  2. [§5.3] The final argument proves that X_C is K-polystable: Theorem 2.6.3 gives β(E)>0 for G-invariant divisors, hence K-polystability. To conclude K-stability rather than merely K-polystability, one must also know that Aut(X_C) is finite; this follows from Theorem 4.3.1, but that theorem is not cited in the paragraph. Please add the explicit step using Corollary 2.2.5.
minor comments (4)
  1. [§5.2, Proposition 5.2.3] In the case split at the end of the proof, the final sentence says “if P∈E and C is irreducible”; the coefficients 1/22, 19/22 and 9/88 come from the reducible case, so “irreducible” should read “reducible”.
  2. [§5.2, Proposition 5.2.4] The computation at the end of the proof states S(W^{S,G}_{•,•,•};O)=37/88+9/88=23/4; the correct sum is 23/44, which is what the preceding inequalities require.
  3. [§5.2, Proposition 5.2.4] The proof asserts without proof or reference that when P lies in Γ∩L_2, the three curves L_1, L_2 and Γ are pairwise transverse at P and satisfy L_1+L_2+Γ∼−K_S. This geometric input is used for the subsequent blow-up Zariski decompositions; please add a justification or a precise citation.
  4. [§4.3, Theorem 4.3.1] The classification proof relies on Magma computations and a final manual verification; the code is included in the appendices and is publicly available, but the manual verification step is described only briefly. A short explanation of how the potential inclusion of one group in another was ruled out in the remaining cases would improve reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation found: K-stability is derived from external criteria and from δ-invariant estimates that are computed from scratch; the load-bearing local bounds are cited from published [27], not defined into existence.

full rationale

The paper's Main Theorem is not obtained by assuming its conclusion. The proof in §5.3 reduces K-polystability of X_C to positivity of β-invariants via the external theorems of Fujita–Li, Blum–Xu, and Zhuang ([50, 69, 104]), then estimates the local stability threshold δ_P(X) using the Abban–Zhuang admissible flag machinery ([3, 7]). The quantities S_X(S) = 13/22, the Zariski decompositions of P(u)|_S, and the displayed coefficients γ = 176/161, 176/169, 88/85, 88/89, 176/171, 22/19 are all computed explicitly in Propositions 5.2.3, 5.2.4, and 5.2.6; they are not fitted to the conclusion. The inequalities (5.1) and (5.2) for δ_P(S, D_t) are quoted from [27, Lemma 24], a published paper whose authors overlap with the present authors. This is a self-citation, but it is not circular: the lemma is an independent external result, and the present paper does not define δ_P(X) in terms of it. The paper does not verify the hypotheses of [27, Lemma 24] in every case that arises, which is a correctness risk rather than a circularity. The automorphism group classification (Theorem 4.3.1) is computed from the pencil of quadrics and verified with Magma code plus a manual check; it is not imported as a uniqueness theorem to force the K-stability conclusion. Minor self-citations such as [1, Lemma 5.7] and [2, Lemma 4.5] appear only in corollaries and are not load-bearing for the Main Theorem. Overall, no step reduces by construction to its own input, so the paper is not circular; at most there are minor non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard external theorems in K-stability and birational geometry, and on one auxiliary lemma from the authors' earlier work [27]. No numerical constants are fitted to data: gamma, 176/171, and 22/19 are proven lower bounds, not fitted values. No new entities are introduced.

assumptions (6)
  • standard math Yau-Tian-Donaldson equivalence: a smooth Fano manifold is Kähler-Einstein iff K-polystable (Theorems 2.3.1, 2.5.5).
    Invoked in Section 2.3 and throughout; external theorem.
  • standard math Fujita-Li valuative criterion and Abban-Zhuang admissible flag estimates (Theorems 2.4.2, 5.1.1, 5.1.3, 5.1.4, 5.1.6).
    Used to convert K-stability to local delta-invariant lower bounds; no proof reproduced.
  • standard math Zhuang's equivariant K-stability results [104], especially Theorems 2.6.3 and 2.6.4.
    Foundational for reducing K-polystability of X_C to G-invariant divisors over X.
  • domain assumption Fano threefolds are Mori dream spaces, hence effective divisors have CKM Zariski decompositions on X (Corollary 3.2.7, Theorem 3.2.10).
    Needed for volume and Zariski decomposition computations in Section 5.2.
  • domain assumption Lemma 24 of [27] gives lower bounds for delta_P(S,D_t) on del Pezzo surfaces of degree 4 with a conic (equations (5.1), (5.2)).
    External lemma with overlapping authors; central to Proposition 5.2.3 estimates.
  • standard math For a non-K-polystable Fano variety, there exists a destabilizing divisor whose center is not a surface ([49]).
    Used in Section 5.3 to restrict possible centers Z.

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Pith. "Pith review of On K-stability of Fano's last Fanos." pith.science (2026). https://pith.science/paper/W3FM4B6P

@misc{pith2026250708528,
  author       = {Pith},
  title        = {Pith review of: On K-stability of Fano's last Fanos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W3FM4B6P}},
  note         = {Machine review of arXiv:2507.08528}
}
abstract

We study K-stability of smooth Fano threefolds of Picard rank $2$ and degree $22$ which can be obtained by blowing up a smooth complete intersection of two quadrics in $\mathbb{P}^5$ along a conic. We also describe the automorphism groups of these threefolds.

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Works this paper leans on

104 extracted references · 77 canonical work pages

  1. [27]

    Cheltsov, K

    I. Cheltsov, K. Fujita, T. Kishimoto, T. Okada,K-stable divisors inP1 ×P 1 ×P 2 of degree(1,1,2), Nagoya Mathematical Journal251(2023), 686–714

    work page 2023

  2. [1]

    K-stability of pointless del Pezzo surfaces and Fano 3-folds

    H. Abban, I. Cheltsov, T. Kishimoto, F. Mangolte,K-stability of pointless del Pezzo surfaces and Fano 3-folds, preprint, arXiv:2411.00767, 2024

    work page Pith review arXiv 2024

  3. [2]

    Abban, I

    H. Abban, I. Cheltsov, T. Kishimoto, F. Mangolte,K-stability of Fano 3-folds in the World of Null-A, preprint, arXiv:2505.04330, 2025

    arXiv 2025

  4. [3]

    Abban, Z

    H. Abban, Z. Zhuang,K-stability of Fano varieties via admissible flags, Forum of Mathematics Pi10(2022), 1–43

    2022

  5. [4]

    Abban, Z

    H. Abban, Z. Zhuang,Seshadri constants and K-stability of Fano manifolds, Duke Mathematical Journal172 (2023) 1109–1144

    2023

  6. [5]

    Alper, H

    J. Alper, H. Blum, D. Halpern-Leistner, Ch. Xu,Reductivity of the automorphism group ofK-polystable Fano varieties, Invent. Math.222(2020), 995–1032

    2020

  7. [6]

    Andreatta, R

    M. Andreatta, R. Pignatelli,Fano’s last Fano, Rendiconti Acc. Naz. Lincei34(2023), 359–381

    2023

  8. [7]

    Araujo, A.-M

    C. Araujo, A.-M. Castravet, I. Cheltsov, K. Fujita, A.-S. Kaloghiros, J. Martinez-Garcia, C. Shramov, H. Süß, N. Viswanathan,The Calabi problem for Fano 3-folds, Lecture Notes in Mathematics, Cambridge University Press,485(2023)

    2023

Show all 104 references
  1. [8]

    Artebani, A

    M. Artebani, A. Laface,Hypersurfaces in Mori dream spaces, J. Algebra371(2012), 26–37

    2012

  2. [9]

    Arzhantsev, U

    I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface,Cox rings, Cambridge Studies in Advanced Mathematics, Vol. 144, Cambridge University Press, 2015

    2015

  3. [10]

    Avilov,Automorphisms of threefolds that can be represented as an intersection of two quadrics, Sbornik: Mathematics207(2016), 315–330

    A. Avilov,Automorphisms of threefolds that can be represented as an intersection of two quadrics, Sbornik: Mathematics207(2016), 315–330

    2016

  4. [11]

    Aubin,Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull

    Th. Aubin,Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math.102 (1978), 63–95

    1978

  5. [12]

    Bauer,A simple proof for the existence of Zariski decompositions on surfaces, J

    Th. Bauer,A simple proof for the existence of Zariski decompositions on surfaces, J. Algebraic Geom.18 (2009), 789–793

    2009

  6. [13]

    Bauer, A

    Th. Bauer, A. Küronya, T. Szemberg,Zariski chambers, volumes, and stable base loci, Journal für die reine und angewandte Mathematik576(2004), 209–233

    2004

  7. [14]

    Belousov, K

    G. Belousov, K. Loginov,K-stability of Fano threefolds of rank4and degree24, European Journal of Mathe- matics9(2023), no. 3, Paper No. 80. 73

    2023

  8. [15]

    Berman, S

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

    2021

  9. [16]

    Birkar, P

    C. Birkar, P. Cascini, Paolo, C. D. Hacon, J. McKernan,Existence of minimal models for varieties of log general type, Journal of the American Mathematical Society23(2010), no. 2, 405–468

    2010

  10. [17]

    H. Blum, D. Halpern-Leistner, Y. Liu, and C. Xu,On properness of K-moduli spaces and optimal degenerations of Fano varieties, Selecta Math.27(2021), article number 73

    2021

  11. [18]

    H. Blum, M. Jonsson,Thresholds, valuations, and K-stability, Advances in Mathematics365(2020), article ID 107062

    2020

  12. [19]

    H. Blum, Y. Liu,Openness of uniform K-stability in families ofQ-Fano varieties, Annales Scientifiques de l’Ecole Normale Superieure55(2022), 1–41

    2022

  13. [20]

    H. Blum, Y. Liu, and C. Xu,Openness of K-semistability for Fano varieties, Duke Math. J.171(2022), 2753–2797

    2022

  14. [21]

    Blum, Ch.Xu,Uniqueness of K-polystable degenerations of Fano varieties, Ann

    H. Blum, Ch.Xu,Uniqueness of K-polystable degenerations of Fano varieties, Ann. of Math.190(2019), 609–656

    2019

  15. [22]

    J. P. Bourguignon,Métriques d’Einstein-Kähler sur les variétés de Fano: obstructions et existence, Astérisque 245(1997), 277–305, Séminaire Bourbaki, Exp. No. 830

    1997

  16. [23]

    Cascini, V

    P. Cascini, V. Lazić,New outlook on the minimal model program. I., Duke Mathematical Journal161(2012), no. 12., 2415–2467

    2012

  17. [24]

    Cheltsov,Log canonical thresholds on hypersurfaces, Sbornik: Mathematics192(2001), 1241–1257

    I. Cheltsov,Log canonical thresholds on hypersurfaces, Sbornik: Mathematics192(2001), 1241–1257

    2001

  18. [25]

    Cheltsov,Log canonical thresholds of del Pezzo surfaces, Geometric and Functional Analysis11(2008), 1118–1144

    I. Cheltsov,Log canonical thresholds of del Pezzo surfaces, Geometric and Functional Analysis11(2008), 1118–1144

    2008

  19. [26]

    Cheltsov, E

    I. Cheltsov, E. Denisova, K. Fujita,K-stable smooth Fano threefolds of Picard rank two, Forum Math. Sigma 12(2024), Paper No. e41

    2024

  20. [28]

    Cheltsov, J

    I. Cheltsov, J. Park,Total log-canonical thresholds and generalized Eckardt points, Sbornik: Mathematics193 (2002), 779–789

    2002

  21. [29]

    Cheltsov, J

    I. Cheltsov, J. Park,K-stable Fano threefolds of rank2and degree30, European Journal of Mathematics8 (2022), no. 3, 834-852

    2022

  22. [30]

    Cheltsov, V

    I. Cheltsov, V. Przyjalkowski, C. Shramov,Fano threefolds with infinite automorphism groups, Izvestiya: Math- ematics83(2019), 860–907

    2019

  23. [31]

    Cheltsov, C

    I. Cheltsov, C. Shramov,Log canonical thresholds of smooth Fano threefolds, Russian Mathematical Surveys 63(2008), 859–958

    2008

  24. [32]

    X. Chen, S. Donaldson, S. Sun,Kähler-Einstein metrics on Fano manifolds. I, II, III, J. Amer. Math. Soc.28 (2015), 183–278

    2015

  25. [33]

    Codogni, Z

    G. Codogni, Z. Patakfalvi,Positivity of the CM line bundle for families of K-stable klt Fano varieties, Invent. Math.223(2021), 811–894

    2021

  26. [34]

    Cutkosky,Zariski decomposition of divisors on algebraic varieties, Duke Math

    S. Cutkosky,Zariski decomposition of divisors on algebraic varieties, Duke Math. J.53, 149–156 (1986)

    1986

  27. [35]

    De Biase, E

    L. De Biase, E. Fatighenti, F. Tanturri,Fano 3-folds from homogeneous vector bundles over Grassmannians, Revista Matemetica Complutense35(2022), 649–710

    2022

  28. [36]

    de Fernex, A

    T. de Fernex, A. Küronya, R. Lazarsfeld,Higher cohomology of divisors on a projective variety, Math. Ann. 337(2007), no. 2., 443–445

    2007

  29. [37]

    Delcroix,Examples of K-unstable Fano manifolds, Annales de l’Institut Fourier72(2022), 2079–2108

    T. Delcroix,Examples of K-unstable Fano manifolds, Annales de l’Institut Fourier72(2022), 2079–2108

    2022

  30. [38]

    Denisova,On K-stability ofP3 blown up along the disjoint union of a twisted cubic curve and a line, preprint, arXiv:2202.04421, 2022

    E. Denisova,On K-stability ofP3 blown up along the disjoint union of a twisted cubic curve and a line, preprint, arXiv:2202.04421, 2022

    2022 arXiv

  31. [39]

    Derenthal, J

    U. Derenthal, J. Hausen, A. Heim, S. Keicher, A. Laface,Cox rings of cubic surfaces and Fano threefolds, J. Algebra436(2015), 228–276

    2015

  32. [40]

    Dolgachev, V

    I. Dolgachev, V. Iskovskikh,Finite subgroups of the plane Cremona group, Progress in Mathematics269(2009), 443–548

    2009

  33. [41]

    Dolgachev,Classical algebraic geometry

    I. Dolgachev,Classical algebraic geometry. A modern view, Cambridge University Press, 2012

    2012

  34. [42]

    Donaldson,Scalar curvature and stability of toric varieties, J

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

    2002

  35. [43]

    Donaldson,Algebraic families of constant scalar curvature Kähler metrics, Surveys in Differential Geometry 19(2015), 111–137

    S. Donaldson,Algebraic families of constant scalar curvature Kähler metrics, Surveys in Differential Geometry 19(2015), 111–137

    2015

  36. [44]

    Eyssidieux, V

    Ph. Eyssidieux, V. Guedj, A.Zeriahi,Singular Kähler–Einstein metrics, J. Amer. Math. Soc.22(2009), 607– 639. 74

    2009

  37. [45]

    Eyssidieux,Métriques de Kähler-Einstein sur les variétés de Fano [d’après Chen-Donaldson-Sun et Tian], Astérisque (2016), no

    Ph. Eyssidieux,Métriques de Kähler-Einstein sur les variétés de Fano [d’après Chen-Donaldson-Sun et Tian], Astérisque (2016), no. 380, Exp. No. 1095, 207–229

    2016

  38. [46]

    Fano,Su una particolare varieta algebrica a tre dimensioni aventi curve sezioni canoniche, Rendiconti Acc

    G. Fano,Su una particolare varieta algebrica a tre dimensioni aventi curve sezioni canoniche, Rendiconti Acc. Naz. Lincei6(1949), 151–156

    1949

  39. [47]

    Belmans,Fanography,https://fanography.info, 2025

    P. Belmans,Fanography,https://fanography.info, 2025

    2025

  40. [48]

    Fujita,On singular Del Pezzo varieties, In Algebraic Geometry

    T. Fujita,On singular Del Pezzo varieties, In Algebraic Geometry. Lecture Notes in Mathematics, vol 1417. Springer, Berlin, Heidelberg, (1990) 117–128

    1990

  41. [49]

    Fujita,On K-stability and the volume functions ofQ-Fano varieties, Proceedings of the LMS113(2016), 541–582

    K. Fujita,On K-stability and the volume functions ofQ-Fano varieties, Proceedings of the LMS113(2016), 541–582

    2016

  42. [50]

    Fujita,A valuative criterion for uniform K-stability ofQ-Fano varieties, Journal für die Reine und Ange- wandte Mathematik751(2019), 309–338

    K. Fujita,A valuative criterion for uniform K-stability ofQ-Fano varieties, Journal für die Reine und Ange- wandte Mathematik751(2019), 309–338

    2019

  43. [51]

    Fujita,On K-stability for Fano threefolds of rank3and degree28, International Mathematics Research Notices, IMRN 2023, no

    K. Fujita,On K-stability for Fano threefolds of rank3and degree28, International Mathematics Research Notices, IMRN 2023, no. 15, 12601-12784

    2023

  44. [52]

    Fujita, Y

    K. Fujita, Y. Odaka,On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J.70 (2018), 511–521

    2018

  45. [53]

    Hausen, A

    J. Hausen, A. Laface, Ch. Mauz,On smooth Fano fourfolds of Picard number two, Revista Matematica Iberoamericana38(2022), 53–93

    2022

  46. [54]

    Jiang,Boundedness ofQ-Fano varieties with degrees and alpha-invariants bounded from below, Ann

    C. Jiang,Boundedness ofQ-Fano varieties with degrees and alpha-invariants bounded from below, Ann. Sci. Ec. Norm. Super. (4)53(2020), 1235–1248

    2020

  47. [55]

    Kaloghiros, A

    A.-S. Kaloghiros, A. Küronya, V. Lazić,Finite generation and geography of models, Adv. Stud. Pure Math.70 (2016), 215–245

    2016

  48. [56]

    Kaloghiros, Y

    A.-S. Kaloghiros, Y. Liu, A. Petracci, J .Zhao,The boundary of K-moduli of prime Fano threefolds of genus twelve, preprint, available at arXiv:2603.29827

    arXiv

  49. [57]

    Kawamata, K

    Yu. Kawamata, K. Matsuda, K. Matsuki,Introduction to the minimal model problem, Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 283-360 (1987)

    1987

  50. [58]

    det” and “Div

    F. Knudsen, D. Mumford,The projectivity of the moduli space of stable curves. I: Preliminaries on “det” and “Div”, Math. Scand.39, 19-55 (1976)

    1976

  51. [59]

    Kollár, Sh

    J. Kollár, Sh. Mori,Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Trans- lated from the 1998 Japanese original

    1998

  52. [60]

    Kollár,Singularities of pairs, Algebraic geometry—Santa Cruz 1995, Proc

    J. Kollár,Singularities of pairs, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Part 1, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287

    1995

  53. [61]

    Kollár,Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol

    J. Kollár,Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013, With a collaboration of Sándor Kovács

    2013

  54. [62]

    Kollár,Families of varieties of general type, Cambridge Tracts in Mathematics, vol

    J. Kollár,Families of varieties of general type, Cambridge Tracts in Mathematics, vol. 231, Cambridge Uni- versity Press, Cambridge, 2023, With the collaboration of Klaus Altmann and Sándor J. Kovács

    2023

  55. [63]

    Küronya,Asymptotic cohomological functions on projective varieties, Amer

    A. Küronya,Asymptotic cohomological functions on projective varieties, Amer. J. Math.128(2006), no. 6., 1475–1519

    2006

  56. [64]

    Küronya, V

    A. Küronya, V. Lozovanu,Geometric aspects of Newton-Okounkov bodies, Banach Center Publications116 (2018), 137–212

    2018

  57. [65]

    Küronya, V

    A. Küronya, V. Lozovanu, C. Maclean,Volume functions of linear series, Mathematische Annalen,356(2013), 635–652

    2013

  58. [66]

    Kuznetsov, Yu

    A. Kuznetsov, Yu. Prokhorov, C. Shramov,Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Japanese Journal of Mathematics13(2018), 109–185

    2018

  59. [67]

    Lazarsfeld,Positivity in algebraic geometry

    R. Lazarsfeld,Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 48, Springer Verlag, Berlin, 2004

    2004

  60. [68]

    Lazarsfeld, M

    R. Lazarsfeld, M. Mustaţă,Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér.42(2009), 783–835

    2009

  61. [69]

    Li,K-semistability is equivariant volume minimization, Duke Mathematical Journal166(2017), 3147–3218

    C. Li,K-semistability is equivariant volume minimization, Duke Mathematical Journal166(2017), 3147–3218

    2017

  62. [70]

    C. Li, X. Wang, C. Xu,Algebraicity of the metric tangent cones and equivariant K-stability, J. Amer. Math. Soc.34(2021), 1175–1214

    2021

  63. [71]

    C. Li, Ch. Xu,Special test configuration and K-stability of Fano varieties, Ann. of Math. (2)180(2014), no. 1, 197–232

    2014

  64. [72]

    Liu,The volume of singular Kähler-Einstein Fano varieties, Compos

    Y. Liu,The volume of singular Kähler-Einstein Fano varieties, Compos. Math.154(2018), no. 6, 1131–1158

    2018

  65. [73]

    Liu,K-stability of cubic fourfolds, J

    Y. Liu,K-stability of cubic fourfolds, J. Reine Angew. Math.786(2022), 55-77. 75

    2022

  66. [74]

    Liu,K-stability of Fano threefolds of rank2and degree14as double covers, Mathematische Zeitschrift303 (2023), no

    Y. Liu,K-stability of Fano threefolds of rank2and degree14as double covers, Mathematische Zeitschrift303 (2023), no. 2, Paper No. 38

    2023

  67. [75]

    Y. Liu, C. Xu,K-stability of cubic threefolds, Duke Math. J.168(2019), no. 11, 2029–2073

    2019

  68. [76]

    Y. Liu, C. Xu, Z. Zhuang,Finite generation for valuations computing stability thresholds and applications to K-stability, Annals of Mathematics196(2022), 507–566

    2022

  69. [77]

    Y. Liu, J. Zhao,K-moduli of Fano threefolds and genus four curves, J. Reine Angew. Math. 2025.https: //doi.org/10.1515/crelle-2025-0016

    2025 doi

  70. [78]

    Matsushima,Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne, Nagoya Math

    Y. Matsushima,Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne, Nagoya Math. J.11(1957), 145–150

    1957

  71. [79]

    Mukai,Biregular classification of Fano 3-folds and Fano manifolds of coindex3, PProceedings of the National Academy of Sciences of the USA86(1989), 3000–3002

    S. Mukai,Biregular classification of Fano 3-folds and Fano manifolds of coindex3, PProceedings of the National Academy of Sciences of the USA86(1989), 3000–3002

    1989

  72. [80]

    Nakayama,Zariski-decomposition and abundance, MSJ Memoirs vol

    N. Nakayama,Zariski-decomposition and abundance, MSJ Memoirs vol. 14, Mathematical Society of Japan, Tokyo, 2004

    2004

  73. [81]

    Odaka,On the moduli of Kähler–Einstein Fano manifolds, Proceeding of Kinosaki Symposium (2013), 112–126

    Y. Odaka,On the moduli of Kähler–Einstein Fano manifolds, Proceeding of Kinosaki Symposium (2013), 112–126

    2013

  74. [82]

    Odaka,The GIT stability of polarized varieties via discrepancy, Ann

    Y. Odaka,The GIT stability of polarized varieties via discrepancy, Ann. of Math.177(2013), no. 2, 645–661

    2013

  75. [83]

    Odaka,On the moduli of Kähler-Einstein Fano manifolds, Proc

    Y. Odaka,On the moduli of Kähler-Einstein Fano manifolds, Proc. Kinosaki symposium, 2013, pp. 112–126

    2013

  76. [84]

    Odaka, Y

    Y. Odaka, Y. Sano,Alpha invariant and K-stability ofQ-Fano varieties, Advances in Mathematics229(2012), 2818–2834

    2012

  77. [85]

    Odaka, C

    Y. Odaka, C. Spotti and S. Sun,Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics, J. Differential Geom.102(2016), no. 1, 127–172

    2016

  78. [86]

    Yu. G. Prokhorov,On the Zariski decomposition problem, in: Birational geometry: Linear systems and finitely generated algebras. Collected papers. Transl. from the Russian, Moskva: Maik Nauka/Interperiodika, 2003, 37–65

    2003

  79. [87]

    Spotti, S

    C. Spotti, S. Sun,Explicit Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds, Pure Appl. Math. Q.13(2017), no. 3, 477–515

    2017

  80. [88]

    Szekelyhidi,Kähler–Einstein metrics, Modern geometry: a celebration of the work of Simon Donaldson, Proc

    G. Szekelyhidi,Kähler–Einstein metrics, Modern geometry: a celebration of the work of Simon Donaldson, Proc. Sympos. Pure Math.99(2018), 331–361

    2018

  81. [89]

    Tian,On Kähler–Einstein metrics on certain Kähler manifolds withc1(M)>0, Inventiones Mathematicae 89(1987), 225–246

    G. Tian,On Kähler–Einstein metrics on certain Kähler manifolds withc1(M)>0, Inventiones Mathematicae 89(1987), 225–246

    1987

  82. [90]

    Tian,Kähler–Einstein metrics with positive scalar curvature, Inventiones Mathematicae130(1997), 1–37

    G. Tian,Kähler–Einstein metrics with positive scalar curvature, Inventiones Mathematicae130(1997), 1–37

    1997

  83. [91]

    Tian,Kähler-Einstein manifolds of positive scalar curvature, Surveys in differential geometry: essays on Einstein manifolds, Surv

    G. Tian,Kähler-Einstein manifolds of positive scalar curvature, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., vol. 6, Int. Press, Boston, MA, 1999, pp. 67–82

    1999

  84. [92]

    Tian,K-stability and Kähler-Einstein metrics, Comm

    G. Tian,K-stability and Kähler-Einstein metrics, Comm. Pure Appl. Math.68(2015), 1085–1156

    2015

  85. [93]

    Tian,Corrigendum: K-stability and Kähler-Einstein metrics, Comm

    G. Tian,Corrigendum: K-stability and Kähler-Einstein metrics, Comm. Pure Appl. Math.68(2015), 2082– 2083

    2015

  86. [94]

    Wisniewski,Fano 4-folds of index 2 withb2 ⩾2

    J. Wisniewski,Fano 4-folds of index 2 withb2 ⩾2. A contribution to Mukai classification, Bulletin of the Polish Academy of Sciences, Mathematics38, (1990), 173–184

    1990

  87. [95]

    C.Xu,K-stability of Fano varieties: an algebro-geometric approach, TheEMSSurveysinMathematicalSciences 8(2021), 265–354

    2021

  88. [96]

    Xu,K-stability of Fano varieties, Princeton University, New Jersey, 2025

    C. Xu,K-stability of Fano varieties, Princeton University, New Jersey, 2025

    2025

  89. [97]

    C. Xu, Y. Liu,K-stability of cubic threefolds, Duke Mathematical Journal168(2019), 2029–2073

    2019

  90. [98]

    C. Xu, Z. Zhuang,On positivity of the CM line bundle on K-moduli spaces, Ann. of Math. (2)192(2020), 1005–1068

    2020

  91. [99]

    C. Xu, Z. Zhuang,Uniqueness of the minimizer of the normalized volume function, Camb. J. Math.9(2021), 149–176

    2021

  92. [100]

    Sh. T. Yau,On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math.31(1978), 339–411

    1978

  93. [101]

    Zariski,The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface, Ann

    O. Zariski,The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math.76(1962), 560–615

    1962

  94. [102]

    Zhao,K-stability of Thaddeus’ moduli of stable bundle pairs on genus two curves, Forum Math

    J. Zhao,K-stability of Thaddeus’ moduli of stable bundle pairs on genus two curves, Forum Math. Sigma, to appear

  95. [103]

    Zhuang,Product theorem for K-stability, Adv

    Z. Zhuang,Product theorem for K-stability, Adv. Math.371(2020), 107250, 18

    2020

  96. [104]

    Zhuang,Optimal destabilizing centers and equivariant K-stability, Inventiones Mathematicae226(2021), 195–223

    Z. Zhuang,Optimal destabilizing centers and equivariant K-stability, Inventiones Mathematicae226(2021), 195–223. 76 Iv an Cheltsov University of Edinburgh, Edinburgh, Scotland i.cheltsov@ed.ac.uk DongSeon Hw ang Center for Complex Geometry, Institute for Basic Science, Daejeon...

    2021

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