Pith. sign in

REVIEW 2 major objections 3 minor 85 references

The sharp volume gap for K\"ahler manifolds with positive Ricci curvature

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper proves a sharp volume gap: a compact Kähler manifold with Ric(ω) ≥ (n+1)ω that is not biholomorphic to P^n has volume at most (2 n^n/(n+1)^n) times the volume of projective space, with equality exactly on the quadric…

desk verdict This paper determines the sharp volume gap in the Kähler almost rigidity theorem; the constant looks right, and the place to focus referee scrutiny is Proposition 2.7. read the letter →

arxiv 2608.08193 v1 pith:NPQ27CWJ submitted 2026-08-08 math.DG math.AGmath.MG

Chi Li , Minghao Miao , Kewei Zhang This is my paper · ORCID
classification math.DGmath.AGmath.MG MSC 53C5514J4532Q2014M25
keywords KählermanifoldspositiveRiccicurvaturevolumegapβ-volumeFanominimalrationalcurvesK-semistabilitytoriclogpairs
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 determines the exact volume threshold that separates projective space from all other Kähler manifolds with positive Ricci curvature. It proves that if (X^n, ω) is compact Kähler with Ric(ω) ≥ (n+1)ω and X is not biholomorphic to P^n, then vol(X, ω) ≤ (2 n^n/(n+1)^n) vol(P^n, ω_FS), with equality precisely for the quadric hypersurface Q^n and the product $P^{1}$ × $P^{{n-1}}$ carrying their standard Kähler–Einstein metrics. This makes the previously implicit constant in the almost-rigidity theorem equal to 1 − 2 n^n/(n+1)^n. The proof works through the rescalable β-volume β(X,L)^n vol(L), establishing a sharp gap 2 n^n for Fano manifolds with any ample R-line bundle, and extends to K-semistable toric log Fano pairs.

What carries the argument

Two pieces carry the argument. First, the β-volume β_η(X,L)^n · vol(L), a rescaling-invariant quantity that controls the actual Kähler volume through the inequality β(X,L) ≥ n+1. Second, the weighted-blowup estimates of Proposition 2.7, which give explicit lower bounds for vol(π^*L − xE) when blowing up a minimal rational curve of degree d: the three cases (embedded d=2, singular d=2, and 3 ≤ d ≤ n+1) produce the functions φ_d(x) that drive the volume comparison. These estimates are combined with Okounkov-body techniques to control equality cases, and with classifications of Fano manifolds by their minimal rational curve degrees.

What would settle it

One concrete check is to compute the left side of Proposition 2.7 directly in a nontrivial case, for example a Fano threefold with a minimal rational curve of degree d = 3 and an ample R-line bundle L, evaluating vol(π^*L − xE) for some x > d and comparing it with the claimed lower bound vol(L) − φ_3(x); a violation would falsify the main theorem. A second check is to search for any smooth Fano manifold X not isomorphic to P^n and any ample R-line bundle L with β(X,L)^n · vol(L) = 2 n^n other than the quadric or $P^{1}$ × $P^{{n-1}}$, since the paper asserts no such example exists.

Watch

Extended reading notes

Core claim

The central assertion is a sharp volume gap: on any compact Kähler manifold (X^n, ω) with Ric(ω) ≥ (n+1)ω, if X is not biholomorphic to P^n, then vol(X, ω) is at most (2 n^n/(n+1)^n) vol(P^n, ω_FS), and equality occurs only when (X, ω) is biholomorphically isometric to Q^n or $P^{1}$ × $P^{{n-1}}$ with their standard Kähler–Einstein metrics. Equivalently, in the almost-rigidity theorem the optimal constant is ε(n) = 1 − 2 n^n/(n+1)^n. The result is obtained by proving a sharper gap for the algebraically defined β-volume: for any smooth Fano manifold X not isomorphic to P^n and any ample R-line bundle L, β(X,L)^n · vol(L) ≤ 2 n^n, with equality only for X ≅ Q^n or X ≅ $P^{1}$ × $P^{{n-1}}$.

Load-bearing premise

The load-bearing premise is that the three volume-loss formulas for blowing up minimal rational curves, stated with only a proof outline and with detailed calculations deferred to another preprint, hold exactly in all the ranges where Sections 3 and 4 use them.

Editorial extensions

If this is right

  • If correct, the sharp constant in the almost-rigidity theorem for Kähler manifolds is exactly ε(n) = 1 − 2 n^n/(n+1)^n, with no gap between the volume threshold and the second-largest possible volume.
  • Any compact Kähler manifold with Ric ≥ (n+1)ω that is not P^n has volume no larger than that of Q^n or P^1 × P^{n-1}, so the second-largest volume is attained and the extremal spaces are rigid.
  • The same bound β(X,L)^n · vol(L) ≤ 2 n^n holds for all smooth Fano manifolds (except P^n) and all ample R-line bundles, which yields a purely algebro-geometric gap independent of a fixed Kähler class.
  • For K-semistable toric log Fano pairs, the anticanonical degree (−K_X − Δ)^n is at most 2 n^n, with equality only for (P^1 × P^{n-1}, 0).
  • The twisted version (Theorem 1.5) implies that if a twisted Fano pair has β-volume above 2 n^n, then every minimal rational curve has anti-canonical degree at least n, forcing a restricted list of possibilities.

Reading between the lines

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

  • The result suggests that the volume-gap constant is universal across all Fano Kähler classes, not just the anticanonical class, which may inform the search for sharp constants in K-stability and height bounds for Fano varieties.
  • The equality characterization relies on large Seshadri constants forcing a product structure; a natural test is whether similar rigidity holds for singular Fano varieties or for twisted pairs with nonzero current η.
  • The proof's dependence on the classification of degree-n minimal rational curves means that a full classification of uniruled manifolds whose minimal curves all have degree at least n (posed as a question in Remark 5.2) could extend the equality analysis to further cases.
  • The toric result resolves the logarithmic gap hypothesis, which in turn may tighten arithmetic applications such as sharp height bounds and Manin–Peyre estimates for K-semistable Fano varieties.
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.

Referee Report

2 major / 3 minor

Summary. The paper proves a sharp volume gap for Kähler manifolds with Ric(ω) ≥ (n+1)ω: any such compact Kähler manifold not biholomorphic to P^n has vol(X,ω) ≤ 2n^n/(n+1)^n vol(P^n,ω_FS), with equality precisely for the quadric Q^n and P^1×P^{n-1} with their standard Kähler-Einstein metrics. The proof establishes a stronger β-volume gap theorem (Theorem 1.4): for any Fano manifold X not P^n and any ample R-line bundle L, β(X,L)^n vol(L) ≤ 2n^n, and equality occurs only for Q^n and P^1×P^{n-1}. The argument uses the β-volume, minimal rational curves, weighted blowups, Okounkov bodies, and a classification result for Fano manifolds with minimal rational curves of degree n. In addition, the paper proves sharp volume estimates for K-semistable toric log Fano pairs (Theorem 1.6), resolving a conjecture of Andreasson-Berman.

Significance. If correct, this is a substantial result. It determines the optimal constant ε(n) in Liu's almost rigidity theorem for Kähler manifolds, with an explicit parameter-free constant and a complete equality case characterization. The proof is conceptually coherent and builds on recent advances in K-stability and the Yau-Tian-Donaldson conjecture. The toric application (Theorem 1.6) is also significant, as it settles the logarithmic gap hypothesis for toric log Fano pairs. The paper is generally well written and gives credit to prior work. However, as detailed below, the central numerical bound rests on technical estimates whose proofs are largely deferred to an unpublished preprint by the same authors.

major comments (2)
  1. [§2.2, Proposition 2.7] This proposition is the quantitative engine of the paper; it supplies the lower bounds vol(π^*L−xE) ≥ vol(L)−φ_d(x) used in Propositions 3.2, 3.3, and Theorem 4.1 to derive the sharp constant 2n^n. The proof is only an outline: the three cases are stated, but the asymptotic expansions of the sums in Case II and Case III are asserted with 'by the same calculation as in [LM25]', and the R-line bundle extension is by continuity. Since [LM25] is an unpublished same-authors preprint, the manuscript does not on its own establish the central estimate. The authors should either include the complete proof of Proposition 2.7, or state it as a theorem from [LM25] with explicit verification that the hypotheses of [LM25] are satisfied in the present setting (including the degree ranges and the nef threshold condition s_η(X,L)≥1).
  2. [§4 and §5] Theorem 4.1's proof relies directly on [LM25, Proposition 4.8] for the strict inequality β_η^n vol(L) < 2n^n in the case 3≤d≤n−1, and Proposition 5.3 (Case II) relies on [LM25, Lemma 4.13] for the volume computation (−K_X)^n < 2n^n. These are load-bearing for the gap theorem in the cases 3≤d≤n−1 and for blow-ups of P^n. The present text should either prove these statements or give precise references (including the statements) so that the reader can verify the claims without consulting an unpublished preprint.
minor comments (3)
  1. [§5, Proposition 5.3] In the proof of Case I, the line 'yielding (1+√2)/((√3/2)^3) < 2' appears to contain a typo: the denominator should be (√(3/2))^3 (i.e., w_3^3) rather than (√3/2)^3; as written, the inequality is false.
  2. [Title page] The title page contains the word 'CURV ATURE' with a line break; if this is not merely a PDF rendering artifact, please correct it.
  3. [§3.2, Proposition 3.3] The expression for φ(x) for x>2 is written as (n−1)x^n + (x−2)^n − n(x−2)x^{n−1}, which is equivalent to the formula in Proposition 2.7 Case (2) but may confuse the reader; consider using the same expression in both places.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the sharp gap is derived from an independent beta-volume upper bound, and the same-author prior results used are parameter-free inputs rather than restatements of the target theorem.

full rationale

The derivation chain is not circular. Theorem 1.3 follows from Theorem 1.4 by the elementary observation that Ric(ω)≥(n+1)ω implies β(X,L)≥n+1; the inequality vol(L)≤2n^n/(n+1)^n is then a rescaling of the beta-volume bound, not a restatement of the Ricci assumption or of the volume conclusion. The load-bearing Proposition 2.7 supplies lower bounds vol(π^*L−xE)≥vol(L)−φ_d(x) from weighted-blowup valuations attached to minimal rational curves; its three φ_d formulas are stated in the paper with a proof outline and detailed calculations deferred to [LM25]. This is reliance on a prior same-author preprint, but [LM25] is a parameter-free computation whose assumptions (s_η(X,L)≥1, d=deg C, adapted weighted blowups) do not include the present sharp gap, so it counts as independent support rather than a self-citation chain forcing the conclusion. The equality characterization uses the external classification Theorem 5.1 and the volume comparison in Proposition 5.3; no parameter is fitted to the target value 2n^n. The toric theorem similarly invokes [CFH14] and [LM25] as prior structural and computational inputs. There are no definitions of the target in terms of the inputs, no fitted quantities relabeled as predictions, and no uniqueness assertion imported from the authors to forbid alternatives. The deferred proof of Proposition 2.7 is a verification risk but not a circularity: the formulas are not derived from Theorem 1.3 or Theorem 1.4. Accordingly no circular step is identified.

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

No fitted constants appear; the sharp volume constants are computed from the projective space, quadric, and product volumes. The main external inputs are the twisted Yau-Tian-Donaldson characterization, the [LM25] volume estimates, the classification of Fano manifolds with minimal rational curve degree n, and standard Okounkov body and Mori theory.

assumptions (7)
  • domain assumption β_η(X,L) = min{s_η(X,L), δ_η(X,L)}, and the easier inequality β_η ≤ min{s_η, δ_η} (Proposition 2.5) that converts existence of twisted Kähler metrics with Ric_η ≥ sω into algebraic threshold inequalities.
    Invoked in §2.1, Theorem 2.4 and Proposition 2.5; a deep recent theorem of K-stability theory cited to [BBJ21], [CRZ19], and [Zha21], not proved in this paper.
  • domain assumption The weighted-blowup volume estimates of Proposition 2.7, including the three explicit formulas for φ_d(x), and their extension to R-line bundles.
    Used in Propositions 3.2, 3.3, and Theorem 4.1 to bound S(L;E); the proof is only outlined and the calculations are deferred to the same-authors preprint [LM25].
  • domain assumption Classification of smooth Fano manifolds X with l_X = n: X ≅ Q^n or X = Bl_A P^n with A = H ∩ S_d, 1 ≤ d ≤ n, contained in a hyperplane (Theorem 5.1(2)).
    Needed to rule out all degree-n minimal rational curve cases except Q^n in the equality analysis; cited from [Miy04], [CD15], and [DH17].
  • standard math Okounkov body facts: vol_Rn(Δ(L)) = vol(L)/n!, slicing identities (13)-(14), and Theorem 2.8 connecting nefness to the origin being in the Okounkov body.
    Used in §3.3 to prove Proposition 3.4 for arbitrary R-line bundles; cited from [LM09] and [KL17].
  • standard math Mori theory: uniruled Fano manifolds are covered by minimal rational curves with normal bundle splitting (7), with degree d in {2, ..., n+1}.
    Basis for the case division by l_X; cited from [Mor79] and [Kol96].
  • domain assumption Local-to-global volume comparison in the log setting and the logarithmic Moraga-Süss estimate volhat(x; X, Δ) ≤ 2(n-1)^n (Proposition A.1).
    Used in Proposition 7.2 for singular toric log Fano pairs; Proposition A.1 is only sketched and relies on [LL19] and [MS24].
  • domain assumption Optimal bend-and-break bound for extremal contractions (JLR26, Theorem 1.2), used in Appendix B to show the contraction is of fiber type.
    A recent preprint result; without it Proposition B.1's proof does not rule out birational contractions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The sharp volume gap for K\"ahler manifolds with positive Ricci curvature." pith.science (2026). https://pith.science/paper/NPQ27CWJ

@misc{pith2026260808193,
  author       = {Pith},
  title        = {Pith review of: The sharp volume gap for K\"ahler manifolds with positive Ricci curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NPQ27CWJ}},
  note         = {Machine review of arXiv:2608.08193}
}
abstract

We prove a sharp volume gap estimate: if an $n$-dimensional compact K\"ahler manifold $(X, \omega)$ satisfies $\mathrm{Ric}(\omega)\ge (n+1)\omega$ and $X\not\cong \mathbb{P}^n$, then $\mathrm{vol}(X, \omega)\le \frac{2n^n}{(n+1)^n}\mathrm{vol}(\mathbb{P}^n,\omega_{\mathrm{FS}})=\frac{2^{n+1} \, \pi^n \, n^n}{(n+1)^n}$. Moreover $\mathrm{vol}(X, \omega)= \frac{2^{n+1} \, \pi^n \, n^n}{(n+1)^n}$ occurs if and only if $(X, \omega)$ is biholomorphically isometric to the K\"ahler-Einstein metric on the quadric hypersurface $Q^n$ or on the product $\mathbb{P}^1\times \mathbb{P}^{n-1}$. We also obtain sharp volume gap estimates for K-semistable toric log Fano pairs.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

85 extracted references · 55 canonical work pages

  1. [1]

    Discrete

    Alexander, Matthew and Fradelizi, Matthieu and Zvavitch, Artem , TITLE =. Discrete. Comput. Geom. , FJOURNAL =. 2019 , PAGES =. doi:, URL =

    2019

  2. [2]

    , TITLE =

    Andreasson, Rolf and Berman, Robert J. , TITLE =. Compos. Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.1112/S0010437X2400736X , URL =

    work page doi:10.1112/s0010437x2400736x 2024

  3. [3]

    , TITLE =

    Andreasson, Rolf and Berman, Robert J. , TITLE =. J. \'Ec. polytech. Math. , FJOURNAL =. 2025 , PAGES =. doi:10.5802/jep.304 , URL =

    work page doi:10.5802/jep.304 2025

  4. [4]

    Bauer, Thomas and Szemberg, Tomasz , TITLE =. J. Pure Appl. Algebra , VOLUME =. 2009 , NUMBER =

    2009

  5. [5]

    , TITLE =

    Berman, Robert J. , TITLE =. Pure Appl. Math. Q. , FJOURNAL =. 2025 , NUMBER =. doi:10.4310/pamq.250115041532 , URL =

    work page doi:10.4310/pamq.250115041532 2025

  6. [6]

    The pseudo-effective cone of a compact

    Boucksom, S\'ebastien and Demailly, Jean-Pierre and P. The pseudo-effective cone of a compact. J. Algebraic Geom. , FJOURNAL =. 2013 , NUMBER =. doi:10.1090/S1056-3911-2012-00574-8 , URL =

    doi:10.1090/s1056-3911-2012-00574-8 2013

  7. [7]

    Osamu Fujino , TITLE =. Publ. Res. Inst. Math.Sci. , FJOURNAL =. 2011 , PAGES =

    2011

  8. [8]

    Forum of Mathematics, Pi , FJOURNAL =

    Jovinelly, Eric and Lehmann, Brian and Riedl, Eric , TITLE =. Forum of Mathematics, Pi , FJOURNAL =. 2026 , PAGES =

    2026

Show all 85 references
  1. [9]

    Semicontinuity of complex singularity exponents and K\"

    Jean-Pierre Demailly and J\'. Semicontinuity of complex singularity exponents and K\". Ann. Sci. \'. 2001 , PAGES =

    2001

  2. [10]

    Jean-Pierre Demailly , TITLE =. J. Differential Geom. , VOLUME =. 1993 , PAGES =

    1993

  3. [11]

    Honda, Shouhei and Mondino, Andrea , title =. Indag. Math., New Ser. , issn =. 2026 , language =. doi:10.1016/j.indag.2025.03.008 , keywords =

    2026 doi

  4. [12]

    Locally unsplit families of rational curves of large anticanonical degree on

    Casagrande, Cinzia and Druel, St\'. Locally unsplit families of rational curves of large anticanonical degree on. Int. Math. Res. Not. IMRN , FJOURNAL =. 2015 , NUMBER =

    2015

  5. [13]

    Chen, Yifei and Fu, Baohua and Hwang, Jun-Muk , TITLE =. Proc. Edinb. Math. Soc. (2) , FJOURNAL =. 2014 , NUMBER =. doi:10.1017/S001309151300093X , URL =

    2014 doi

  6. [14]

    , Title =

    Cho, Koji and Miyaoka, Yoichi and Shepherd-Barron, N.I. , Title =. Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math., 25 , ISBN =. 2002 , Publisher =

    1997

  7. [15]

    Numerical characterisation of quadrics , JOURNAL =

    Dedieu, Thomas and H\". Numerical characterisation of quadrics , JOURNAL =. 2017 , NUMBER =. doi:, URL =

    2017

  8. [16]

    Liu, Yuchen , TITLE =. Compos. Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.1112/S0010437X18007042 , URL =

    2018 doi

  9. [17]

    Li, Chi and Liu, Yuchen , TITLE =. Adv. Math. , FJOURNAL =. 2019 , NUMBER =

    2019

  10. [19]

    , TITLE =

    Miyaoka, Y. , TITLE =. Complex Analysis in Several Variables-Memorial Conference of Kiyoshi Oka's Centennial Birthday , SERIES =. 2004 , MRCLASS =

    2004

  11. [20]

    Mori, Shigefumi , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1979 , NUMBER =. doi:10.2307/1971241 , URL =

    1979 doi

  12. [21]

    Birational geometry of algebraic varieties , SERIES =

    Koll\'. Birational geometry of algebraic varieties , SERIES =. 1998 , PAGES =. doi:10.1017/CBO9780511662560 , URL =

    1998 doi

  13. [22]

    Miyaoka, Yoichi and Mori, Shigefumi , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1986 , NUMBER =. doi:10.2307/1971387 , URL =

    1986 doi

  14. [23]

    Zhang, Kewei , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2022 , NUMBER =. doi:10.1093/imrn/rnaa295 , URL =

    2022 doi

  15. [24]

    Zhang, Kewei , TITLE =. Adv. Math. , FJOURNAL =. 2021 , PAGES =. doi:10.1016/j.aim.2021.107888 , URL =

    2021

  16. [25]

    Convex bodies associated to linear series , JOURNAL =

    Lazarsfeld, Robert and Musta. Convex bodies associated to linear series , JOURNAL =. 2009 , NUMBER =. doi:10.24033/asens.2109 , URL =

    2009 doi

  17. [26]

    Bounding toric singularities with normalized volume , JOURNAL =

    Moraga, Joaqu\'. Bounding toric singularities with normalized volume , JOURNAL =. 2024 , NUMBER =

    2024

  18. [27]

    , TITLE =

    Perelman, G. , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 1994 , NUMBER =. doi:10.2307/2152760 , URL =

    1994 doi

  19. [28]

    , TITLE =

    Cheeger, Jeff and Colding, Tobias H. , TITLE =. J. Differential Geom. , FJOURNAL =. 1997 , NUMBER =

    1997

  20. [29]

    K\"uronya, Alex and Lozovanu, Victor , TITLE =. Doc. Math. , FJOURNAL =. 2017 , PAGES =

    2017

  21. [30]

    On the volume of a line bundle

    Boucksom, S. On the volume of a line bundle. , fjournal =. Int. J. Math. , issn =. 2002 , language =. doi:10.1142/S0129167X02001575 , keywords =

    2002 doi

  22. [31]

    1987 , language =

    Bando, Shigetoshi and Mabuchi, Toshiki , title =. 1987 , language =

    1987

  23. [32]

    and Boucksom, S

    Berman, Robert J. and Boucksom, S. A variational approach to the. J. Am. Math. Soc. , issn =. 2021 , language =. doi:10.1090/jams/964 , keywords =

    2021 doi

  24. [33]

    and Rubinstein, Yanir A

    Cheltsov, Ivan A. and Rubinstein, Yanir A. and Zhang, Kewei , title =. Sel. Math., New Ser. , issn =. 2019 , language =. doi:10.1007/s00029-019-0473-z , keywords =

    2019 doi

  25. [34]

    Fujita, Kento and Odaka, Yuji , title =. T. 2018 , language =. doi:10.2748/tmj/1546570823 , keywords =

    2018

  26. [35]

    Blum, Harold and Jonsson, Mattias , title =. Adv. Math. , issn =. 2020 , language =. doi:10.1016/j.aim.2020.107062 , keywords =

    2020

  27. [36]

    Rational curves on algebraic varieties , fseries =

    Koll. Rational curves on algebraic varieties , fseries =. 1996 , publisher =

    1996

  28. [37]

    , title =

    Rubinstein, Yanir A. , title =. Adv. Math. , issn =. 2008 , language =. doi:10.1016/j.aim.2008.03.017 , keywords =

    2008 doi

  29. [38]

    Tian, Gang , title =. Int. J. Math. , issn =. 1992 , language =. doi:10.1142/S0129167X92000175 , keywords =

    1992 doi

  30. [39]

    Greatest lower bounds on the

    Sz. Greatest lower bounds on the. Compos. Math. , issn =. 2011 , language =. doi:10.1112/S0010437X10004938 , keywords =

    2011 doi

  31. [40]

    Song, Jian and Wang, Xiaowei , title =. Geom. Topol. , issn =. 2016 , language =. doi:10.2140/gt.2016.20.49 , keywords =

    2016 doi

  32. [41]

    Li, Chi , title =. Adv. Math. , issn =. 2011 , language =. doi:10.1016/j.aim.2010.12.023 , keywords =

    2011 doi

  33. [42]

    2004 , publisher =

    Lazarsfeld, Robert , title =. 2004 , publisher =

    2004

  34. [43]

    Liu, Yuchen and Zhuang, Ziquan , title =. Math. Z. , issn =. 2018 , language =. doi:10.1007/s00209-017-1941-9 , keywords =

    2018 doi

  35. [44]

    Li, Chi , title =. J. Reine Angew. Math. , issn =. 2017 , language =. doi:10.1515/crelle-2014-0156 , keywords =

    2017 doi

  36. [45]

    Fujita, Kento , title =. Am. J. Math. , issn =. 2018 , language =. doi:10.1353/ajm.2018.0009 , keywords =

    2018

  37. [46]

    Rolf Andreasson and Robert J. Berman. Sharp bounds on the height of K -semistable F ano varieties I , the toric case. Compos. Math. , 160(10):2366--2406, 2024

    2024

  38. [47]

    Rolf Andreasson and Robert J. Berman. Sharp bounds on the height of K -semistable F ano varieties II , the log case. J. \'Ec. polytech. Math. , 12:983--1018, 2025

    2025

  39. [48]

    Polytopes of maximal volume product

    Matthew Alexander, Matthieu Fradelizi, and Artem Zvavitch. Polytopes of maximal volume product. Discrete. Comput. Geom. , 62:583--600, 2019

    2019

  40. [49]

    Berman, S \'e bastien Boucksom, and Mattias Jonsson

    Robert J. Berman, S \'e bastien Boucksom, and Mattias Jonsson. A variational approach to the Yau - Tian - Donaldson conjecture. J. Am. Math. Soc. , 34(3):605--652, 2021

    2021

  41. [50]

    The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension

    S\'ebastien Boucksom, Jean-Pierre Demailly, Mihai P a un, and Thomas Peternell. The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension. J. Algebraic Geom. , 22(2):201--248, 2013

    2013

  42. [51]

    Robert J. Berman. On K -stability, height bounds and the M anin- P eyre conjecture. Pure Appl. Math. Q. , 21(3):931--970, 2025

    2025

  43. [52]

    Thresholds, valuations, and K -stability

    Harold Blum and Mattias Jonsson. Thresholds, valuations, and K -stability. Adv. Math. , 365:57, 2020. Id/No 107062

    2020

  44. [53]

    Uniqueness of Einstein K \"a hler metrics modulo connected group actions

    Shigetoshi Bando and Toshiki Mabuchi. Uniqueness of Einstein K \"a hler metrics modulo connected group actions. Algebraic geometry, Proc . Symp ., Sendai / Jap . 1985, Adv . Stud . Pure Math . 10, 11-40 (1987)., 1987

    1987

  45. [54]

    On the volume of a line bundle

    S \'e bastien Boucksom. On the volume of a line bundle. Int. J. Math. , 13(10):1043--1063, 2002

    2002

  46. [55]

    Seshadri constants and the generation of jets

    Thomas Bauer and Tomasz Szemberg. Seshadri constants and the generation of jets. J. Pure Appl. Algebra , 213(11):2134–2140, 2009

    2009

  47. [56]

    Jeff Cheeger and Tobias H. Colding. On the structure of spaces with R icci curvature bounded below. I . J. Differential Geom. , 46(3):406--480, 1997

    1997

  48. [57]

    Locally unsplit families of rational curves of large anticanonical degree on F ano manifolds

    Cinzia Casagrande and St\' e phane Druel. Locally unsplit families of rational curves of large anticanonical degree on F ano manifolds. Int. Math. Res. Not. IMRN , (21):10756--10800, 2015

    2015

  49. [58]

    Minimal rational curves on complete toric manifolds and applications

    Yifei Chen, Baohua Fu, and Jun-Muk Hwang. Minimal rational curves on complete toric manifolds and applications. Proc. Edinb. Math. Soc. (2) , 57(1):111--123, 2014

    2014

  50. [59]

    Shepherd-Barron

    Koji Cho, Yoichi Miyaoka, and N.I. Shepherd-Barron. Characterizations of projective space and applications to complex symplectic manifolds. In Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math., 25 , pages 1--88. Mathematical Society of Japan, 2002

    1997

  51. [60]

    Cheltsov, Yanir A

    Ivan A. Cheltsov, Yanir A. Rubinstein, and Kewei Zhang. Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces. Sel. Math., New Ser. , 25(2):36, 2019. Id/No 34

    2019

  52. [61]

    Numerical characterisation of quadrics

    Thomas Dedieu and Andreas H\" o ring. Numerical characterisation of quadrics. Algebraic Geometry , 4(1):120--135, 2017

    2017

  53. [62]

    On the K -stability of Fano varieties and anticanonical divisors

    Kento Fujita and Yuji Odaka. On the K -stability of Fano varieties and anticanonical divisors. T \^o hoku Math. J. (2) , 70(4):511--521, 2018

    2018

  54. [63]

    Fundamental theorems for the log minimal model program

    Osamu Fujino. Fundamental theorems for the log minimal model program. Publ. Res. Inst. Math.Sci. , 47:727--789, 2011

    2011

  55. [64]

    Optimal bounds for the volumes of K \"a hler - Einstein Fano manifolds

    Kento Fujita. Optimal bounds for the volumes of K \"a hler - Einstein Fano manifolds. Am. J. Math. , 140(2):391--414, 2018

    2018

  56. [65]

    Gap phenomena under curvature restrictions

    Shouhei Honda and Andrea Mondino. Gap phenomena under curvature restrictions. Indag. Math., New Ser. , 37(3):744--763, 2026

    2026

  57. [66]

    Optimal bounds in bend-and-break

    Eric Jovinelly, Brian Lehmann, and Eric Riedl. Optimal bounds in bend-and-break. Forum of Mathematics, Pi , 14:e16:arXiv:2509.08065, 2026

    2026

  58. [67]

    Positivity of line bundles and N ewton- O kounkov bodies

    Alex K\"uronya and Victor Lozovanu. Positivity of line bundles and N ewton- O kounkov bodies. Doc. Math. , 22:1285--1302, 2017

    2017

  59. [68]

    Rational curves on algebraic varieties , volume 32 of Ergeb

    J \'a nos Koll \'a r. Rational curves on algebraic varieties , volume 32 of Ergeb. Math. Grenzgeb., 3. Folge . Berlin: Springer-Verlag, 1996

    1996

  60. [69]

    Positivity in algebraic geometry

    Robert Lazarsfeld. Positivity in algebraic geometry. I . Classical setting: line bundles and linear series , volume 48 of Ergeb. Math. Grenzgeb., 3. Folge . Berlin: Springer, 2004

    2004

  61. [70]

    Greatest lower bounds on Ricci curvature for toric Fano manifolds

    Chi Li. Greatest lower bounds on Ricci curvature for toric Fano manifolds. Adv. Math. , 226(6):4921--4932, 2011

    2011

  62. [71]

    Yau- Tian - Donaldson correspondence for K -semistable Fano manifolds

    Chi Li. Yau- Tian - Donaldson correspondence for K -semistable Fano manifolds. J. Reine Angew. Math. , 733:55--85, 2017

    2017

  63. [72]

    K\" a hler- E instein metrics and volume minimization

    Chi Li and Yuchen Liu. K\" a hler- E instein metrics and volume minimization. Adv. Math. , 341:440--492, 2019

    2019

  64. [73]

    Convex bodies associated to linear series

    Robert Lazarsfeld and Mircea Musta t a . Convex bodies associated to linear series. Ann. Sci. \' E c. Norm. Sup\' e r. (4) , 42(5):783--835, 2009

    2009

  65. [74]

    On the volume of K -semistable F ano manifolds

    Chi Li and Minghao Miao. On the volume of K -semistable F ano manifolds. arXiv preprint arXiv:2506.17420 , 2025

    2025 arXiv

  66. [75]

    Characterization of projective spaces by Seshadri constants

    Yuchen Liu and Ziquan Zhuang. Characterization of projective spaces by Seshadri constants. Math. Z. , 289(1-2):25--38, 2018

    2018

  67. [76]

    Y. Miyaoka. Numerical characterisations of hyperquadrics. In Complex Analysis in Several Variables-Memorial Conference of Kiyoshi Oka's Centennial Birthday , volume 42 of Adv. Stud. Pure Math. , pages 209--235. Math. Soc. Japan, Tokyo, 2004

    2004

  68. [77]

    A numerical criterion for uniruledness

    Yoichi Miyaoka and Shigefumi Mori. A numerical criterion for uniruledness. Ann. of Math. (2) , 124(1):65--69, 1986

    1986

  69. [78]

    Projective manifolds with ample tangent bundles

    Shigefumi Mori. Projective manifolds with ample tangent bundles. Ann. of Math. (2) , 110(3):593--606, 1979

    1979

  70. [79]

    Bounding toric singularities with normalized volume

    Joaqu\' i n Moraga and Hendrik S\" u ss. Bounding toric singularities with normalized volume. Bull. Lond. Math. Soc. , 56(6):2212--2229, 2024

    2024

  71. [80]

    Perelman

    G. Perelman. Manifolds of positive R icci curvature with almost maximal volume. J. Amer. Math. Soc. , 7(2):299--305, 1994

    1994

  72. [81]

    Rubinstein

    Yanir A. Rubinstein. Some discretizations of geometric evolution equations and the Ricci iteration on the space of K \"a hler metrics. Adv. Math. , 218(5):1526--1565, 2008

    2008

  73. [82]

    The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality

    Jian Song and Xiaowei Wang. The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality. Geom. Topol. , 20(1):49--102, 2016

    2016

  74. [83]

    Greatest lower bounds on the Ricci curvature of Fano manifolds

    G \'a bor Sz \'e kelyhidi. Greatest lower bounds on the Ricci curvature of Fano manifolds. Compos. Math. , 147(1):319--331, 2011

    2011

  75. [84]

    On stability of the tangent bundles of Fano varieties

    Gang Tian. On stability of the tangent bundles of Fano varieties. Int. J. Math. , 3(3):401--413, 1992

    1992

  76. [85]

    Continuity of delta invariants and twisted K \"ahler- E instein metrics

    Kewei Zhang. Continuity of delta invariants and twisted K \"ahler- E instein metrics. Adv. Math. , 388:Paper No. 107888, 25, 2021

    2021

  77. [86]

    On the optimal volume upper bound for K \"ahler manifolds with positive R icci curvature (with an appendix by Y uchen L iu)

    Kewei Zhang. On the optimal volume upper bound for K \"ahler manifolds with positive R icci curvature (with an appendix by Y uchen L iu). Int. Math. Res. Not. IMRN , (8):6135--6156, 2022

    2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.