Pith. sign in

REVIEW 2 major objections 9 minor 53 references

On the K\"ahler MMP and the transcendental base-point-free theorem

T0 review · 2 major / 9 minor · reviewed 2026-07-31 · grok-4.5

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

Pith's one-line read The minimal model program works for big gklt Kähler pairs, and a nef class of that form is semiample via a Moishezon contraction.

desk verdict Major completion of the Kähler MMP for big gklt pairs and Tosatti’s BPF, but the non-big half hangs on an unpublished Paun personal communication for the moduli part. read the letter →

arxiv 2607.24986 v1 pith:CS4RYQ53 submitted 2026-07-27 math.AG

Christopher Hacon , Lingyao Xie This is my paper · ORCID
classification math.AG MSC 14E3032J2714J40
keywords KählerMMPtranscendentalbase-point-freegeneralizedkltpairsMoishezoncontractionmodifiedbigmultiplieridealscanonicalbundleformula
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 classical minimal model program (MMP) tells you how to simplify a projective variety by contracting curves and flipping until the canonical class becomes nef, then (under abundance) semiample. This paper extends that program to compact Kähler varieties, which need not be projective, for generalized klt pairs whose boundary-plus-b-divisor is modified big. The main theorems give a Mori fiber space when the canonical class is not pseudo-effective, and a good log terminal model when it is. As a direct consequence, any nef class of the form K_X + B + β_X is semiample: it is the pullback of a Kähler class under a Moishezon contraction to a compact Kähler space. That settles Tosatti’s transcendental base-point-free conjecture in this setting and supplies the missing contraction and termination steps needed to run the Kähler MMP with scaling.

What carries the argument

An inductive package linking a non-gklt contraction theorem (Theorem 1.6), the transcendental base-point-free theorem (Theorem 1.4), and MMP-with-scaling termination (Theorem 1.5). The engine is successive jumping numbers of multiplier ideals along a Kähler current with singularities along Null(α), together with the canonical-bundle formula that reduces dimension when the pair is not big.

What would settle it

Exhibit a compact Kähler gklt pair with B+β_X modified big and α = K_X+B+β_X nef such that no Moishezon contraction realizes α as the pullback of a Kähler class, or produce an infinite sequence of flips with scaling that does not terminate.

Watch

Extended reading notes

Core claim

For a compact Kähler gklt pair (X, B+β) with B+β_X modified big, the MMP holds: if K_X+B+β_X is not pseudo-effective there is a Mori fiber space; if it is pseudo-effective (or big) there is a good log terminal model. In particular, when α = K_X+B+β_X is nef it is semiample, realized by a Moishezon contraction f : X → Y onto a compact Kähler space with α ≡ f*γ for a Kähler class γ on Y (and f projective when X is strongly Q-factorial).

Load-bearing premise

Everything depends on the boundary-plus-b-divisor being modified big; without that hypothesis the vanishing, canonical-bundle formula, and termination arguments do not go through.

Editorial extensions

If this is right

  • Tosatti’s transcendental base-point-free conjecture holds for big gklt Kähler pairs.
  • Flips and divisorial contractions of strongly Q-factorial big gklt Kähler pairs preserve the Kähler condition.
  • Good log terminal models and Mori fiber spaces exist, so the Kähler MMP with scaling can be run to completion in the big case.
  • Several earlier conjectures on the Kähler–Ricci flow and collapsing (Tosatti–Zhang, Tosatti) follow immediately.
  • Relative versions over a base and geography of weak log canonical models are available when the pair is modified big over the base.

Reading between the lines

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

  • The modified-bigness hypothesis is the precise barrier to a full Kähler abundance statement; removing it would finish the program the authors leave as open conjectures.
  • The same inductive skeleton (jumping numbers + canonical bundle formula + NQC triviality) is likely to adapt to glc pairs once a suitable non-klt contraction theorem is in place.
  • Existence of the Moishezon contraction gives a concrete analytic substitute for the usual projective linear system, which should feed directly into analytic constructions of Kähler–Einstein metrics on the resulting models.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 9 minor

Summary. The paper proves the minimal model program for compact Kähler generalized klt pairs (X, B+β) with B+β_X modified big: if K_X+B+β_X is not pseudo-effective there is a Mori fiber space, and if it is pseudo-effective (with B+β_X big, or itself big) there is a good log terminal model (Theorem 1.1). The central new input is the transcendental base-point-free theorem (Theorem 1.4): nef α = K_X+B+β_X is semiample, realized by a Moishezon contraction with α ≡ f*γ for a Kähler class γ on the base. The proof is an explicit dimension induction linking the contraction theorem (1.6), base-point-freeness (1.4), and the MMP with scaling (1.5). The big case of the contraction theorem (Section 3) proceeds via jumping numbers of multiplier ideals, adjunction onto successive non-klt strata, and gluing of contractions via push-out diagrams, extending the resulting morphism off the null locus using Artin–Fujiki blowing-down. The non-big case uses Ou's uniruledness criterion, the MRC fibration, and a canonical bundle formula (Theorem 2.53) whose key analytic input is a semipositivity statement (Theorem 2.52) attributed to a personal communication of Paun.

Significance. If the arguments hold, the paper establishes the minimal model program for big gklt Kähler pairs (Theorem 1.1/1.5) and proves Tosatti's transcendental base-point-free conjecture for nef K_X+B+β_X with modified-big boundary (Theorem 1.4), together with the Kähler criterion (Theorem 7.1), geography of log canonical models (Theorem 5.15), and several consequences in the Kähler–Ricci flow literature (Theorems 1.12, 1.13). This is the natural extension of [BCHM10] to the transcendental setting and would be a landmark result. Particular strengths: the proof is a genuine induction (1.6_{n−1} ⇒ 1.6_{n,big} ⇒ 1.4_n ⇒ 1.5_n ⇒ 1.6_n) with the big and non-big cases cleanly separated; Section 3 (the big-case contraction theorem via jumping numbers, adjunction on non-reduced strata, and Artin–Fujiki extension) is substantial new technical work; and the dependence on prior results (flips [DH23], cone theorem [HP24, HLX26], Ou's uniruledness criterion) is as black-box lemmas with independent proofs, not circular restatements. The paper also states its boundary precisely: the modified-big hypothesis is flagged as essential, with the general case left as Conjecture 1.14.

major comments (2)
  1. [§2.12, Theorem 2.52 / Theorem 2.53; used in §4 (Thm 4.1 Case 2), §5.2 (Prop 5.13), §6 (Prop 6.1)] Theorem 2.52 is cited only as 'M. Paun, personal communication, July 2026, to appear' [Pau26]; there is no public preprint. This is not a peripheral input. Theorem 2.53 (the canonical bundle formula producing a gklt pair (Z, ∆+δ) with δ modified big) is proved by applying Theorem 2.52 to the perturbed pairs (X′, B′_δ), β′_δ to obtain a positive current with zero Lelong numbers in the moduli class. Theorem 2.53 is in turn the load-bearing input for the entire non-big half of the induction: Theorem 4.1 Case 2 (semiample-ness of nef non-big α, p. 31), Proposition 5.13 (existence of log terminal models when K_X+B+β_X is psef but not big, p. 41), and Proposition 6.1 Case 2 (p. 47) all reduce, via Ou's theorem and the MRC fibration, to a lower-dimensional gklt pair on the base whose moduli part must be modified big for the inductive hypotheses 1.4_{n−1} and 1.5_{n−1} to apply — and those hypot
  2. [Theorem 1.7; used in §4, proof of Theorem 4.1, Case 1] Theorem 1.7 (the Kähler criterion: for a gklt pair with α = K_X+B+β_X nef and big, α·C > 0 for all rational curves implies α Kähler) is stated in the introduction with no proof and no citation, but it is invoked at a critical point in the proof of Theorem 4.1, Case 1 (p. 30: 'By Theorem 1.7, [γ] is a Kähler class'). The in-paper variant, Theorem 7.1, cannot substitute: its proof uses Theorem 1.5, which depends on Theorem 1.4_n, so invoking it inside Theorem 4.1 would be circular. Presumably Theorem 1.7 is imported from [HLX26], but this needs to be stated explicitly, with the hypotheses matched (note that Theorem 1.7 as stated requires no NQC assumption, unlike Theorem 7.1), and the dependency made clear in the inductive scheme of §1.1.
minor comments (9)
  1. [References] Reference [Pau26]: 'peronal communication' → 'personal communication'.
  2. [Proof of Theorem 2.53] Typo: 'satisfy the hypotesis of Theorem 2.52' → 'hypothesis'.
  3. [Theorem 2.3 (statement)] Typo: 'and and D a Cartier divisor'.
  4. [Proof of Lemma 2.20] Typo: 'log terminimal model' → 'log terminal model'.
  5. [Theorem 1.13 (statement)] Grammar: 'Conjectures 1.1, 1.2, and 1.3 of [TZ18] holds' → 'hold'.
  6. [Definition 2.8] The first sentence ('α is big if α ≥ ω where ω is a Kähler current') is garbled and only the second formulation (existence of a Kähler current T with [T] ∈ α) is meaningful; please rewrite.
  7. [Lemma 2.4 (proof)] The notation E ∧ Z = 0 (no common components) is used without having been defined; a one-line definition would help.
  8. [§3, opening paragraph vs. Theorem 3.1] The section preamble lists Theorems 1.4_{n−1}, 1.5_{n−1}, and 1.6_{n−1} as assumptions, while Theorem 3.1 states that 1.6_{n−1} alone implies 1.6_{n,big}. Please align the two, and check whether the results cited inside Proposition 3.2 (e.g. the relative vanishing inputs) require the additional hypotheses.
  9. [§1.3, Conjecture 1.16] 'Next we recall our famous Abundance conjecture' — the phrasing is odd for a standard conjecture; suggest neutral wording.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the argument is a genuine dimension induction using prior independent lemmas; self-citations and one unpublished external lemma are load-bearing but not definitional or self-proving.

full rationale

The paper's central claims (Theorems 1.1/1.4/1.5) are proved by a carefully ordered induction on dimension (Section 1.1): 1.6_{n-1} implies the big contraction theorem, which with lower-dimensional base-point-free and MMP-with-scaling yields the transcendental base-point-free theorem in dimension n, which yields MMP-with-scaling in dimension n, which closes the non-big contraction theorem. Base cases in dimension 1 are trivial. No quantity is defined in terms of the target semi-ampleness or good models; no parameters are fitted to data and then re-predicted; no uniqueness theorem is imported solely to forbid alternatives. Self-citations (DH23 flips, HP24 cone/Null locus, HLX26 dlt/Q-factorial models and Kähler criterion, DH24 projective g-pairs) supply earlier theorems used as black-box lemmas; those statements have independent proofs and do not presuppose the present results. The non-big half relies on the canonical-bundle formula (Theorem 2.53), whose nef/modified-big moduli part comes from Paun's Theorem 2.52 (personal communication, to appear). That is an external unpublished input, not a self-citation and not a reduction of the conclusion to its own premises by construction; it is a verifiability/correctness risk, not circularity under the stated criteria. The derivation is therefore self-contained against the circularity patterns, with only ordinary (non-load-bearing-for-circularity) dependence on the authors' prior work.

Assumptions & free parameters 0 free parameters · 7 assumptions · 2 invented entities

The paper works entirely inside standard complex-analytic and birational geometry. Load-bearing external inputs are prior theorems on flips, cone theorems, vanishing, uniruledness, and the canonical bundle formula under projectivity; the only essential domain restriction invented for the present work is the standing “modified big” hypothesis on B+β_X. No numerical free parameters appear.

assumptions (7)
  • domain assumption Existence of flips for Q-factorial compact Kähler gklt pairs (DH23 Thm 5.12)
    Taken as black box; used whenever a flipping contraction appears in the MMP with scaling.
  • domain assumption Cone theorem for compact Kähler gklt pairs, with finiteness when B+β is big (HP24, HLX26)
    Supplies the extremal rays that are contracted; cited as Theorem 1.3.
  • domain assumption Ou’s characterization: compact Kähler manifold is uniruled iff K_X is not pseudo-effective
    Opens the non-big case via the MRC fibration (Theorem 1.9).
  • domain assumption Null(α)=E_nK(α) for nef and big classes on Fujiki class C spaces (HP24/CT15)
    Lets the authors concentrate singularities of a Kähler current along an analytic set in the big case.
  • standard math Nadel and relative Kawamata–Viehweg vanishing for generalized pairs on analytic spaces
    Used repeatedly to produce short exact sequences of multiplier ideals and to glue contractions.
  • ad hoc to paper B+β_X is modified big (standing hypothesis of all main theorems)
    Essential for relative bigness, canonical-bundle formula with big moduli part, and geography of models; explicitly left open when dropped (Conjecture 1.14).
  • domain assumption Canonical bundle formula under projective fibration + modified bigness (Thm 2.53, using Paun)
    Reduces the non-big case to a lower-dimensional gklt pair on the base of the MRC/Mori fiber space.
invented entities (2)
  • Moishezon contraction defined by a nef class (EMC) independent evidence
    purpose: Analytic replacement for the morphism defined by a semiample line bundle; allows the base-point-free statement to be phrased purely in Bott–Chern cohomology.
    Definition 2.34 packages curve-connectedness, Moishezon-ness, and numerical triviality; the paper proves existence rather than postulating a new physical object.
  • NQC (nef Q-Cartier combination) classes in H^{1,1}_BC independent evidence
    purpose: Guarantees that MMP steps with large scaling remain α-trivial (Lemma 2.46), enabling reduction to lower dimension.
    Extension of the projective NQC notion to Bott–Chern classes; existence for K_X+B+β+ω is proved from the cone theorem (Lemma 2.45).

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the K\"ahler MMP and the transcendental base-point-free theorem." pith.science (2026). https://pith.science/paper/CS4RYQ53

@misc{pith2026260724986,
  author       = {Pith},
  title        = {Pith review of: On the K\"ahler MMP and the transcendental base-point-free theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CS4RYQ53}},
  note         = {Machine review of arXiv:2607.24986}
}
read the original abstract

In this article, we establish the minimal model program for big gklt K\"ahler pairs, and in particular we prove Tosatti's transcendental base-point-free conjecture.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 6 linked inside Pith

  1. [1]

    Artin, Algebraization of formal moduli, II: Existence of modifications , Ann

    M. Artin, Algebraization of formal moduli, II: Existence of modifications , Ann. of Math. (2) 91 (1970), 88–135

    1970

  2. [2]

    ahler–Einstein metrics and the K\

    R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, K\"ahler–Einstein metrics and the K\"ahler–Ricci flow on log Fano varieties , J. Reine Angew. Math. 751 (2019), 27–89

    2019

  3. [3]

    Birkar, On existence of log terminal models , Compositio Mathematica 146(4), 919–928 (2010)

    C. Birkar, On existence of log terminal models , Compositio Mathematica 146(4), 919–928 (2010)

    2010

  4. [4]

    Birkar, On existence of log minimal models and weak Zariski decompositions , Math

    C. Birkar, On existence of log minimal models and weak Zariski decompositions , Math. Annalen, Volume 354, pages 787–799, (2012)

    2012

  5. [5]

    Birkar, P

    C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23(2), 405–468 (2010)

    2010

  6. [6]

    Boucksom, Divisorial Zariski decompositions on compact complex manifolds , Ann

    S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds , Ann. Sci. \'Ecole Norm. Sup. (4) 37(1), 45–76 (2004)

    2004

  7. [7]

    oring, Rational curves on compact K\

    J. Cao and A. H\"oring, Rational curves on compact K\"ahler manifolds , J. Differential Geom., 114, (2020), No. 1, 1–39

    2020

  8. [8]

    Claudon and A

    B. Claudon and A. H\"oring, Projectivity criteria for Kähler morphisms , https://arxiv.org/abs/2404.13927

    arXiv

Show all 53 references
  1. [9]

    T. C. Collins and V. Tosatti, K\"ahler currents and null loci, Invent. Math. 202(3), 1167-1198 (2015)

    2015

  2. [10]

    Das and C

    O. Das and C. Hacon, On the Minimal Model Program for K\"ahler 3-folds , arXiv e-prints, arXiv:2306.11708 (June 2023), 2306.11708

    2023 arXiv

  3. [11]

    Das and C

    O. Das and C. Hacon, Transcendental Minimal Model Program for Projective Varieties , arXiv:2412.07650 (December 2024), 2412.07650

    2024 arXiv

  4. [12]

    Das and C

    O. Das and C. Hacon, The log minimal model program for K\"ahler 3-folds , J. Differential Geom. 130(1): 151-207 (May 2025)

    2025

  5. [13]

    Das and C

    O. Das and C. Hacon, K\"ahler 4-fold MMP , in preparation

  6. [14]

    O. Das, C. Hacon and M. Paun, On the 4-dimensional minimal model program for K\"ahler varieties , Adv. Math. 443, Paper No. 109615 (2024)

    2024

  7. [15]

    O. Das, C. Hacon and J. I. Yanez, MMP for Generalized Pairs on K\"ahler 3-folds , arXiv e-prints , arXiv:2305.00524v1 (April 2023), (to appear in Annali Scuola Norm. Sup.)

    2023 arXiv

  8. [16]

    Das and W

    O. Das and W. Ou, On the log abundance for compact K\"ahler threefolds , Manuscripta Math. 173 (2024), no. 1-2, 341–404

    2024

  9. [17]

    Das and W

    O. Das and W. Ou, On the log abundance for compact K\"ahler threefolds II , Proceedings of the London Mathematical Society: Volume 132, Issue 3 (2026)

    2026

  10. [18]

    Essydieux, V

    P. Essydieux, V. Guedj, and A. Zeriahi: Singular Kähler-Einstein metrics , J. Amer. Math. Soc. 22 (2009), 607-639

    2009

  11. [19]

    Filip and V

    S. Filip and V. Tosatti, Smooth and Rough Positive Currents , Ann. Inst. Fourier, Tome 68, no 7 (2018), p. 2981-2999

    2018

  12. [20]

    Fujiki, On the blowing down of analytic spaces , Publ

    A. Fujiki, On the blowing down of analytic spaces , Publ. Res. Inst. Math. Sci. 10, 473–507 (1974/75)

    1974

  13. [21]

    Fujino, Special termination and reduction to pl flips, in Flips for 3-folds and 4-folds , volume 35 of Oxford Lecture Ser

    O. Fujino, Special termination and reduction to pl flips, in Flips for 3-folds and 4-folds , volume 35 of Oxford Lecture Ser. Math. Appl., pages 63–75, Oxford Univ. Press, Oxford, 2007

    2007

  14. [22]

    Fujino, Minimal model program for projective morphisms between complex analytic spaces , (2022)

    O. Fujino, Minimal model program for projective morphisms between complex analytic spaces , (2022)

    2022

  15. [23]

    Fujino, T

    O. Fujino, T. Fujisawa, Variations of mixed Hodge structure and semipositivity theorems , Publ. Res. Inst. Math. Sci. 50 (2014), no. 4, 589–661

    2014

  16. [24]

    Graber, J

    T. Graber, J. Harris, and J. Starr, Families of rationally connected varieties , Journal of AMS 16 (2003), 57-67

    2003

  17. [25]

    Guenancia, Families of conic Kahler-Einstein metrics , Math

    H. Guenancia, Families of conic Kahler-Einstein metrics , Math. Ann. 376 (2020), no. 1-2, 1–37

    2020

  18. [26]

    Hacon, Y

    C. Hacon, Y. Li, and L. Xie, Fujiki class C varieties and a K\"ahler Criterion , preprint, 2026

    2026

  19. [27]

    Hacon and James McKernan, On Shokurov’s rational connectedness conjecture, Duke Math

    Christopher D. Hacon and James McKernan, On Shokurov’s rational connectedness conjecture, Duke Math. J. 138 (2007), no. 1, 119–136

    2007

  20. [28]

    Hacon and M

    C. Hacon and M. Paun, On the canonical bundle formula and adjunction for generalized Kaehler pairs , 2024, arXiv:2404.12007

    2024 arXiv

  21. [29]

    Hacon, Y

    C. Hacon, Y. Li, and S. Rao, On Pseudo-Effectivity and Volumes of Adjoint Classes in Kähler Families with Projective Central Fiber , https://arxiv.org/pdf/2602.04158 v2

  22. [30]

    Han and Z

    J. Han and Z. Li, Weak Zariski decompositions and log terminal models for generalized polarized pairs, Math. Z. 302 (2022), 707--741

    2022

  23. [31]

    Hashizume, Minimal model program for normal pairs along log canonical locus

    K. Hashizume, Minimal model program for normal pairs along log canonical locus. Forum Math. Sigma 13 (2025), Paper No. e143, 71 pp

    2025

  24. [32]

    oring and T. Peternell, Minimal models for K\

    A. H\"oring and T. Peternell, Minimal models for K\"ahler threefolds , Invent. Math. 203(1), 217–264 (2016)

    2016

  25. [33]

    Kawamata, Flops Connect Minimal Models

    Y. Kawamata, Flops Connect Minimal Models. Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, pp. 419–423

    2008

  26. [34]

    Kebekus and C

    S. Kebekus and C. Schnell, Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities. J. Am. Math. Soc., 34(2):315– 368, 2021

    2021

  27. [35]

    Koll\'ar, Higher direct images of dualizing sheaves

    J. Koll\'ar, Higher direct images of dualizing sheaves. I. Ann. of Math. (2), 123(1):11-42, 1986

    1986

  28. [36]

    Koll\'ar, Quotient spaces modulo algebraic groups , Ann

    J. Koll\'ar, Quotient spaces modulo algebraic groups , Ann. of Math. (2) 145:1 (1997), 33–79

    1997

  29. [37]

    Koll\'ar, Kodaira's canonical bundle formula and adjunction , in Flips for 3-folds and 4-folds, Oxford Lecture Series in Mathematics and Its Applications, Volume 35, pp

    J. Koll\'ar, Kodaira's canonical bundle formula and adjunction , in Flips for 3-folds and 4-folds, Oxford Lecture Series in Mathematics and Its Applications, Volume 35, pp. 134–162 (Oxford University Press, 2007)

    2007

  30. [38]

    Koll\'ar, Singularities of the minimal model program, Cambridge Tracts in Math

    J. Koll\'ar, Singularities of the minimal model program, Cambridge Tracts in Math. 200 (2013), Cambridge Univ. Press. With a collaboration of S. Kov\'acs

    2013

  31. [39]

    Koll\'ar and S

    J. Koll\'ar and S. Mori, Classification of Three-Dimensional Flips , Journal of the American Mathematical Society 5, 533–704 (1992)

    1992

  32. [40]

    Koll\'ar and S

    J. Koll\'ar and S. Mori, Birational geometry of algebraic varieties , volume 134 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original

    1998

  33. [41]

    Lazi\'c, T

    V. Lazi\'c, T. Peternell, On Generalised Abundance, I. Publ. Res. Inst. Math. Sci. 56 (2020), no. 2, pp. 353–389

    2020

  34. [42]

    Nakayama, The lower semicontinuity of the plurigenera of complex varieties , In Algebraic geometry, Sendai, 1985, volume 10 of Adv

    N. Nakayama, The lower semicontinuity of the plurigenera of complex varieties , In Algebraic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pages 551–590. North-Holland, Amsterdam, 1987

    1985

  35. [43]

    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

  36. [44]

    Namikawa, Projectivity criterion of Moishezon spaces and density of projective symplectic varieties , International Journal of Mathematics Vol

    Y. Namikawa, Projectivity criterion of Moishezon spaces and density of projective symplectic varieties , International Journal of Mathematics Vol. 13, No. 02, 2002

    2002

  37. [45]

    Ou, A characterization of uniruled compact K\"ahler manifolds , 2025, arXiv:2501.18088

    W. Ou, A characterization of uniruled compact K\"ahler manifolds , 2025, arXiv:2501.18088

    2025 arXiv

  38. [46]

    Paun, A canonical bundle formula , peronal communication, July 2026, to appear

    M. Paun, A canonical bundle formula , peronal communication, July 2026, to appear

    2026

  39. [47]

    Song and G

    J. Song and G. Tian, The K\"ahler-Ricci flow through singularities , Invent. Math. 207 (2017), no. 2, 519–595

    2017

  40. [48]

    Takegoshi, Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper K\"ahler morphisms

    K. Takegoshi, Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper K\"ahler morphisms . Math. Ann., 303(3):389–416, 1995

    1995

  41. [49]

    Tosatti, KAWA lecture notes on the K\"ahler-Ricci flow

    V. Tosatti, KAWA lecture notes on the K\"ahler-Ricci flow. Ann. Fac. Sci. Toulouse Math. (6) 27 no. 2, (2018) 285–376

    2018

  42. [50]

    Tosatti, Semipositive Line Bundles and (1, 1)-classes

    V. Tosatti, Semipositive Line Bundles and (1, 1)-classes. Acta. Math. Sin. English Ser. (2024)

    2024

  43. [51]

    Tosatti and Y

    V. Tosatti and Y. Zhang , Finite time collapsing of the K\"ahler-Ricci flow on threefolds . Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 no. 1, (2018) 105-118

    2018

  44. [52]

    Wang, On the Iitaka conjecture C_ n,m for K\"ahler fibre spaces , Ann

    J. Wang, On the Iitaka conjecture C_ n,m for K\"ahler fibre spaces , Ann. Fac. Sci. Toulouse Math. (6) 30 (2021), no. 4, 813-897

    2021

  45. [53]

    Zhang, Weak transcendental base-point freeness and diameter lower bounds for the K\"ahler-Ricci flow , https://arxiv.org/pdf/2511.14735

    J. Zhang, Weak transcendental base-point freeness and diameter lower bounds for the K\"ahler-Ricci flow , https://arxiv.org/pdf/2511.14735

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.