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On the boundedness of elliptic Calabi-Yau 4-folds

T0 review · 2 major / 5 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Elliptic Calabi–Yau 4-folds that are not product-type quotients belong to only finitely many algebraic families.

desk verdict Solid upgrade of birational boundedness to actual boundedness for non-product elliptic CY 4-folds; the architecture is clean and the soft spot is the expected external cone-conjecture input, not an internal gap. read the letter →

arxiv 2607.27048 v1 pith:OBHCAEH5 submitted 2026-07-29 math.AG

Stefano Filipazzi , Fulin Xu This is my paper · ORCID
classification math.AG MSC 14E3014J2714J3214D06
keywords Calabi-Yau4-foldsellipticfibrationsboundednessminimalmodelprogramMorrison-KawamataconeconjecturelogpairsindexIitakavolume
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

This paper proves that a vast class of elliptic Calabi–Yau fourfolds is bounded: they appear as the fibers of only finitely many algebraic families. The excluded cases are those crepant-birational to a quotient of a product of a Calabi–Yau threefold and an elliptic curve; in particular the result covers every elliptic Calabi–Yau fourfold whose fibration is not isotrivial. Boundedness immediately implies only finitely many topological types, a question with direct consequences in string theory and in the broader classification of Calabi–Yau varieties. Along the way the authors obtain partial evidence that middle Betti numbers of Calabi–Yau threefolds are bounded, settle the index conjecture for fibered K-trivial fourfolds, and prove that Iitaka volumes of log-canonical fourfolds satisfy the descending chain condition.

What carries the argument

The relative Morrison–Kawamata cone conjecture applied to the rationally connected log Calabi–Yau 3-fold bases produced by the canonical bundle formula. Once those bases are shown to be log-bounded, existing birational boundedness of the fourfolds upgrades to actual boundedness.

What would settle it

Produce an infinite sequence of pairwise non-isomorphic elliptic Calabi–Yau 4-folds, none crepant to a product-type quotient, that cannot appear as fibers of any finite collection of algebraic families; or exhibit one of the base families constructed in the paper for which the relative cone conjecture fails.

Watch

Extended reading notes

Core claim

Any elliptic Calabi–Yau 4-fold that is not crepant birational to a quotient (Y × E)/G, where E is an elliptic curve and Y a Calabi–Yau 3-fold, belongs to finitely many algebraic families. Equivalently, the set of elliptic Calabi–Yau 4-folds not of product type is bounded; the statement applies whenever the elliptic fibration is not isotrivial.

Load-bearing premise

The argument requires that the relative movable-cone conjecture holds for the families of rationally connected three-dimensional bases after étale base change; if it fails for some such family, the upgrade from birational to actual boundedness collapses.

Editorial extensions

If this is right

  • Non-product-type elliptic Calabi–Yau 4-folds have only finitely many topological types.
  • The index of any fibered log-canonical K-trivial 4-fold is bounded by a universal constant.
  • Iitaka volumes of log-canonical 4-fold pairs form a DCC set.
  • The Hodge bundle of a fibration in log-canonical K-trivial 3-folds has uniformly bounded Cartier index, giving partial evidence that middle Betti numbers of Calabi–Yau 3-folds are bounded.
  • Boundedness extends to the product-type case whenever the base has positive augmented irregularity.

Reading between the lines

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

  • The remaining open case for full boundedness of elliptic Calabi–Yau 4-folds is precisely when the base is a quotient of a Calabi–Yau 3-fold, so the 4-fold problem is now tightly tied to the still-open cone conjecture in dimension 3.
  • The same reduction—canonical bundle formula plus relative cone conjecture on the base—can be expected to yield boundedness for elliptic Calabi–Yau n-folds once the cone conjecture is known for the (n−1)-dimensional bases.
  • Uniform boundedness of the Cartier index of the Hodge bundle is a concrete numerical shadow of bounded Betti numbers; reversing the classical Fujino–Mori relation would turn the paper’s integrality result into an actual Betti-number bound.
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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 / 5 minor

Summary. The paper proves that elliptic Calabi–Yau 4-folds that are not crepant birational to a K-trivial orbibundle whose base has vanishing augmented irregularity form a bounded family (Theorem 1.1/4.1). The argument upgrades birational boundedness of fibered K-trivial varieties [EFG+25, Thm A] via the relative cone-conjecture reduction of [FHS25, Thm 6.18], after establishing log boundedness of the rationally connected log Calabi–Yau 3-fold bases (Theorem 3.4/Corollary 3.5). Product-type cases are isolated via the canonical bundle formula and κ(−K)=0 (Lemma 2.12, Proposition 3.1). Secondary results give a uniform b-Cartier index for the moduli part of lc-trivial fibrations of relative dimension 3 (Theorem 4.4), the index conjecture for fibered lc K-trivial 4-folds (Theorem 4.6), and the DCC for Iitaka volumes of lc 4-folds when κ ge d−3 (Theorem 4.7).

Significance. Boundedness of elliptic Calabi–Yau 4-folds outside the product-type locus is a substantial advance on the Reid–Yau finiteness question in dimension 4 and has direct consequences for the finiteness of topological types relevant to string compactifications. The paper cleanly separates the isotrivial/product case (where the cone conjecture for CY 3-folds remains open) from the non-isotrivial case, and the secondary results on the moduli part, the index, and Iitaka volumes are of independent interest; Theorem 4.4 in particular supplies the first general evidence that middle Betti numbers of CY 3-folds should be bounded, by replacing a Betti bound with the known index conjecture for slc 3-folds. The modular structure (birational boundedness + relative cone conjecture + deformation of cones + MMP specialization) is a natural and reusable template.

major comments (2)
  1. [§3.2, Theorem 3.4] Theorem 3.4 (and thus the passage to Theorem 4.1 via [FHS25, Thm 6.18]) relies on the relative movable cone conjecture holding for the families of rationally connected log Calabi–Yau 3-fold bases after étale base change and Noetherian stratification. The manuscript invokes [LZ25a, Thm 6.6] together with Xu’s global results [Xu24, Cor 2] and [Xu25, Thm 1.2], plus the deformation comparison of cones (Lemma 2.20) and BB-decomposition in families (Proposition 2.29). This chain is load-bearing: if the relative statement fails for some of the families that arise, log-boundedness in codimension 1 does not upgrade to actual log boundedness of the bases. The citations appear to cover the needed cases (rationally connected bases, positive augmented irregularity when κ(−K)=0), but the paper should add an explicit paragraph verifying that every family produced by the stratification of W′ satisfies t
  2. [§3.2, Theorem 3.4, Case 2] In Case 2 of the proof of Theorem 3.4 the authors assert that a movable divisor D obtained by deforming f^*(A) remains big on the total space because its restriction to the generic fiber is big (by deformation invariance of self-intersection) and that any MMP for D consists only of small maps. The argument is plausible but terse: it should be spelled out that D lies in the interior of the relative pseudo-effective cone (or at least that its volume is positive on the generic fiber and remains positive after the base change), and that the relative movable cone is full-dimensional so that a small perturbation stays movable. Without this, the appeal to [HMX18, Lem 3.1] and the subsequent identification of the ample model with the original pair is not fully justified.
minor comments (5)
  1. [throughout] Typographical errors: “compelx” (p. 1), “intruduced” (p. 2), “Boudnedness” (heading of §3.2), “parituclar” (Prop. 2.29), “dimesional” (Prop. 2.13), “codimenison” (Lem. 2.12). A global spell-check is needed.
  2. [§2.5] Definition 2.4 of Calabi–Yau variety requires K_X∼0 and vanishing of intermediate reflexive forms on all quasi-étale covers; Remark 2.6 correctly notes that the property is not crepant-birationally invariant. It would help the reader to flag explicitly, when the term is used later (e.g., Prop. 3.1), whether a crepant model or the original singular space is intended.
  3. [§4.2, Theorem 4.4] In the proof of Theorem 4.4, Case 1, the appeal to [Bir19, Prop. 6.3] after obtaining m(K_X+tf^*c)∼0 on a neighbourhood of c is correct but compressed; a one-sentence reminder of how the vertical divisor V is absorbed into the moduli part would improve readability.
  4. [§2.14] Proposition 2.29 (BB decomposition in families) is used crucially for the augmented-irregularity condition in Theorem 3.4(d). The argument via topological local triviality and Grauert–Remmert is standard, but a forward reference from Corollary 3.5 would make the logical dependence clearer.
  5. [§4.2] The constant I in Theorems 4.4–4.5 is stated to depend only on the horizontal multiplicities of ∆; it would be useful to record that the dependence is effective once the index bound for slc 3-folds is fixed, even if no numerical value is given.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: standard research-program stack of independent prior theorems, not a self-referential derivation

full rationale

The central claim (Thm 1.1/4.1) is obtained by composing three independent inputs: birational boundedness of fibered K-trivial varieties [EFG+25, Thm A], the relative/elliptic cone-conjecture machine of [FHS25, Thm 6.18], and log-boundedness of the rationally connected log-CY 3-fold bases (Thm 3.4), itself built from Xu’s global cone results [Xu24, Xu25], Li’s relative criteria, and deformation comparison of cones (Lem 2.20). None of these inputs is defined in terms of the 4-fold boundedness conclusion; none is a fit renamed as a prediction; and none is a uniqueness theorem that forbids alternatives by author fiat. Self-citations (FHS25, Fil20, Fil24, Xu24, Xu25, EFG+25) are ordinary black-box use of prior theorems whose statements do not include the target. Product-type exclusion (Lem 2.12, Prop 3.1), MMP specialization (HMX18 Lem 3.1, Lem 2.24), and index/moduli-part arguments (Thm 4.4–4.6) are self-contained reductions. No equation forces the boundedness conclusion by construction. Score 0 is the honest finding.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

Pure characteristic-0 algebraic geometry. No empirical free parameters. The load-bearing external inputs are standard MMP/canonical-bundle-formula machinery plus several deep recent theorems (birational boundedness of fibered K-trivial varieties, low-dimensional cone conjecture, index conjecture for slc 3-folds, relative cone criteria). Product-type / orbibundle language is definitional, not a new physical entity.

assumptions (7)
  • domain assumption Birational boundedness of fibered K-trivial varieties (EFG+25 Theorem A) supplies the starting birational families for elliptic CY 4-folds.
    Invoked at the start of the proof of Theorem 4.1; without it the upgrade from birational to actual boundedness has nothing to act on.
  • domain assumption Relative Morrison–Kawamata movable cone conjecture holds for the relevant families of rationally connected log CY 3-fold pairs after étale base change (via Li23 + Xu24/Xu25).
    Core of Theorem 3.4; used to produce finitely many small Q-factorial modifications and birational contractions in family.
  • domain assumption FHS25 Theorem 6.18: relative cone conjecture on the base plus birational boundedness imply boundedness of elliptic Calabi–Yau total spaces.
    The bridge from log-bounded bases to bounded 4-folds in Theorem 4.1.
  • domain assumption Index conjecture for semi-log canonical 3-folds (Xu20b): only finitely many possible indices.
    Replaces a bound on middle Betti numbers in the proof that a uniform multiple of the moduli b-divisor is b-Cartier (Theorem 4.4).
  • standard math Standard MMP in characteristic 0: existence of minimal models, flips, good minimal models for klt/dlt pairs in the dimensions used (BCHM, Birkar, HX, etc.).
    Used throughout reductions (orbibundle recognition, dlt modifications, MMP with scaling over the base).
  • standard math Canonical bundle formula for lc-trivial fibrations: induces a generalized pair (Y,B_Y,M) with M b-semiample (Ambro, Fujino–Mori, BFMT25).
    Defines the bases whose log boundedness is proved and supplies the Hodge/moduli part in Theorems 4.4–4.7.
  • domain assumption Beauville–Bogomolov decomposition for klt K-trivial varieties and its behavior in locally stable families (HP19, MW25, Proposition 2.29).
    Needed to treat augmented irregularity in families and to separate product-type cases with vanishing vs positive eq.
invented entities (1)
  • Product type (elliptic CY crepant to (Z×F)/G with componentwise finite group action) independent evidence
    purpose: Delimit the exceptional class excluded from the main boundedness theorem, corresponding to isotrivial/orbibundle situations where the cone conjecture on CY 3-fold bases is unavailable.
    Definitional organizing device built from Kollár’s orbibundles; not a new geometric object with independent ontology.

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Pith. "Pith review of On the boundedness of elliptic Calabi-Yau 4-folds." pith.science (2026). https://pith.science/paper/OBHCAEH5

@misc{pith2026260727048,
  author       = {Pith},
  title        = {Pith review of: On the boundedness of elliptic Calabi-Yau 4-folds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBHCAEH5}},
  note         = {Machine review of arXiv:2607.27048}
}
abstract

In this work, we settle the boundedness of a vast class of elliptic Calabi-Yau 4-folds. In particular, we show that any elliptic Calabi-Yau 4-fold that is not crepant to the quotient of a product $Y \times E$, where $E$ is an elliptic curve and $Y$ a Calabi-Yau 3-fold, belongs to finitely many algebraic families. In particular, the statement applies whenever the elliptic fibration of the Calabi-Yau 4-fold is not isotrivial. We also provide partial evidence for the boundedness of the middle Betti number of Calabi-Yau 3-folds and study the index of fibered $K$-trivial 4-folds.

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Works this paper leans on

127 extracted references · 43 canonical work pages

  1. [1]

    Gachet, C\'ecile , TITLE =. J. \'Ec. polytech. Math. , FJOURNAL =. 2024 , PAGES =. doi:10.5802/jep.277 , URL =

    work page doi:10.5802/jep.277 2024

  2. [2]

    Higher-dimensional complex varieties (

    Oguiso, Keiji , TITLE =. Higher-dimensional complex varieties (. 1996 , ISBN =

    1996

  3. [3]

    Namikawa, Yukihiko , TITLE =. Math. Ann. , FJOURNAL =. 1985 , NUMBER =. doi:10.1007/BF01456182 , URL =

    work page doi:10.1007/bf01456182 1985

  4. [4]

    Sterk, Hans , TITLE =. Math. Z. , FJOURNAL =. 1985 , NUMBER =. doi:10.1007/BF01168156 , URL =

    work page doi:10.1007/bf01168156 1985

  5. [5]

    Shokurov, V. V. , TITLE =. J. Math. Sci. , FJOURNAL =. 1996 , NUMBER =. doi:10.1007/BF02362335 , URL =

    doi:10.1007/bf02362335 1996

  6. [6]

    arXiv e-prints , keywords =

    Positivity of the Moduli Part. arXiv e-prints , keywords =. doi:10.48550/arXiv.2111.00423 , archivePrefix =. 2111.00423 , primaryClass =

    arXiv doi:10.48550/arxiv.2111.00423

  7. [7]

    Martucci, Luca and Risso, Nicol\`o and Weigand, Timo , TITLE =. J. High Energy Phys. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/jhep03(2023)197 , URL =

    work page doi:10.1007/jhep03(2023)197 2023

  8. [8]

    Flips for 3-folds and 4-folds , SERIES =

    Koll\'ar, J\'anos , TITLE =. Flips for 3-folds and 4-folds , SERIES =. 2007 , ISBN =. doi:10.1093/acprof:oso/9780198570615.003.0008 , URL =

    arXiv 2007

Show all 127 references
  1. [9]

    Wittenberg, Olivier , TITLE =. Math. Ann. , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s00208-007-0170-7 , URL =

    2008 doi

  2. [10]

    Nagoya Math

    Fujino, Osamu , TITLE =. Nagoya Math. J. , FJOURNAL =. 2003 , PAGES =. doi:10.1017/S0027763000008679 , URL =

    2003 doi

  3. [11]

    Moduli of boundary polarized

    Ascher, Kenneth and Bejleri, Dori and Blum, Harold and DeVleming, Kristin and Inchiostro, Giovanni and Liu, Yuchen and Wang, Xiaowei , journal=. Moduli of boundary polarized

  4. [12]

    Threefold thresholds , JOURNAL =

    McKernan, James and Prokhorov,. Threefold thresholds , JOURNAL =. 2004 , NUMBER =. doi:10.1007/s00229-004-0457-x , URL =

    2004 doi

  5. [13]

    arXiv e-prints , keywords =

    Well-clipped cones behave themselves under all finite quotients, the cone conjecture under most. arXiv e-prints , keywords =. doi:10.48550/arXiv.2504.01753 , archivePrefix =. 2504.01753 , primaryClass =

    doi arXiv

  6. [14]

    arXiv e-prints , keywords =

    On the cone conjecture for Enriques manifolds. arXiv e-prints , keywords =. doi:10.48550/arXiv.2303.07095 , archivePrefix =. 2303.07095 , primaryClass =

    doi arXiv

  7. [15]

    Kawamata, Yujiro , TITLE =. Math. Ann. , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF01459135 , URL =

    1986 doi

  8. [16]

    , TITLE =

    Morrison, David R. , TITLE =. Math. Ann. , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF01459136 , URL =

    1986 doi

  9. [17]

    Complex and differential geometry , SERIES =

    Markman, Eyal , TITLE =. Complex and differential geometry , SERIES =. 2011 , ISBN =. doi:10.1007/978-3-642-20300-8\_15 , URL =

    2011 doi

  10. [18]

    arXiv e-prints , keywords =

    The Kawamata-Morrison Cone Conjecture for Generalized Hyperelliptic Variety. arXiv e-prints , keywords =. doi:10.48550/arXiv.2403.13156 , archivePrefix =. 2403.13156 , primaryClass =

    doi arXiv

  11. [19]

    Masamura, Yuto , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2025 , NUMBER =. doi:10.1093/imrn/rnaf114 , URL =

    2025 doi

  12. [20]

    Algebra Number Theory , FJOURNAL =

    Jiang, Chen and Liu, Haidong , TITLE =. Algebra Number Theory , FJOURNAL =. 2021 , NUMBER =. doi:10.2140/ant.2021.15.547 , URL =

    2021 doi

  13. [21]

    Hashizume, Kenta and Hu, Zheng-Yu , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2020 , PAGES =. doi:10.1515/crelle-2019-0032 , URL =

    2020 doi

  14. [22]

    Jiang, Chen , TITLE =. J. Algebraic Geom. , FJOURNAL =. 2021 , NUMBER =. doi:10.1090/jag/759 , URL =

    2021 doi

  15. [23]

    Lazi\'c, Vladimir and Peternell, Thomas , TITLE =. Math. Res. Lett. , FJOURNAL =. 2013 , NUMBER =. doi:10.4310/MRL.2013.v20.n6.a9 , URL =

    2013 doi

  16. [24]

    Oguiso, Keiji , TITLE =. J. Algebraic Geom. , FJOURNAL =. 2014 , NUMBER =. doi:10.1090/S1056-3911-2014-00642-1 , URL =

    2014 doi

  17. [25]

    Prendergast-Smith, Artie , TITLE =. J. Math. Sci. Univ. Tokyo , FJOURNAL =. 2012 , NUMBER =

    2012

  18. [26]

    Duke Math

    Totaro, Burt , TITLE =. Duke Math. J. , FJOURNAL =. 2010 , NUMBER =. doi:10.1215/00127094-2010-039 , URL =

    2010 doi

  19. [27]

    and Lian, B

    Klemm, A. and Lian, B. and Roan, S.-S. and Yau, S.-T. , TITLE =. Nuclear Phys. B , FJOURNAL =. 1998 , NUMBER =. doi:10.1016/S0550-3213(97)00798-0 , URL =

    1998 doi

  20. [28]

    Filipazzi, Stefano and Svaldi, Roberto , TITLE =. Mat. Contemp. , FJOURNAL =. 2020 , PAGES =

    2020

  21. [29]

    Internat

    Kawamata, Yujiro , TITLE =. Internat. J. Math. , FJOURNAL =. 1997 , NUMBER =. doi:10.1142/S0129167X97000354 , URL =

    1997 doi

  22. [30]

    , TITLE =

    Morrison, David R. , TITLE =. Ast\'erisque , FJOURNAL =. 1993 , PAGES =

    1993

  23. [31]

    , TITLE =

    Morrison, David R. , TITLE =. Proceedings of the. 1996 , MRCLASS =

    1996

  24. [32]

    Kawamata, Yujiro , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 2008 , NUMBER =. doi:10.2977/prims/1210167332 , URL =

    2008

  25. [33]

    arXiv e-prints , keywords =

    Baily--Borel compactifications of period images and the b-semiampleness conjecture. arXiv e-prints , keywords =. doi:10.48550/arXiv.2508.19215 , archivePrefix =. 2508.19215 , primaryClass =

    doi

  26. [34]

    arXiv e-prints , keywords =

    The movable fan of the Horrocks-Mumford quintic. arXiv e-prints , keywords =. doi:10.48550/arXiv.math/0102055 , archivePrefix =. math/0102055 , primaryClass =

    doi arXiv

  27. [35]

    Duke Math

    Gross, Mark , TITLE =. Duke Math. J. , FJOURNAL =. 1994 , NUMBER =. doi:10.1215/S0012-7094-94-07414-0 , URL =

    1994 doi

  28. [36]

    Surveys in differential geometry

    Yau, Shing-Tung , TITLE =. Surveys in differential geometry. 2009 , ISBN =. doi:10.4310/SDG.2008.v13.n1.a9 , URL =

    2009 doi

  29. [37]

    Reid, Miles , TITLE =. Math. Ann. , FJOURNAL =. 1987 , NUMBER =. doi:10.1007/BF01458074 , URL =

    1987 doi

  30. [38]

    Verbitsky, Misha , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00039-009-0037-z , URL =

    2010 doi

  31. [39]

    Algebraic integrability of foliations with numerically trivial canonical bundle , JOURNAL =

    H\". Algebraic integrability of foliations with numerically trivial canonical bundle , JOURNAL =. 2019 , NUMBER =. doi:10.1007/s00222-018-00853-2 , URL =

    2019 doi

  32. [40]

    A decomposition theorem for singular spaces with trivial canonical class of dimension at most five , JOURNAL =

    Druel, St\'. A decomposition theorem for singular spaces with trivial canonical class of dimension at most five , JOURNAL =. 2018 , NUMBER =. doi:10.1007/s00222-017-0748-y , URL =

    2018 doi

  33. [42]

    Greb, Daniel and Guenancia, Henri and Kebekus, Stefan , TITLE =. Geom. Topol. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/gt.2019.23.2051 , URL =

    2019 doi

  34. [43]

    Beauville, Arnaud , TITLE =. J. Differential Geom. , FJOURNAL =. 1983 , NUMBER =

    1983

  35. [44]

    Bogomolov, F. A. , TITLE =. Mat. Sb. (N.S.) , FJOURNAL =. 1974 , PAGES =

    1974

  36. [45]

    , TITLE =

    Kodaira, K. , TITLE =. Amer. J. Math. , FJOURNAL =. 1968 , PAGES =. doi:10.2307/2373289 , URL =

    1968 doi

  37. [46]

    , TITLE =

    Kodaira, K. , TITLE =. Amer. J. Math. , FJOURNAL =. 1968 , PAGES =. doi:10.2307/2373426 , URL =

    1968 doi

  38. [47]

    , TITLE =

    Kodaira, K. , TITLE =. Amer. J. Math. , FJOURNAL =. 1966 , PAGES =. doi:10.2307/2373150 , URL =

    1966 doi

  39. [48]

    , TITLE =

    Kodaira, K. , TITLE =. Amer. J. Math. , FJOURNAL =. 1964 , PAGES =. doi:10.2307/2373157 , URL =

    1964 doi

  40. [49]

    1949 , PAGES =

    Enriques, Federigo , TITLE =. 1949 , PAGES =

    1949

  41. [50]

    Manuscripta Math

    Xu, Jinsong , TITLE =. Manuscripta Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s00229-019-01137-6 , URL =

    2020 doi

  42. [51]

    Chen, Meng and Zhang, Qi , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2013 , NUMBER =. doi:10.4171/JEMS/406 , URL =

    2013 doi

  43. [52]

    Internat

    Alexeev, Valery , TITLE =. Internat. J. Math. , FJOURNAL =. 1994 , NUMBER =. doi:10.1142/S0129167X94000395 , URL =

    1994 doi

  44. [53]

    1988 , PAGES =

    Goresky, Mark and MacPherson, Robert , TITLE =. 1988 , PAGES =. doi:10.1007/978-3-642-71714-7 , URL =

    1988 doi

  45. [54]

    1996 , PAGES =

    Koll\'ar, J\'anos , TITLE =. 1996 , PAGES =. doi:10.1007/978-3-662-03276-3 , URL =

    1996 doi

  46. [55]

    and Hacon, Christopher D

    Chen, Jungkai A. and Hacon, Christopher D. , TITLE =. Invent. Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/s002220000111 , URL =

    2001 doi

  47. [56]

    Schreieder, Stefan and Tasin, Luca , TITLE =. Math. Ann. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00208-017-1615-2 , URL =

    2019 doi

  48. [57]

    , TITLE =

    Nori, Madhav V. , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1983 , NUMBER =

    1983

  49. [58]

    Matsumura, Shin-ichi and Wang, Juanyong , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2025 , note =. doi:10.4171/JEMS/1702 , URL =

    2025 doi

  50. [59]

    Koll\'ar, J\'anos and Miyaoka, Yoichi and Mori, Shigefumi , TITLE =. J. Algebraic Geom. , FJOURNAL =. 1992 , NUMBER =

    1992

  51. [60]

    Minimal models and extremal rays (

    Greb, Daniel and Kebekus, Stefan and Peternell, Thomas , TITLE =. Minimal models and extremal rays (. 2016 , ISBN =. doi:10.2969/aspm/07010067 , URL =

    2016

  52. [61]

    Koll\'ar, J\'anos , TITLE =. Invent. Math. , FJOURNAL =. 1993 , NUMBER =. doi:10.1007/BF01244307 , URL =

    1993 doi

  53. [62]

    1977 , PAGES =

    Hartshorne, Robin , TITLE =. 1977 , PAGES =

    1977

  54. [63]

    arXiv e-prints , keywords =

    On finiteness of fiber space structures of klt Calabi-Yau pairs in dimension 3. arXiv e-prints , keywords =. doi:10.48550/arXiv.2501.10239 , archivePrefix =. 2501.10239 , primaryClass =

    doi arXiv

  55. [64]

    Graber, Tom and Harris, Joe and Starr, Jason , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2003 , NUMBER =. doi:10.1090/S0894-0347-02-00402-2 , URL =

    2003 doi

  56. [65]

    and Mckernan, James , TITLE =

    Hacon, Christopher D. and Mckernan, James , TITLE =. Duke Math. J. , FJOURNAL =. 2007 , NUMBER =. doi:10.1215/S0012-7094-07-13813-4 , URL =

    2007 doi

  57. [66]

    Moduli , FJOURNAL =

    Filipazzi, Stefano and Spicer, Calum , TITLE =. Moduli , FJOURNAL =. 2025 , PAGES =. doi:10.1112/mod.2025.10004 , URL =

    2025

  58. [67]

    Internat

    Takayama, Shigeharu , TITLE =. Internat. J. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1142/S0129167X0300196X , URL =

    2003 doi

  59. [68]

    arXiv e-prints , keywords =

    Boundedness of some fibered K-trivial varieties. arXiv e-prints , keywords =. doi:10.48550/arXiv.2507.00973 , archivePrefix =. 2507.00973 , primaryClass =

    doi arXiv

  60. [69]

    arXiv e-prints , keywords =

    On structures and discrepancies of klt Calabi--Yau pairs. arXiv e-prints , keywords =. doi:10.48550/arXiv.2405.13321 , archivePrefix =. 2405.13321 , primaryClass =

    doi arXiv

  61. [70]

    Forum Math

    Figueroa, Fernando and Filipazzi, Stefano and Moraga, Joaqu\'in and Peng, Junyao , TITLE =. Forum Math. Sigma , FJOURNAL =. 2025 , PAGES =. doi:10.1017/fms.2024.69 , URL =

    2025 doi

  62. [71]

    1996 , PAGES =

    Beauville, Arnaud , TITLE =. 1996 , PAGES =. doi:10.1017/CBO9780511623936 , URL =

    1996 doi

  63. [72]

    Fujino, Osamu and Mori, Shigefumi , TITLE =. J. Differential Geom. , FJOURNAL =. 2000 , NUMBER =

    2000

  64. [73]

    Complements on

    Xu, Yanning , year=. Complements on. doi:10.17863/CAM.58891 , school=

    doi

  65. [74]

    and McKernan, James and Xu, Chenyang , TITLE =

    Hacon, Christopher D. and McKernan, James and Xu, Chenyang , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2014 , NUMBER =. doi:10.4007/annals.2014.180.2.3 , URL =

    2014 doi

  66. [75]

    Filipazzi, Stefano , TITLE =. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , FJOURNAL =. 2020 , PAGES =

    2020

  67. [76]

    Towards the second main theorem on complements , JOURNAL =

    Prokhorov,. Towards the second main theorem on complements , JOURNAL =. 2009 , NUMBER =. doi:10.1090/S1056-3911-08-00498-0 , URL =

    2009 doi

  68. [77]

    Fujino, Osamu and Gongyo, Yoshinori , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2014 , NUMBER =. doi:10.5802/aif.2894 , URL =

    2014 doi

  69. [78]

    Ambro, Florin , TITLE =. J. Differential Geom. , FJOURNAL =. 2004 , NUMBER =

    2004

  70. [79]

    Birkar, Caucher and Zhang, De-Qi , TITLE =. Publ. Math. Inst. Hautes \'Etudes Sci. , FJOURNAL =. 2016 , PAGES =. doi:10.1007/s10240-016-0080-x , URL =

    2016 doi

  71. [80]

    Internat

    Floris, Enrica , TITLE =. Internat. J. Math. , FJOURNAL =. 2023 , NUMBER =. doi:10.1142/S0129167X23500805 , URL =

    2023 doi

  72. [81]

    2023 , PAGES =

    Koll\'ar, J\'anos , TITLE =. 2023 , PAGES =

    2023

  73. [82]

    and McKernan, James , TITLE =

    Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2010 , NUMBER =. doi:10.1090/S0894-0347-09-00649-3 , URL =

    2010 doi

  74. [83]

    Recent advances in algebraic geometry , SERIES =

    Koll\'ar, J\'anos , TITLE =. Recent advances in algebraic geometry , SERIES =. 2015 , ISBN =

    2015

  75. [84]

    1998 , PAGES =

    Koll\'ar, J\'anos and Mori, Shigefumi , TITLE =. 1998 , PAGES =. doi:10.1017/CBO9780511662560 , URL =

    1998 doi

  76. [85]

    and Xu, Chenyang , TITLE =

    Hacon, Christopher D. and Xu, Chenyang , TITLE =. Invent. Math. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00222-012-0409-0 , URL =

    2013 doi

  77. [86]

    Birkar, Caucher , TITLE =. Publ. Math. Inst. Hautes \'Etudes Sci. , FJOURNAL =. 2012 , PAGES =. doi:10.1007/s10240-012-0039-5 , URL =

    2012 doi

  78. [87]

    Ambro, Florin , TITLE =. Compos. Math. , FJOURNAL =. 2005 , NUMBER =. doi:10.1112/S0010437X04001071 , URL =

    2005 doi

  79. [88]

    arXiv Mathematics e-prints , keywords =

    The Adjunction Conjecture and its applications. arXiv Mathematics e-prints , keywords =. doi:10.48550/arXiv.math/9903060 , archivePrefix =. math/9903060 , primaryClass =

    doi arXiv

  80. [89]

    and McKernan, James and Xu, Chenyang , TITLE =

    Hacon, Christopher D. and McKernan, James and Xu, Chenyang , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2013 , NUMBER =. doi:10.4007/annals.2013.177.3.6 , URL =

    2013 doi

  81. [90]

    Filipazzi, Stefano , TITLE =. Algebr. Geom. , FJOURNAL =. 2024 , NUMBER =

    2024

  82. [91]

    Duke Math

    Keel, Sean and Matsuki, Kenji and McKernan, James , TITLE =. Duke Math. J. , FJOURNAL =. 1994 , NUMBER =. doi:10.1215/S0012-7094-94-07504-2 , URL =

    1994 doi

  83. [92]

    Fujita, Takao , TITLE =. J. Math. Soc. Japan , FJOURNAL =. 1986 , NUMBER =. doi:10.2969/jmsj/03810019 , URL =

    1986

  84. [93]

    Di Cerbo, Gabriele and Svaldi, Roberto , TITLE =. Compos. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1112/S0010437X2100717X , URL =

    2021 doi

  85. [94]

    Li, Zhan , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2024 , NUMBER =. doi:10.1112/jlms.12871 , URL =

    2024 doi

  86. [95]

    arXiv e-prints , keywords =

    On the relative Morrison--Kawamata cone conjecture (II). arXiv e-prints , keywords =. doi:10.48550/arXiv.2309.04673 , archivePrefix =. 2309.04673 , primaryClass =

    doi arXiv

  87. [96]

    and Svaldi, Roberto , TITLE =

    Filipazzi, Stefano and Hacon, Christopher D. and Svaldi, Roberto , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2025 , NUMBER =. doi:10.4171/jems/1467 , URL =

    2025 doi

  88. [97]

    Birkar, Caucher and Di Cerbo, Gabriele and Svaldi, Roberto , TITLE =. J. Differential Geom. , FJOURNAL =. 2024 , NUMBER =. doi:10.4310/jdg/1727712887 , URL =

    2024

  89. [98]

    Algebra, arithmetic, and geometry: in honor of

    Koll\'ar, J\'anos and Larsen, Michael , TITLE =. Algebra, arithmetic, and geometry: in honor of. 2009 , ISBN =. doi:10.1007/978-0-8176-4747-6\_6 , URL =

    2009 doi

  90. [99]

    arXiv e-prints , keywords =

    On the cone conjecture for certain pairs of dimension at most 4. arXiv e-prints , keywords =. doi:10.48550/arXiv.2405.20899 , archivePrefix =. 2405.20899 , primaryClass =

    doi arXiv

  91. [100]

    Li, Zhan and Zhao, Hang , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2025 , NUMBER =. doi:10.1112/plms.70099 , URL =

    2025 doi

  92. [101]

    arXiv e-prints , keywords =

    Birkar, Caucher , title = ". arXiv e-prints , keywords =. doi:10.48550/arXiv.2209.08797 , archivePrefix =. 2209.08797 , primaryClass =

    doi arXiv

  93. [102]

    Hacon and James McKernan and Chenyang Xu , TITLE =

    Christopher D. Hacon and James McKernan and Chenyang Xu , TITLE =. J. Eur. Math. Soc. , FJOURNAL =. 2018 , NUMBER =

    2018

  94. [103]

    Singularities of the Minimal Model Program , publisher=

    Koll\'ar, J\'anos , year=. Singularities of the Minimal Model Program , publisher=

  95. [104]

    arXiv e-prints , keywords =

    Low degree Hodge theory for klt varieties. arXiv e-prints , keywords =. doi:10.48550/arXiv.1612.01919 , archivePrefix =. 1612.01919 , primaryClass =

    doi arXiv

  96. [105]

    Mathematical Proceedings of the Cambridge Philosophical Society , author=

    The fundamental group of the orbit space of a discontinuous group , volume=. Mathematical Proceedings of the Cambridge Philosophical Society , author=. 1968 , pages=. doi:10.1017/S0305004100042845 , number=

    1968 doi

  97. [106]

    2001 , publisher=

    Higher-Dimensional Algebraic Geometry , author=. 2001 , publisher=

    2001

  98. [107]

    2000 , journal=

    Abundance theorem for semi log canonical threefolds , author=. 2000 , journal=

    2000

  99. [108]

    Anti-pluricanonical systems on

    Birkar, Caucher , year =. Anti-pluricanonical systems on. Annals of Mathematics , doi =

  100. [109]

    Higher Direct Images of Dualizing Sheaves

    János Kollár , journal =. Higher Direct Images of Dualizing Sheaves

  101. [110]

    1992 , zbl =

    Flips and abundance for algebraic threefolds -. 1992 , zbl =

    1992

  102. [111]

    Flips and abundance for algebraic threefolds -

    Corti, Alessio , title =. Flips and abundance for algebraic threefolds -. 1992 , pages =

    1992

  103. [112]

    Flips and abundance for algebraic threefolds -

    Koll\'ar, J\'anos , title =. Flips and abundance for algebraic threefolds -. 1992 , pages =

    1992

  104. [113]

    Manuscripta Mathematica , volume =

    Homogeneous fibrations on log Calabi--Yau varieties. Manuscripta Mathematica , volume =

  105. [114]

    Proceedings of the Japan Academy, Series A, Mathematical Sciences , number =

    Osamu Fujino , title =. Proceedings of the Japan Academy, Series A, Mathematical Sciences , number =. 2009 , doi =

    2009

  106. [115]

    International Mathematics Research Notices , year=

    Inductive Approach to Effective b-Semiampleness , author=. International Mathematics Research Notices , year=

  107. [116]

    Minimal models and the Kodaira dimension of algebraic fiber spaces

    Yujiro Kawamata , pages =. Minimal models and the Kodaira dimension of algebraic fiber spaces. , title =. Journal für die reine und angewandte Mathematik , doi =. 1985 , lastchecked =

    1985

  108. [117]

    arXiv e-prints , keywords =

    The geometric cone conjecture in relative dimension two. arXiv e-prints , keywords =. doi:10.48550/arXiv.2409.13068 , archivePrefix =. 2409.13068 , primaryClass =

    doi arXiv

  109. [118]

    arXiv e-prints , keywords =

    The Morrison Cone Conjecture under Deformation. arXiv e-prints , keywords =. doi:10.48550/arXiv.2410.05949 , archivePrefix =. 2410.05949 , primaryClass =

    doi arXiv

  110. [119]

    Algebraic methods in the global theory of complex spaces , NOTE =

    B. Algebraic methods in the global theory of complex spaces , NOTE =. 1976 , PAGES =

    1976

  111. [120]

    2003 , PAGES =

    Rev\^. 2003 , PAGES =

    2003

  112. [121]

    Bakker, Benjamin and Guenancia, Henri and Lehn, Christian , TITLE =. Invent. Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00222-022-01096-y , URL =

    2022 doi

  113. [122]

    and Peternell, Thomas , TITLE =

    Greb, Daniel and Kebekus, Stefan and Kov\'acs, S\'andor J. and Peternell, Thomas , TITLE =. Publ. Math. Inst. Hautes \'Etudes Sci. , FJOURNAL =. 2011 , PAGES =. doi:10.1007/s10240-011-0036-0 , URL =

    2011 doi

  114. [123]

    Bakker, Benjamin and Lehn, Christian , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2022 , PAGES =. doi:10.1515/crelle-2022-0033 , URL =

    2022 doi

  115. [124]

    arXiv e-prints , keywords =

    On the Morrison--Kawamata dream space and its applications. arXiv e-prints , keywords =. doi:10.48550/arXiv.2512.01516 , archivePrefix =. 2512.01516 , primaryClass =

    doi arXiv

  116. [125]

    arXiv e-prints , keywords =

    The effective cone conjecture for Calabi--Yau pairs. arXiv e-prints , keywords =. doi:10.48550/arXiv.2406.07307 , archivePrefix =. 2406.07307 , primaryClass =

    doi

  117. [126]

    Journal of the London Mathematical Society , volume =

    Chen, Guodu and Han, Jingjun and Liu, Wenfei , title =. Journal of the London Mathematical Society , volume =. doi:https://doi.org/10.1112/jlms.70132 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms.70132 , year =

    doi

  118. [127]

    Han, Jingjun and Liu, Jihao and Shokurov, V. V. , title =. Peking Mathematical Journal , volume =

  119. [128]

    arXiv e-prints , keywords =

    Boundedness and volume of generalised pairs. arXiv e-prints , keywords =. doi:10.48550/arXiv.2103.14935 , archivePrefix =. 2103.14935 , primaryClass =

    doi arXiv

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