@article{EinKuchleLazarsfeld1995,
  author = {Ein, Lawrence and K{\"u}chle, Oliver and Lazarsfeld, Robert},
  title = {Local positivity of ample line bundles},
  journal = {Journal of Differential Geometry},
  volume = {42}, number = {2}, year = {1995}, pages = {193--219},
  doi = {10.4310/jdg/1214457231},
  url = {https://www.math.stonybrook.edu/~roblaz/Reprints/Ein.Kuchle.Laz.LPALB.pdf},
  note = {Proposition 2.3}
}

@misc{OpenAIIitaka2026,
  author = {{OpenAI}},
  title = {{Orbifold and logarithmic Iitaka subadditivity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf}{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026}},
  year = {2026},
  note = {Corollaries 6.2 and 6.3}
}

@article{ClaudonHoeringKollar2013,
  author = {Claudon, Beno{\^i}t and H{\"o}ring, Andreas and Koll{\'a}r, J{\'a}nos},
  title = {Algebraic varieties with quasi-projective universal cover},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume = {679},
  year = {2013},
  pages = {207--221},
  doi = {10.1515/crelle.2012.017},
  url = {https://math.univ-cotedazur.fr/~hoering/articles/a13-universal.pdf}
}

@article{Hashizume2018,
  author = {Hashizume, Kenta},
  title = {On the non-vanishing conjecture and existence of log minimal models},
  journal = {Publications of the Research Institute for Mathematical Sciences},
  volume = {54},
  number = {1},
  year = {2018},
  pages = {89--104},
  doi = {10.4171/PRIMS/54-1-3},
  eprint = {1609.00121},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://arxiv.org/abs/1609.00121v4},
  note = {Theorem 1.4 in arXiv:1609.00121v4}
}

@incollection{FujinoGongyoRings2014,
  author = {Fujino, Osamu and Gongyo, Yoshinori},
  title = {On log canonical rings},
  booktitle = {Higher Dimensional Algebraic Geometry in honour of Professor Yujiro Kawamata's sixtieth birthday},
  series = {Advanced Studies in Pure Mathematics},
  volume = {74},
  publisher = {Mathematical Society of Japan},
  address = {Tokyo},
  year = {2017},
  pages = {159--169},
  eprint = {1302.5194},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://www.math.kyoto-u.ac.jp/~fujino/log-ring-aspm.pdf},
  note = {Conjectures A--C, Theorem 1.1, Corollary 1.4, and Lemma 3.5}
}

@article{Birkar2011II,
  author = {Birkar, Caucher},
  title = {On existence of log minimal models {II}},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume = {658},
  year = {2011},
  pages = {99--113},
  doi = {10.1515/crelle.2011.062},
  eprint = {0907.4170},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://arxiv.org/abs/0907.4170v1},
  note = {Remark 3.1 and Proposition 3.2(3) in arXiv:0907.4170v1}
}

@article{Birkar2010,
  author = {Birkar, Caucher},
  title = {On existence of log minimal models},
  journal = {Compositio Mathematica},
  volume = {146},
  number = {4},
  year = {2010},
  pages = {919--928},
  doi = {10.1112/S0010437X09004564},
  eprint = {0706.1792},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://arxiv.org/abs/0706.1792v3},
  note = {Lemma 3.6 in arXiv:0706.1792v3}
}

@article{Birkar2012Special,
  author = {Birkar, Caucher},
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  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume = {115},
  year = {2012},
  pages = {325--368},
  doi = {10.1007/s10240-012-0039-5},
  eprint = {1104.4981},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  note = {Theorem 1.9, restated as Theorem 4.1}
}

@article{FujinoGongyo2014,
  author = {Fujino, Osamu and Gongyo, Yoshinori},
  title = {Log pluricanonical representations and the abundance conjecture},
  journal = {Compositio Mathematica},
  volume = {150},
  number = {4},
  year = {2014},
  pages = {593--620},
  doi = {10.1112/S0010437X13007495},
  url = {https://www.math.kyoto-u.ac.jp/~fujino/fg-comp-final.pdf},
  note = {Theorem 1.5, restated as Theorem 4.3}
}

@article{Ambro2005,
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}

@article{BCHM2010,
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  number = {2},
  year = {2010},
  pages = {405--468},
  doi = {10.1090/S0894-0347-09-00649-3},
  eprint = {math/0610203},
  archivePrefix = {arXiv},
  primaryClass = {math.AG}
}

@misc{Fujino2017,
  author = {Fujino, Osamu},
  title = {Introduction to the log minimal model program for log canonical pairs},
  year = {2009},
  howpublished = {Unpublished manuscript},
  eprint = {0907.1506},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://www.math.kyoto-u.ac.jp/~fujino/MMP21.pdf},
  note = {Version 6.01, January 8, 2009; Theorem 3.74(3), p. 109}
}

@article{GongyoLehmann2013,
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  doi = {10.1112/S0010437X12000553},
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  note = {Read with the numerical-dimension erratum on Lehmann's research page}
}

@article{Fujino2019Numerical,
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  doi = {10.2969/jmsj/82568256},
  eprint = {1904.11639}, archivePrefix = {arXiv}, primaryClass = {math.AG},
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}

@article{KovacsPatakfalvi2017,
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  journal = {Journal of the American Mathematical Society},
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  doi = {10.1090/jams/871},
  url = {https://sites.math.washington.edu/~kovacs/current/papers/Kovacs_Patakfalvi__Projectivity.pdf},
  note = {Theorem 9.9 in the author version}
}

@article{BDPP2013,
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@article{LazicPeternell2017,
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}

@misc{BrunebarbeCampana2015,
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  eprint = {1510.07922}, archivePrefix = {arXiv}, primaryClass = {math.AG},
  url = {https://arxiv.org/abs/1510.07922},
  note = {Version 2}
}

@article{Campana1995,
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@article{CampanaPaun2015,
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  title = {Orbifold generic semi-positivity: an application to families of canonically polarized manifolds},
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  year = {2015}, volume = {65}, number = {2}, pages = {835--861},
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}

@article{Langer2004,
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@article{Sun2008,
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}

@article{FujinoHashizume2023,
  author = {Fujino, Osamu and Hashizume, Kenta},
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  note = {Paper No. 13, 32 pp.; Theorem 2.10 in the accepted author version},
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@article{FujinoGongyoModuli2014,
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@incollection{HMX2018,
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  booktitle = {Algebraic Geometry: Salt Lake City 2015, Part 1},
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@incollection{Coste2007,
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@book{SGA1,
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@article{MustataPopa2022,
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@article{KollarKovacs2010,
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}

@incollection{BauerEtAl2002,
  author = {Bauer, Thomas and Campana, Fr{\'e}d{\'e}ric and Eckl, Thomas and Kebekus, Stefan and Peternell, Thomas and Rams, S{\l}awomir and Szemberg, Tomasz and Wotzlaw, Lorenz},
  title = {A reduction map for nef line bundles},
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  doi = {10.1007/978-3-642-56202-0_2},
  url = {https://arxiv.org/abs/math/0106147v1},
  note = {Theorem 2.1 in arXiv:math/0106147v1}
}

@misc{StacksExistence,
  author = {{The Stacks Project Authors}},
  title = {The {Stacks} Project: {Grothendieck}'s existence theorem, {I}},
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  howpublished = {\url{https://stacks.math.columbia.edu/tag/087V}},
  note = {Tag 087V, Lemmas 30.24.1 and 30.24.3; accessed September 26, 2026}
}

@article{Taji2014,
  author = {Taji, Behrouz},
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  doi = {10.5802/aif.2845},
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  note = {With an appendix by Fr{\'e}d{\'e}ric Campana}
}

@article{Hanamura1988,
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  note = {Theorems 2.1--2.2; statements also reproduced in Blanc, Theorems 3.8--3.9}
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@incollection{Blanc2017,
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@article{Siu1974,
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@book{Demailly2012,
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  address = {Somerville, MA, and Beijing},
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}

@article{MiyaokaMori1986,
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  title = {A numerical criterion for uniruledness},
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@article{Paun2007,
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}

@article{Miyaoka1988,
  author = {Miyaoka, Yoichi},
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@article{Kawamata1992,
  author = {Kawamata, Yujiro},
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  volume = {108}, year = {1992}, pages = {229--246},
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@article{KeelMatsukiMcKernan1994,
  author = {Keel, Sean and Matsuki, Kenji and McKernan, James},
  title = {Log abundance theorem for threefolds},
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  doi = {10.1215/S0012-7094-94-07504-2},
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@article{KeelMatsukiMcKernan2004Correction,
  author = {Keel, Sean and Matsuki, Kenji and McKernan, James},
  title = {Corrections to ``Log abundance theorem for threefolds''},
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  doi = {10.1215/S0012-7094-04-12236-5},
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}

@misc{Matsuki2003Correction,
  author = {Matsuki, Kenji},
  title = {A correction to the paper ``Log abundance theorem for threefolds''},
  year = {2003},
  eprint = {math/0302360v1}, archivePrefix = {arXiv}, primaryClass = {math.AG},
  url = {https://arxiv.org/abs/math/0302360v1},
  note = {arXiv:math/0302360v1}
}

@article{Fujino2000,
  author = {Fujino, Osamu},
  title = {Abundance theorem for semi log canonical threefolds},
  journal = {Duke Mathematical Journal},
  volume = {102}, number = {3}, year = {2000}, pages = {513--532},
  doi = {10.1215/S0012-7094-00-10237-2},
  url = {https://www.math.kyoto-u.ac.jp/~fujino/Abundance.pdf}
}

@article{Gongyo2013,
  author = {Gongyo, Yoshinori},
  title = {Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs},
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  doi = {10.1090/S1056-3911-2012-00593-1},
  eprint = {1005.2796v2}, archivePrefix = {arXiv}, primaryClass = {math.AG},
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@article{DemaillyHaconPaun2013,
  author = {Demailly, Jean-Pierre and Hacon, Christopher D. and P{\u a}un, Mihai},
  title = {Extension theorems, non-vanishing and the existence of good minimal models},
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@book{EsnaultViehweg1992,
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@article{Deligne1971,
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@misc{OpenAIMinimalModels2026,
  author = {{OpenAI}},
  title = {{Minimal models and Mori fibre spaces for generalized log canonical $\mathbb{Q}$-pairs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf}{OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026}},
  year = {2026},
  note = {Corollary 1.2}
}

@misc{LiuXu2025,
  author = {Liu, Jihao and Xu, Zheng},
  title = {Non-vanishing implies numerical dimension one abundance},
  year = {2025},
  eprint = {2505.05250v2}, archivePrefix = {arXiv}, primaryClass = {math.AG},
  url = {https://arxiv.org/abs/2505.05250v2},
  note = {arXiv:2505.05250v2, July 30, 2025; Theorems 3.1, 5.1 and 5.4}
}

@article{KitadaiSumihiro2008,
  author = {Kitadai, Yukinori and Sumihiro, Hideyasu},
  title = {Canonical filtrations and stability of direct images by {Frobenius} morphisms},
  journal = {Tohoku Mathematical Journal},
  series = {2}, volume = {60}, number = {2}, year = {2008},
  pages = {287--301}, doi = {10.2748/tmj/1215442876},
  url = {https://doi.org/10.2748/tmj/1215442876},
  note = {Definition 3.1, Remark 3.2, and Corollary 3.5}
}

@article{MustataSchwede2014,
  author = {Musta{\c t}{\u a}, Mircea and Schwede, Karl},
  title = {A {Frobenius} variant of {Seshadri} constants},
  journal = {Mathematische Annalen},
  volume = {358}, number = {3-4}, year = {2014}, pages = {861--878},
  doi = {10.1007/s00208-013-0976-4},
  eprint = {1203.1081v1}, archivePrefix = {arXiv}, primaryClass = {math.AG},
  url = {https://arxiv.org/abs/1203.1081v1},
  note = {Section 2 and Proposition 2.12 in arXiv:1203.1081v1}
}

@article{JoshiRamananXiaYu2006,
  author = {Joshi, Kirti and Ramanan, S. and Xia, Eugene Z. and Yu, Jiu-Kang},
  title = {On vector bundles destabilized by {Frobenius} pull-back},
  journal = {Compositio Mathematica},
  volume = {142}, number = {3}, year = {2006}, pages = {616--630},
  doi = {10.1112/S0010437X05001788},
  url = {https://doi.org/10.1112/S0010437X05001788},
  note = {Section 5.3}
}

@misc{ChanChoi2021,
 author = {Chan, Tsz On Mario and Choi, Young-Jun},
 title = {Extension with log-canonical measures and an improvement to the plt extension of {Demailly--Hacon--P\u{a}un}},
 year = {2021}, eprint = {1912.08076v5}, archivePrefix = {arXiv},
 url = {https://arxiv.org/abs/1912.08076v5},
 doi = {10.1007/s00208-021-02152-3},
 note = {arXiv:1912.08076v5; Theorem 1.6.1}
}


@book{Kollar2013,
  author = {Koll{\'a}r, J{\'a}nos},
  title = {Singularities of the Minimal Model Program},
  series = {Cambridge Tracts in Mathematics}, volume = {200},
  publisher = {Cambridge University Press}, address = {Cambridge}, year = {2013},
  note = {With the collaboration of S{\'a}ndor Kov{\'a}cs}
}

@article{MehtaRamanathan1982,
  author = {Mehta, V. B. and Ramanathan, A.},
  title = {Semistable sheaves on projective varieties and their restriction to curves},
  journal = {Mathematische Annalen}, volume = {258}, year = {1982},
  pages = {213--224}, note = {Theorem 6.1 and Remark 6.2}
}

@misc{StacksSectionModules,
  author = {{The Stacks Project Authors}},
  title = {The {Stacks} Project: Coherent sheaves on {Proj}, {I}},
  year = {2026},
  howpublished = {\url{https://stacks.math.columbia.edu/tag/01YS}},
  note = {Tag 01YS, Lemma 30.14.1(5); accessed September 27, 2026}
}

@misc{FujinoRelativeAnalytic2025,
  author = {Fujino, Osamu},
  title = {On finiteness of relative log pluricanonical representations},
  year = {2025},
  howpublished = {Author manuscript},
  url = {https://www.math.kyoto-u.ac.jp/~fujino/analytic-abundance7.pdf},
  note = {Version 0.14, May 29, 2025; Theorem 1.10 and its proof on p. 30}
}

@misc{FujinoAnalyticMMP2022,
  author = {Fujino, Osamu},
  title = {Minimal model program for projective morphisms between complex analytic spaces},
  year = {2022},
  eprint = {2201.11315v1}, archivePrefix = {arXiv}, primaryClass = {math.AG},
  url = {https://arxiv.org/abs/2201.11315v1},
  note = {Definitions 2.28, 2.32 and 2.48}
}

@misc{CaoPaunAbelian2016,
  author = {Cao, Junyan and P{\u a}un, Mihai},
  title = {{Kodaira} dimension of algebraic fiber spaces over abelian varieties},
  year = {2016},
  eprint = {1504.01095v3},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1504.01095v3},
  note = {Version 3; Theorem 1.1 gives the abelian-base case}
}

@misc{HaconPopaSchnell2017,
  author = {Hacon, Christopher D. and Popa, Mihnea and Schnell, Christian},
  title = {Algebraic fiber spaces over abelian varieties: around a recent theorem by {Cao} and {P{\u a}un}},
  year = {2017},
  eprint = {1611.08768v2},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1611.08768v2},
  note = {Version 2; Theorem 1.1}
}