Bibliography

Eleonora Di Nezza, Vincent Guedj, and Chinh Hoang Lu. Geometric smoothing by the Kähler–Ricci flow. 2026. arXiv:2607.08325v1. URL: https://arxiv.org/abs/2607.08325, arXiv:2607.08325.

Quang-Tuan Dang. Hermitian null loci. arXiv:2404.01126v2, 2026. Accepted for publication in Transactions of the American Mathematical Society. Locators refer to arXiv:2404.01126v2, 17 June 2026; Theorem 1.1 assumes nefness and bigness, whereas Theorems A and C retain the stated volume and metric hypotheses. URL: https://arxiv.org/abs/2404.01126.

Vladimir G. Berkovich. Étale cohomology for non-Archimedean analytic spaces. Inst. Hautes Études Sci. Publ. Math., pages 5–161 (1994), 1993. URL: http://www.numdam.org/item?id=PMIHES_1993__78__5_0.

Masakazu Takakura. A generalized $L^2$ division theorem and the strong openness property of multiplier ideal sheaves. 2026. arXiv:2609.36420v1, 29 September 2026. Numbered statements refer to this version. URL: https://arxiv.org/abs/2609.36420v1, arXiv:2609.36420v1.

Antoine Ducros. Réduction en famille d'espaces affinoïdes. Bull. Soc. Math. France, 149(4):547–611, 2021. URL: https://doi.org/10.24033/bsmf.2838, doi:10.24033/bsmf.2838.

Gang Tian. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom., 32(1):99–130, 1990. URL: http://projecteuclid.org/euclid.jdg/1214445039.

Tamás Darvas. The Mabuchi completion of the space of Kähler potentials. American Journal of Mathematics, 139(5):1275–1313, 2017. Numbered statements and page locators refer to arXiv:1401.7318v3. URL: https://arxiv.org/abs/1401.7318v3.

Nurbek Kh. Narzillaev. Weighted regularity and plurithinness in $\mathbb C^n$. 2026. arXiv:2609.23530v1. URL: https://arxiv.org/abs/2609.23530v1, arXiv:2609.23530.

Michael Francis Atiyah. The index of elliptic operators on compact manifolds. In Séminaire Bourbaki, volume 8, 159–169. 1964. Exposé no. 253. URL: https://www.numdam.org/item/SB_1962-1964__8__159_0/.

Xieping Wang. Complete pluripolar sets and removable singularities of plurisubharmonic functions. Indiana Univ. Math. J., 73(6):1929–1936, 2024.

Francesco Antonio Denisi and Andreas Höring. Canonical extensions of ruled surfaces over elliptic curves. 2026. arXiv:2610.03048v1. URL: https://arxiv.org/abs/2610.03048v1, arXiv:2610.03048.

R. E. Greene and H. Wu. $C^\infty $ approximations of convex, subharmonic, and plurisubharmonic functions. Annales Scientifiques de l'École Normale Supérieure, 12(1):47–84, 1979. URL: https://www.numdam.org/item/ASENS_1979_4_12_1_47_0/, doi:10.24033/asens.1361.

W. K. Hayman and P. B. Kennedy. Subharmonic functions. Vol. I. Volume No. 9 of London Mathematical Society Monographs. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976.

Urban Cegrell. The general definition of the complex Monge-Ampère operator. Ann. Inst. Fourier (Grenoble), 54(1):159–179, 2004. URL: https://doi.org/10.5802/aif.2014, doi:10.5802/aif.2014.

Duc Viet Vu and Mingchen Xia. A transcendental Nakamaye theorem. Undated author-supplied manuscript; checked copy retained on 3 October 2026.

Chinh H. Lu, Trong-Thuc Phung, and Tât-Dat Tô. Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds. Annales de l'Institut Fourier, 71(5):2019–2045, 2021. URL: https://aif.centre-mersenne.org/item/10.5802/aif.3436.pdf, doi:10.5802/aif.3436.

S. K. Donaldson. Symmetric spaces, Kähler geometry and Hamiltonian dynamics. In Northern California Symplectic Geometry Seminar, volume 196 of Amer. Math. Soc. Transl. Ser. 2, pages 13–33. Amer. Math. Soc., Providence, RI, 1999. URL: https://doi.org/10.1090/trans2/196/02, doi:10.1090/trans2/196/02.

K. Uhlenbeck and S.-T. Yau. On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Communications on Pure and Applied Mathematics, 39(S1):S257–S293, 1986. doi:10.1002/cpa.3160390714.

Bo Berndtsson. The openness conjecture for plurisubharmonic functions. 2013. arXiv:1305.5781.

Mihai Păun and Shigeharu Takayama. Positivity of twisted relative pluricanonical bundles and their direct images. J. Algebraic Geom., 27(2):211–272, 2018. URL: https://doi.org/10.1090/jag/702, doi:10.1090/jag/702.

Pierre Lelong. Les fonctions plurisousharmoniques. Ann. Sci. École Norm. Sup. (3), 62:301–338, 1945. URL: http://www.numdam.org/item?id=ASENS_1945_3_62__301_0.

Zbigniew Błocki. Several complex variables. 2014. Lecture notes, UAM, Poznań, Fall 2014. URL: https://gamma.im.uj.edu.pl/~blocki/publ/ln/scv-poznan.pdf.

Bent Fuglede. Finely harmonic functions. Volume Vol. 289 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1972.

S. Boucksom, C. Favre, and M. Jonsson. The Non-Archimedean Monge–Ampère Equation, pages 31–49. Springer, 2016.

Anna Fino and Luigi Vezzoni. A note on the pluriclosed flow on balanced manifolds with $c_1=0$. arXiv:2606.03176v1, 2026. 2 June 2026; Conjectures 1.1–1.2. URL: https://arxiv.org/abs/2606.03176.

Ahcène Lamari. Courants kählériens et surfaces compactes. Ann. Inst. Fourier (Grenoble), 49(1):263–285, 1999. URL: https://aif.centre-mersenne.org/item/10.5802/aif.1673.pdf, doi:10.5802/aif.1673.

Sławomir Dinew and Sławomir Kołodziej. Liouville and Calabi–Yau type theorems for complex Hessian equations. American Journal of Mathematics, 139(2):403–415, 2017. Numbered statements refer to arXiv:1203.3995v2. URL: https://arxiv.org/abs/1203.3995v2.

Duc-Viet Vu. Locally pluripolar sets are pluripolar. Internat. J. Math., 30(13):1950029, 13, 2019. arXiv version: arXiv:1812.00465v1, 2 December 2018. URL: https://doi.org/10.1142/S0129167X19500290, doi:10.1142/S0129167X19500290.

Sébastien Boucksom, Vincent Guedj, and Chinh H. Lu. Volumes of Bott–Chern classes. 2024. arXiv:2406.01090.

Julius Ross and David Witt Nyström. Analytic test configurations and geodesic rays. J. Symplectic Geom., 12(1):125–169, 2014. URL: https://doi.org/10.4310/JSG.2014.v12.n1.a5, doi:10.4310/JSG.2014.v12.n1.a5.

Xiaonan Ma and George Marinescu. Holomorphic Morse inequalities and Bergman kernels. Volume 254 of Progress in Mathematics. Birkhäuser Verlag, Basel, 2007. ISBN 978-3-7643-8096-0.

Qi'an Guan and Xiangyu Zhou. Effectiveness of Demailly's strong openness conjecture and related problems. Invent. Math., 202(2):635–676, 2015. URL: https://doi.org/10.1007/s00222-014-0575-3, doi:10.1007/s00222-014-0575-3.

Hoang-Chinh Lu. Pluripotential theory and complex geometry. Habilitation thesis, Université Paris-Saclay, 2022. URL: https://hoangchinh-lu.perso.math.cnrs.fr/HDR_Lu.pdf.

Gunnar Þór Magnússon. The geometry of Kähler cones. arXiv:1211.6934v1, 2012. URL: https://arxiv.org/abs/1211.6934v1.

Michael E. Taylor. Pseudodifferential operators. Author-hosted Chapter 7, Sections 4 and 10; accessed 4 October 2026. URL: https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/chap7.pdf.

Sébastien Boucksom. Singularities of plurisubharmonic functions and multiplier ideals. 2023. Lecture notes, version of 11 September 2023, based on the Orsay 2015–2017 course. URL: https://sebastien.boucksom.perso.math.cnrs.fr/notes/L2.pdf.

Sławomir Dinew. Uniqueness and stability in $\mathcal e(x,\omega )$. 2008. Version 1, 21 April 2008. arXiv:0804.3407.

Tamás Darvas, Eleonora Di Nezza, and Chinh H. Lu. $L^1$ metric geometry of big cohomology classes. Ann. Inst. Fourier (Grenoble), 68(7):3053–3086, 2018. URL: http://aif.cedram.org/item?id=AIF_2018__68_7_3053_0.

Stephen Willard. General topology. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970.

Osamu Fujino. Relative Bertini type theorem for multiplier ideal sheaves. Osaka J. Math., 60(1):207–226, 2023.

Wei-Dong Ruan. Canonical coordinates and Bergmann [Bergman] metrics. Comm. Anal. Geom., 6(3):589–631, 1998. URL: https://doi.org/10.4310/CAG.1998.v6.n3.a5, doi:10.4310/CAG.1998.v6.n3.a5.

Yuta Kusakabe and Shin-ichi Matsumura. Atiyah classes and ellipticity of Oka manifolds. 2026. arXiv:2610.12175v1, 8 October 2026; Lemmas 3.2–3.4, pp.20–21. URL: https://arxiv.org/abs/2610.12175v1, arXiv:2610.12175.

Gerald B. Folland. Real analysis. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition, 1999. ISBN 0-471-31716-0. Modern techniques and their applications, A Wiley-Interscience Publication.

Shin-Yao Jow. Okounkov bodies and restricted volumes along very general curves. Adv. Math., 223(4):1356–1371, 2010. URL: https://doi.org/10.1016/j.aim.2009.09.015, doi:10.1016/j.aim.2009.09.015.

Tamás Darvas and Mingchen Xia. The closures of test configurations and algebraic singularity types. Adv. Math., 397:Paper No. 108198, 56, 2022. URL: https://doi.org/10.1016/j.aim.2022.108198, doi:10.1016/j.aim.2022.108198.

Yifan Chen, Simion Filip, Song Sun, Valentino Tosatti, and Junsheng Zhang. Compact Kähler surfaces with semipositive anticanonical bundle. 2026. arXiv:2609.26716v1, 22 September 2026. URL: https://arxiv.org/abs/2609.26716v1, arXiv:2609.26716.

Xiaojun Wu. Asymptotic behaviour of the sectional ring. Comptes Rendus. Mathématique, 362:1389–1397, 2024. URL: https://comptes-rendus.academie-sciences.fr/mathematique/item/10.5802/crmath.651/, doi:10.5802/crmath.651.

J. J. Duistermaat and G. J. Heckman. On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math., 69(2):259–268, 1982. URL: https://doi.org/10.1007/BF01399506, doi:10.1007/BF01399506.

Liviu Ornea and Misha Verbitsky. Principles of Locally Conformally Kähler Geometry. Volume 354 of Progress in Mathematics. Birkhäuser, 2024. Section numbers refer to arXiv:2208.07188v6. URL: https://arxiv.org/abs/2208.07188v6.

Henri Skoda. Application des techniques $L\sp 2$ à la théorie des idéaux d'une algèbre de fonctions holomorphes avec poids. Ann. Sci. École Norm. Sup. (4), 5:545–579, 1972. URL: http://www.numdam.org/item?id=ASENS_1972_4_5_4_545_0.

Eleonora Di Nezza and Stefano Trapani. The regularity of envelopes. Ann. Sci. Éc. Norm. Supér. (4), 57(5):1347–1370, 2024.

Jiyuan Han. Parabolic Monge–Ampère equations on a family of Calabi–Yau manifolds. 2026. arXiv:2610.01202v1, 1 October 2026. URL: https://arxiv.org/abs/2610.01202v1, arXiv:2610.01202.

Mingchen Xia. Transcendental b-divisors II. \href https://mingchenxia.github.io/Papers/TB2.pdflink, 2025.

Gil R. Cavalcanti. Hodge theory and deformations of SKT manifolds. 2013. arXiv:1203.0493v5; first version 2 March 2012. URL: https://arxiv.org/abs/1203.0493v5, arXiv:1203.0493v5.

Eric Bedford and B. A. Taylor. Fine topology, Šilov boundary, and $(dd^c)^n$. J. Funct. Anal., 72(2):225–251, 1987. URL: https://doi.org/10.1016/0022-1236(87)90087-5, doi:10.1016/0022-1236(87)90087-5.

Toshiki Mabuchi. Some symplectic geometry on compact Kähler manifolds. I. Osaka J. Math., 24(2):227–252, 1987. URL: http://projecteuclid.org/euclid.ojm/1200780161.

Liviu Ornea and Misha Verbitsky. The Lee–Gauduchon cone on complex manifolds. In Real and Complex Geometry: In Honour of Paul Gauduchon, pages 237–250. Springer, 2025. URL: https://arxiv.org/abs/2411.05595v2, doi:10.1007/978-3-031-92297-8_11.

Andrei Okounkov. Why would multiplicities be log-concave? In The orbit method in geometry and physics (Marseille, 2000), volume 213 of Progr. Math., pages 329–347. Birkhäuser Boston, Boston, MA, 2003.

Jean-Pierre Demailly. Analytic methods in algebraic geometry. 2011. Author-hosted manuscript, title page July 2011; retained PDF dated 12 December 2014 in its creation metadata. URL: https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/analmeth_book.pdf.

David Witt Nyström. Test configurations and Okounkov bodies. Compos. Math., 148(6):1736–1756, 2012. URL: https://doi.org/10.1112/S0010437X12000358, doi:10.1112/S0010437X12000358.

Robert J. Berman, Sébastien Boucksom, Vincent Guedj, and Ahmed Zeriahi. A variational approach to complex Monge-Ampère equations. Publ. Math. Inst. Hautes Études Sci., 117:179–245, 2013. URL: https://doi.org/10.1007/s10240-012-0046-6, doi:10.1007/s10240-012-0046-6.

Robert Berman and Jean-Pierre Demailly. Regularity of plurisubharmonic upper envelopes in big cohomology classes. In Perspectives in analysis, geometry, and topology, volume 296 of Progr. Math., pages 39–66. Birkhäuser/Springer, New York, 2012. URL: https://doi.org/10.1007/978-0-8176-8277-4_3, doi:10.1007/978-0-8176-8277-4\_3.

Dan Coman, Vincent Guedj, Sibel Sahin, and Ahmed Zeriahi. Toric pluripotential theory. Ann. Polon. Math., 123(1):215–242, 2019. URL: https://doi.org/10.4064/ap180409-3-7, doi:10.4064/ap180409-3-7.

Sławomir Dinew. An inequality for mixed Monge–Ampère measures. Math. Z., 262(1):1–15, 2009. Theorem locations refer to arXiv:0705.0974v1, 7 May 2007. URL: https://arxiv.org/abs/0705.0974v1, doi:10.1007/s00209-008-0356-z.

Sławomir Dinew. An inequality for mixed Monge–Ampère measures. 2007. Exact first version, 7 May 2007. Later published in Mathematische Zeitschrift 262 (2009), 1–15, DOI: 10.1007/s00209-008-0356-z. URL: https://arxiv.org/abs/0705.0974v1, arXiv:0705.0974v1.

Mingchen Xia. Non-pluripolar products on vector bundles and Chern-Weil formulae. Math. Ann., 390(3):3239–3316, 2024. URL: https://doi.org/10.1007/s00208-024-02838-4, doi:10.1007/s00208-024-02838-4.

Victor Guillemin. Kaehler structures on toric varieties. J. Differential Geom., 40(2):285–309, 1994. URL: http://projecteuclid.org/euclid.jdg/1214455538.

Jean-Pierre Demailly. Regularization of closed positive currents of type (1,1) by the flow of a Chern connection. Undated author-hosted twenty-page manuscript; retrieved 5 October 2026; locators use its own pagination. URL: https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization_flow.pdf.

Subharmonicity on all affine complex lines forces upper semicontinuity. Author-supplied six-page manuscript. Unsigned and undated; exact supplied file retained 3 October 2026. URL: https://mingchenxia.github.io/Papers/PSHD_vibe.pdf.

Nguyen-Bac Dang and Charles Favre. Intersection theory of nef $b$-divisor classes. Compos. Math., 158(7):1563–1594, 2022. URL: https://doi.org/10.1112/s0010437x22007515, doi:10.1112/s0010437x22007515.

Robert J. Berman. From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit. Math. Z., 291(1-2):365–394, 2019. URL: https://doi.org/10.1007/s00209-018-2087-0, doi:10.1007/s00209-018-2087-0.

Andrea Colesanti, Monika Ludwig, and Fabian Mussnig. The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures. Calc. Var. Partial Differential Equations, 61(5):Paper No. 181, 37, 2022. URL: https://doi.org/10.1007/s00526-022-02288-3, doi:10.1007/s00526-022-02288-3.

Junyan Cao. Ohsawa–Takegoshi extension theorem for compact Kähler manifolds and applications. 2017. arXiv:1404.6937v2, posted 3 October 2017; author version dated 4 October 2017. URL: https://arxiv.org/abs/1404.6937v2.

Marcel Brelot. Les étapes et les aspects multiples de la théorie du potentiel. Enseign. Math. (2), 18:1–36, 1972.

Edward Bierstone and Pierre D. Milman. Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math., 128(2):207–302, 1997. URL: https://doi.org/10.1007/s002220050141, doi:10.1007/s002220050141.

John Erik Fornæ ss and Raghavan Narasimhan. The Levi problem on complex spaces with singularities. Math. Ann., 248(1):47–72, 1980. URL: https://doi.org/10.1007/BF01349254, doi:10.1007/BF01349254.

Steve Zelditch. Szegő kernels and a theorem of Tian. Internat. Math. Res. Notices, 1998(6):317–331, 1998. URL: https://doi.org/10.1155/S107379289800021X, doi:10.1155/S107379289800021X.

David Witt Nyström. Transforming metrics on a line bundle to the Okounkov body. Ann. Sci. Éc. Norm. Supér. (4), 47(6):1111–1161, 2014. URL: https://doi.org/10.24033/asens.2235, doi:10.24033/asens.2235.

Lennart Carleson. Selected problems on exceptional sets. In Selected reprints, Wadsworth Math. Ser., pages iv+100. Wadsworth, Belmont, CA, 1983.

Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi. Viscosity solutions to degenerate complex Monge–Ampère equations. 2010. Exact first version, 1 July 2010. Later published in Communications on Pure and Applied Mathematics 64 (2011), no. 8, 1059–1094, DOI: 10.1002/cpa.20364. URL: https://arxiv.org/abs/1007.0076v1, arXiv:1007.0076v1.

Sébastien Boucksom and Huayi Chen. Okounkov bodies of filtered linear series. Compos. Math., 147(4):1205–1229, 2011. URL: https://doi.org/10.1112/S0010437X11005355, doi:10.1112/S0010437X11005355.

Mingchen Xia. Transcendental b-divisors. \href https://mingchenxia.github.io/Papers/TB.pdflink, 2025.

Xin Fu and Zexuan Ouyang. Rigid and nonrigid isotropic nef classes on K3 surfaces. 2026. arXiv:2609.33632v1, 27 September 2026; numbered statements refer to version 1. URL: https://arxiv.org/abs/2609.33632v1, arXiv:2609.33632.

Zbigniew Błocki. The complex Monge–Ampère operator in pluripotential theory. 2002. Unfinished lecture notes based on a 1997 graduate course; last modified November 2002. URL: https://gamma.im.uj.edu.pl/~blocki/publ/ln/wykl.pdf.

Vincent Guedj and Chinh H. Lu. Quasi-plurisubharmonic envelopes 3: solving Monge–Ampère equations on Hermitian manifolds. J. Reine Angew. Math., 800:259–298, 2023. Numbered statements compared with arXiv:2107.01938v1, 5 July 2021. URL: https://arxiv.org/abs/2107.01938v1, doi:10.1515/crelle-2023-0030.

Michel Schweitzer. Autour de la cohomologie de Bott–Chern. 2007. URL: https://arxiv.org/abs/0709.3528v1, arXiv:0709.3528v1.

Lars Hörmander. $L\sp 2$ estimates and existence theorems for the $\bar \partial $ operator. Acta Math., 113:89–152, 1965. URL: https://doi.org/10.1007/BF02391775, doi:10.1007/BF02391775.

Alessio Figalli. The Monge-Ampère equation and its applications. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2017. ISBN 978-3-03719-170-5. URL: https://doi.org/10.4171/170, doi:10.4171/170.

Gábor Székelyhidi, Valentino Tosatti, and Ben Weinkove. Gauduchon metrics with prescribed volume form. Acta Mathematica, 219(1):181–211, 2017. Theorem 1.4 and Sections 3–4 in the published version. URL: https://doi.org/10.4310/ACTA.2017.v219.n1.a6, doi:10.4310/ACTA.2017.v219.n1.a6.

Yum Tong Siu. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math., 27:53–156, 1974.

Mingchen Xia. Foundations of pluripotential theory. Online manuscript, 2026. Revision da2be417c7b38b386d0b688251cf4978df94d682; prerequisite statements and proofs checked against the complete active input graph. URL: https://github.com/MingchenXia/foundations-of-pluripotential-theory.

Xuan Li. Monge–Ampère type equations on compact Hermitian manifolds with bounded mass property. 2026. arXiv:2601.13446v1, 19 January 2026. URL: https://arxiv.org/abs/2601.13446v1.

Maciej Klimek. Pluripotential theory. Volume 6 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York, 1991. ISBN 0-19-853568-6. Oxford Science Publications.

Zhiqin Lu. On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Amer. J. Math., 122(2):235–273, 2000. URL: http://muse.jhu.edu/journals/american_journal_of_mathematics/v122/122.2lu.pdf.

Karl Oeljeklaus and Matei Toma. Non-Kähler compact complex manifolds associated to number fields. Annales de l'Institut Fourier, 55(1):161–171, 2005. doi:10.5802/aif.2093.

Henri Skoda and Joël Briançon. Sur la clôture intégrale d'un idéal de germes de fonctions holomorphes en un point de $\bf C\sp n$. C. R. Acad. Sci. Paris Sér. A, 278:949–951, 1974.

Ryuichi Fukuoka and Anderson Macedo Setti. Mollifier smoothings of strongly convex $C^0$-Finsler structures. 2026. arXiv:2610.12108v1, 8 October 2026; Example 3.18, p.20. URL: https://arxiv.org/abs/2610.12108v1, arXiv:2610.12108.

Jarosław Włodarczyk. Resolution of singularities of analytic spaces. In Proceedings of Gökova Geometry-Topology Conference 2008, 31–63. Gökova Geometry/Topology Conference (GGT), Gökova, 2009.

Jean-Pierre Demailly. Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines. Mém. Soc. Math. France (N.S.), pages 124, 1985.

Sławomir Kołodziej. The complex Monge-Ampère equation and pluripotential theory. Mem. Amer. Math. Soc., 178(840):x+64, 2005. URL: https://doi.org/10.1090/memo/0840, doi:10.1090/memo/0840.

Robert J. Berman, Sébastien Boucksom, and Mattias Jonsson. A variational approach to the Yau-Tian-Donaldson conjecture. J. Amer. Math. Soc., 34(3):605–652, 2021. URL: https://doi.org/10.1090/jams/964, doi:10.1090/jams/964.

Liviu Ornea and Misha Verbitsky. A Calabi–Yau theorem for Vaisman manifolds. Communications in Analysis and Geometry, 32(10):2889–2900, 2024. URL: https://arxiv.org/abs/2206.08808v2.

Yi Li. Mabuchi and Aubin–Yau functionals over complex manifolds. 2010. arXiv:\allowbreak 1004.0553; torsion corrections to energy functionals. URL: https://arxiv.org/abs/1004.0553, arXiv:1004.0553.

Siarhei Finski. Submultiplicative norms and filtrations on section rings. Proc. Lond. Math. Soc. (3), 131(2):Paper No. e70077, 47, 2025. URL: https://doi.org/10.1112/plms.70077, doi:10.1112/plms.70077.

Shouwen Fang, Valentino Tosatti, Ben Weinkove, and Tao Zheng. Inoue surfaces and the Chern–Ricci flow. Journal of Functional Analysis, 271(11):3162–3185, 2016. URL: https://arxiv.org/abs/1501.07578.

Ana María Botero, José Ignacio Burgos Gil, David Holmes, and Robin de Jong. Chern-Weil and Hilbert-Samuel formulae for singular Hermitian line bundles. Doc. Math., 27:2563–2624, 2022. URL: https://doi.org/10.4171/dm/x36, doi:10.4171/dm/x36.

Frédéric Riesz. Sur les Fonctions Subharmoniques et Leur Rapport à la Théorie du Potentiel. Acta Math., 48(3-4):329–343, 1926. URL: https://doi.org/10.1007/BF02565338, doi:10.1007/BF02565338.

Terence Tao. 245C, notes 3: distributions. 2009. Online lecture notes, 19 April 2009. URL: https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/.

Liviu Ornea and Misha Verbitsky. Balanced metrics and Gauduchon cone of locally conformally Kähler manifolds. International Mathematics Research Notices, 2025(3):rnaf014, 2025. URL: https://arxiv.org/abs/2407.04623v2, doi:10.1093/imrn/rnaf014.

Paul Gauduchon. Les métriques standard d'une surface complexe compacte à premier nombre de Betti pair. Astérisque, 126:129–135, 1985. URL: https://www.numdam.org/article/AST_1985__126__129_0.pdf.

Sébastien Boucksom, Charles Favre, and Mattias Jonsson. Valuations and plurisubharmonic singularities. Publ. Res. Inst. Math. Sci., 44(2):449–494, 2008. URL: https://doi.org/10.2977/prims/1210167334, doi:10.2977/prims/1210167334.

Ciro Ciliberto, Michal Farnik, Alex Küronya, Victor Lozovanu, Joaquim Roé, and Constantin Shramov. Newton-Okounkov bodies sprouting on the valuative tree. Rend. Circ. Mat. Palermo (2), 66(2):161–194, 2017. URL: https://doi.org/10.1007/s12215-016-0285-3, doi:10.1007/s12215-016-0285-3.

Valentino Tosatti and Ben Weinkove. The complex Monge–Ampère equation on compact Hermitian manifolds. Journal of the American Mathematical Society, 23(4):1187–1195, 2010. Proof locations refer to arXiv:0910.1390v2, 11 November 2009. URL: https://arxiv.org/abs/0910.1390v2.

Valentino Tosatti. Transcendental Morse inequality on Kähler manifolds. arXiv:2608.30153v2, 2026. 8 September 2026; Theorem 1.1 and Corollary 1.2 in version 2. URL: https://arxiv.org/abs/2608.30153.

Heisuke Hironaka. Flattening theorem in complex-analytic geometry. Amer. J. Math., 97:503–547, 1975. URL: https://doi.org/10.2307/2373721, doi:10.2307/2373721.

Mohamed El Kadiri. Remarks on weak convergence of complex Monge-Ampère measures. 2023. arXiv:2301.09495v3, 25 January 2023; preprint of the published 2024 paper. URL: https://arxiv.org/abs/2301.09495v3.

Michael Taylor. Regularity of solutions to nonlinear overdetermined elliptic PDE. Author-hosted note; accessed 4 October 2026. URL: https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2022/05/overellip.pdf.

Sławomir Dinew. Pluripotential theory on compact Hermitian manifolds. Annales de la Faculté des Sciences de Toulouse. Mathématiques, 25(1):91–139, 2016. URL: https://afst.centre-mersenne.org/item/10.5802/afst.1488.pdf, doi:10.5802/afst.1488.

Sébastien Boucksom and Mattias Jonsson. Addendum to the article global pluripotential theory over a trivially valued field. 2022. arXiv:2206.07183.

C. Hacon, M. Popa, and C. Schnell. Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Păun. In Local and global methods in algebraic geometry, volume 712 of Contemp. Math., pages 143–195. Amer. Math. Soc., [Providence], RI, 2018.

David Witt Nyström. Monotonicity of non-pluripolar Monge-Ampère masses. Indiana Univ. Math. J., 68(2):579–591, 2019. arXiv version: arXiv:1703.01950v1, 6 March 2017. URL: https://doi.org/10.1512/iumj.2019.68.7630, doi:10.1512/iumj.2019.68.7630.

Xiangsen Qin. Prescribed limits and cluster sets of complex Monge–Ampère measures. 2026. arXiv:2609.06568v1, 6 September 2026; numbered statements refer to version 1. URL: https://arxiv.org/abs/2609.06568v1, arXiv:2609.06568.

Terence Tao. 245B, notes 1: signed measures and the Radon-Nikodym-Lebesgue theorem. 2009. Online lecture notes, 4 January 2009. URL: https://terrytao.wordpress.com/2009/01/04/245b-notes-1-signed-measures-and-the-radon-nikodym-lebesgue-theorem/.

Alexander Rashkovskii and Ragnar Sigurdsson. Green functions with singularities along complex spaces. Internat. J. Math., 16(4):333–355, 2005. URL: https://doi.org/10.1142/S0129167X05002904, doi:10.1142/S0129167X05002904.

Jianchun Chu and Bin Zhou. Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds. Sci. China Math., 62(2):371–380, 2019. URL: https://doi.org/10.1007/s11425-017-9173-0, doi:10.1007/s11425-017-9173-0.

Jun Li and Shing Tung Yau. Hermitian–Yang–Mills connection on Non-Kähler manifolds. In Mathematical Aspects of String Theory, 560–573. World Scientific, 1987. doi:10.1142/9789812798411_0027.

Thierry Bouche. Convergence de la métrique de Fubini-Study d'un fibré linéaire positif. Ann. Inst. Fourier (Grenoble), 40(1):117–130, 1990. URL: https://doi.org/10.5802/aif.1206, doi:10.5802/aif.1206.

Tomoyuki Hisamoto. Restricted Bergman kernel asymptotics. Trans. Amer. Math. Soc., 364(7):3585–3607, 2012. URL: https://doi.org/10.1090/S0002-9947-2012-05641-8, doi:10.1090/S0002-9947-2012-05641-8.

Xueyuan Wan. Positivity of the canonical bundle and Hermitian metrics with quasi-negative holomorphic sectional curvature. 2026. arXiv:2610.08111v1, 6 October 2026. URL: https://arxiv.org/abs/2610.08111v1, arXiv:2610.08111.

Ngoc Cuong Nguyen. The complex Monge–Ampère type equation on compact Hermitian manifolds and applications. Advances in Mathematics, 286:240–285, 2016. Numbered statements refer to arXiv:1501.00891v1, 5 January 2015. URL: https://arxiv.org/abs/1501.00891v1, doi:10.1016/j.aim.2015.09.009.

Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. ISBN 0-471-05059-8. Reprint of the 1978 original. URL: https://doi.org/10.1002/9781118032527, doi:10.1002/9781118032527.

Andrei Okounkov. Brunn-Minkowski inequality for multiplicities. Invent. Math., 125(3):405–411, 1996. URL: https://doi.org/10.1007/s002220050081, doi:10.1007/s002220050081.

Lawrence Ein, Robert Lazarsfeld, Mircea Mustaţă, Michael Nakamaye, and Mihnea Popa. Restricted volumes and base loci of linear series. Amer. J. Math., 131(3):607–651, 2009. URL: https://doi.org/10.1353/ajm.0.0054, doi:10.1353/ajm.0.0054.

Yum Tong Siu. The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group. Ann. of Math. (2), 127(3):585–627, 1988. URL: https://doi.org/10.2307/2007006, doi:10.2307/2007006.

Sheng Rao, Song Yang, and Xiangdong Yang. Dolbeault cohomologies of blowing up complex manifolds. J. Math. Pures Appl. (9), 130:68–92, 2019. URL: https://doi.org/10.1016/j.matpur.2019.01.016, doi:10.1016/j.matpur.2019.01.016.

Yum Tong Siu. Invariance of plurigenera. Invent. Math., 134(3):661–673, 1998. URL: https://doi.org/10.1007/s002220050276, doi:10.1007/s002220050276.

Yan He, Johannes Testorf, and Xu Wang. Ross–Witt Nyström correspondence and Ohsawa–Takegoshi extension. 2023. arXiv:2311.03840.

Sławomir Kołodziej and Ngoc Cuong Nguyen. Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity. Calculus of Variations and Partial Differential Equations, 60:93, 2021. Theorem 4.1 in the published version; article number 93. URL: https://doi.org/10.1007/s00526-021-01944-4, doi:10.1007/s00526-021-01944-4.

Jean-Pierre Demailly. Complex analytic and differential geometry. \href https://www-fourier.univ-grenoble-alpes.fr/ demailly/manuscripts/agbook.pdfAuthor's website, 2012. Version of June 21, 2012.

Jian Song and Ben Weinkove. Lecture notes on the Kähler–Ricci flow. 2012. Exact first arXiv version, 15 December 2012. URL: https://arxiv.org/abs/1212.3653v1, arXiv:1212.3653.

Hideyuki Matsumura. Commutative ring theory. Volume 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1989. ISBN 0-521-36764-6. Translated from the Japanese by M. Reid.

The Stacks Project Authors. Stacks project. \url http://stacks.math.columbia.edu, 2020.

Kiyosi Oka. Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes. Tôhoku Math. J., 49:15–52, 1942.

Tomoyuki Hisamoto. On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold. J. Reine Angew. Math., 713:129–148, 2016. URL: https://doi.org/10.1515/crelle-2014-0021, doi:10.1515/crelle-2014-0021.

Daniele Angella, Artūras Dubickas, Alexandra Otiman, and Jonas Stelzig. On metric and cohomological properties of Oeljeklaus–Toma manifolds. Publicacions Matemàtiques, 68(1):219–239, 2024. URL: https://arxiv.org/abs/2201.06377.

Gábor Székelyhidi. Filtrations and test-configurations. Math. Ann., 362(1-2):451–484, 2015. With an appendix by Sebastien Boucksom. URL: https://doi.org/10.1007/s00208-014-1126-3, doi:10.1007/s00208-014-1126-3.

Jean-Pierre Demailly. Regularization of closed positive currents and intersection theory. J. Algebraic Geom., 1(3):361–409, 1992.

Bo Guan and Qun Li. Complex Monge–Ampère equations and totally real submanifolds. Advances in Mathematics, 225(3):1185–1223, 2010. arXiv:0910.1851, doi:10.1016/j.aim.2010.03.019.

Xiangsen Qin. Dolbeault cohomology growth under curvature rank bounds. 2026. arXiv:2610.09295v1, 7 October 2026; numbered statements refer to version 1. URL: https://arxiv.org/abs/2610.09295v1, arXiv:2610.09295.

Daniel Barlet. How to use the cycle space in complex geometry. In Michael Schneider and Yum-Tong Siu, editors, Several Complex Variables, volume 37 of Math. Sci. Res. Inst. Publ., pages 25–42. Cambridge University Press, Cambridge, 2000.

S. K. Donaldson. Scalar curvature and stability of toric varieties. J. Differential Geom., 62(2):289–349, 2002. URL: http://projecteuclid.org/euclid.jdg/1090950195.

Reese Harvey and A. W. Knapp. Positive $(p,\,p)$ forms, Wirtinger's inequality, and currents. In Value distribution theory (Proc. Tulane Univ. Program, Tulane Univ., New Orleans, La., 1972-1973), Part A, volume 25 of Pure Appl. Math., pages 43–62. Dekker, New York, 1974.

Junyan Cao. Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorem on compact Kähler manifolds. Compos. Math., 150(11):1869–1902, 2014. URL: https://doi.org/10.1112/S0010437X14007398, doi:10.1112/S0010437X14007398.

Tamás Darvas. The Mabuchi completion of the space of Kähler potentials. Amer. J. Math., 139(5):1275–1313, 2017. URL: https://doi.org/10.1353/ajm.2017.0032, doi:10.1353/ajm.2017.0032.

Eleonora Di Nezza and Stefano Trapani. Monge–Ampère measures on contact sets. 2019. Exact version 1, submitted 29 December 2019; 11-page preprint. URL: https://arxiv.org/abs/1912.12720v1, arXiv:1912.12720.

Pascal Cherrier. Équations de Monge–Ampère sur les variétés hermitiennes compactes. Bull. Sci. Math. (2), 111(4):343–385, 1987.

Sławomir Kołodziej. The complex Monge-Ampère equation. Acta Math., 180(1):69–117, 1998. URL: https://doi.org/10.1007/BF02392879, doi:10.1007/BF02392879.

Tien-Cuong Dinh and Viêt-Anh Nguyên. The mixed Hodge–Riemann bilinear relations for compact Kähler manifolds. 2005. arXiv:math/0501449v2, 14 April 2005; published in Geom. Funct. Anal. 16 (2006), no. 4, 838–849, DOI 10.1007/s00039-006-0572-9. URL: https://arxiv.org/abs/math/0501449v2.

Tamás Darvas and Kewei Zhang. Twisted Kähler-Einstein metrics in big classes. Comm. Pure Appl. Math., 77(12):4289–4327, 2024. URL: https://doi.org/10.1002/cpa.22206, doi:10.1002/cpa.22206.

Kai Pang, Haoyuan Sun, Zhiwei Wang, and Xiangyu Zhou. Capacity stability of complex Monge–Ampère equations with moving prescribed singularities. 2026. arXiv:2607.12797v1, posted 14 July 2026. URL: https://arxiv.org/abs/2607.12797v1, arXiv:2607.12797.

P. Lelong, P. Dolbeault, and H. Skoda, editors. Séminaire d'analyse P. Lelong-P. Dolbeault-H. Skoda. Années 1983/1984. Volume 1198 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. ISBN 3-540-16762-5.

Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider. Pseudo-effective line bundles on compact Kähler manifolds. Internat. J. Math., 12(6):689–741, 2001. arXiv version: arXiv:math/0006205v2, 10 October 2000. URL: https://doi.org/10.1142/S0129167X01000861, doi:10.1142/S0129167X01000861.

Andrzej Derdziński. Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compositio Mathematica, 49(3):405–433, 1983. URL: https://www.numdam.org/item/CM_1983__49_3_405_0/.

Jiaogen Zhang and Xi Zhang. The interpolation method for the complex Hessian equation on compact Kähler manifolds. 2026. arXiv:2610.07852v1, 6 October 2026. URL: https://arxiv.org/abs/2610.07852v1, arXiv:2610.07852.

Dan Coman, Vincent Guedj, and Ahmed Zeriahi. Extension of plurisubharmonic functions with growth control. J. Reine Angew. Math., 676:33–49, 2013. URL: https://doi.org/10.1515/crelle.2011.185, doi:10.1515/crelle.2011.185.

Tamás Darvas, Eleonora Di Nezza, and Chinh H. Lu. On the singularity type of full mass currents in big cohomology classes. Compos. Math., 154(2):380–409, 2018. URL: https://doi.org/10.1112/S0010437X1700759X, doi:10.1112/S0010437X1700759X.

Askold Georgievich Khovanskiĭ. The Newton polytope, the Hilbert polynomial and sums of finite sets. Funktsional. Anal. i Prilozhen., 26(4):57–63, 96, 1992. URL: https://doi.org/10.1007/BF01075048, doi:10.1007/BF01075048.

Riley Guyett and Ryan Mc Gowan. Monge–Ampère degenerations and conifold contractions in Fujiki class $\mathcal C$. 2026. arXiv:2610.00942v1, 1 October 2026. URL: https://arxiv.org/abs/2610.00942v1, arXiv:2610.00942v1.

Gábor Székelyhidi. Fully non-linear elliptic equations on compact Hermitian manifolds. Journal of Differential Geometry, 109(2):337–378, 2018. Numbered statements refer to arXiv:1501.02762v3, 23 April 2015, unless another arXiv version is specified. URL: https://arxiv.org/abs/1501.02762v3, doi:10.4310/jdg/1527040875.

Mingchen Xia. Mabuchi geometry of big cohomology classes. J. Reine Angew. Math., 798:261–292, 2023. URL: https://doi.org/10.1515/crelle-2023-0019, doi:10.1515/crelle-2023-0019.

David M. J. Calderbank, Liana David, and Paul Gauduchon. The Guillemin formula and Kähler metrics on toric symplectic manifolds. 2003. Exact arXiv:math/0310243v1, posted 16 October 2003; distinct from the published source. URL: https://arxiv.org/abs/math/0310243v1, arXiv:math/0310243.

Robert Berman, Sébastien Boucksom, and David Witt Nyström. Fekete points and convergence towards equilibrium measures on complex manifolds. Acta Math., 207(1):1–27, 2011. URL: https://doi.org/10.1007/s11511-011-0067-x, doi:10.1007/s11511-011-0067-x.

Michael E. Taylor. Functional analysis. Author-hosted notes, Section 7; accessed 4 October 2026. URL: https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/fctnl.pdf.

Hoseob Seo. Analytic multiplier ideals of toric plurisubharmonic functions on a toric complex space. 2026. arXiv:2610.08741v1, 6 October 2026. URL: https://arxiv.org/abs/2610.08741v1, arXiv:2610.08741.

Ya Deng. Transcendental Morse inequality and generalized Okounkov bodies. Algebr. Geom., 4(2):177–202, 2017. URL: https://doi.org/10.14231/AG-2017-009, doi:10.14231/AG-2017-009.

Terence Tao. Lecture notes 4 for 247A. UCLA Math 247A lecture notes, 2006. URL: https://www.math.ucla.edu/~tao/247a.1.06f/notes4.pdf.

Mingchen Xia. Singularities in global pluripotential theory. Lecture manuscript, 2026. Version updated 4 October 2026; accessed 7 October 2026. URL: https://mingchenxia.github.io/Lectures/SGPT_final.pdf.

L. L. Helms. Introduction to Potential Theory. Volume 22 of Pure and Applied Mathematics. Wiley-Interscience, New York, 1969. Complete supplied edition; conventions pp.1–5, Chapters 1–12 pp.6–266, notes pp.267–274.

Hongrong Chen, Guokuan Shao, and Wenxuan Wang. Characterization of Sobolev regularity of plurisubharmonic functions. 2026. arXiv:2609.39136v1, 30 September 2026. URL: https://arxiv.org/abs/2609.39136v1, arXiv:2609.39136v1.

Sławomir Kołodziej and Ngoc Cuong Nguyen. Weak solutions to the complex Monge–Ampère equation on Hermitian manifolds. 2015. arXiv:1312.5491v2, posted 20 January 2015; preprint pagination. URL: https://arxiv.org/abs/1312.5491v2, arXiv:1312.5491, doi:10.1090/conm/644/12775.

Ionuţ Chiose. On the invariance of the total Monge–Ampère volume of Hermitian metrics. Annales de la Faculté des Sciences de Toulouse. Mathématiques, 33(3):575–579, 2024. URL: https://afst.centre-mersenne.org/articles/10.5802/afst.1781/, doi:10.5802/afst.1781.

Robert Lazarsfeld. Positivity in algebraic geometry. I. Volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer-Verlag, Berlin, 2004. ISBN 3-540-22533-1. Classical setting: line bundles and linear series. URL: https://doi.org/10.1007/978-3-642-18808-4, doi:10.1007/978-3-642-18808-4.

Bernard Shiffman. Extension of positive line bundles and meromorphic maps. Invent. Math., 15(4):332–347, 1972. URL: https://doi.org/10.1007/BF01405087, doi:10.1007/BF01405087.

Pietro Mesquita-Piccione. A non-Archimedean theory of complex spaces and the cscK problem. Adv. Math., 481:Paper No. 110543, 83, 2025. URL: https://doi.org/10.1016/j.aim.2025.110543, doi:10.1016/j.aim.2025.110543.

Tristan C. Collins. Concave elliptic equations and generalized Khovanskii–Teissier inequalities. Pure and Applied Mathematics Quarterly, 17(3):1061–1082, 2021. URL: https://doi.org/10.4310/PAMQ.2021.v17.n3.a10, doi:10.4310/PAMQ.2021.v17.n3.a10.

Alexander Grothendieck and Jean Dieudonné. Éléments de géométrie algébrique IV.2. Number 24 in Publications Mathématiques de l'IHÉS. Institut des Hautes Études Scientifiques, 1965.

David Gilbarg and Neil S. Trudinger. Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Berlin, 2 edition, 2001. ISBN 978-3-540-41160-4. URL: https://doi.org/10.1007/978-3-642-61798-0, doi:10.1007/978-3-642-61798-0.

Tristan C. Collins and Valentino Tosatti. Restricted volumes on Kähler manifolds. Ann. Fac. Sci. Toulouse Math. (6), 31(3):907–947, 2022.

Jean-Pierre Demailly. On the cohomology of pseudoeffective line bundles. In Complex geometry and dynamics, volume 10 of Abel Symp., pages 51–99. Springer, Cham, 2015.

You-Cheng Chou and Kuang-Ru Wu. RC-positivity of complex surfaces. 2026. arXiv:2609.33656v1, 27 September 2026. URL: https://arxiv.org/abs/2609.33656v1, arXiv:2609.33656.

Sébastien Boucksom. Cônes positifs des variétés complexes compactes. PhD thesis, Université Joseph-Fourier-Grenoble I, 2002.

Mohamed El Kadiri. A note on the structure of plurifinely open sets and the equality of some complex Monge–Ampère measures. 2023. arXiv:2309.03391.

N. Bourbaki. Algèbre commutative. Springer Science & Business Media, 2007.

Valentino Tosatti, Yu Wang, Ben Weinkove, and Xiaokui Yang. $C^2,\alpha $ estimates for nonlinear elliptic equations in complex and almost complex geometry. Calculus of Variations and Partial Differential Equations, 54(1):431–453, 2015. Theorem locations refer to the published version unless an arXiv version is specified. URL: https://doi.org/10.1007/s00526-014-0791-0, doi:10.1007/s00526-014-0791-0.

Antonio Trusiani. Kähler-Einstein metrics with prescribed singularities on Fano manifolds. J. Reine Angew. Math., 793:1–57, 2022. URL: https://doi.org/10.1515/crelle-2022-0047, doi:10.1515/crelle-2022-0047.

Mingchen Xia. Analytic Bertini theorem. Math. Z., 302(2):1171–1176, 2022. URL: https://doi.org/10.1007/s00209-022-03103-7, doi:10.1007/s00209-022-03103-7.

Daniel Huybrechts. Lectures on K3 surfaces. Cambridge University Press, 2016. Author draft available online. URL: https://www.math.uni-bonn.de/people/huybrech/K3Global.pdf.

Junyan Cao. Additivity of the approximation functional of currents induced by Bergman kernels. C. R. Math. Acad. Sci. Paris, 353(2):117–119, 2015. URL: https://doi.org/10.1016/j.crma.2014.11.004, doi:10.1016/j.crma.2014.11.004.

Dan Abramovich, Kalle Karu, Kenji Matsuki, and Jarosław Włodarczyk. Torification and factorization of birational maps. J. Amer. Math. Soc., 15(3):531–572, 2002. Analytic weak factorization: Theorem 0.3.1 in arXiv:math/9904135v4, 31 May 2000. doi:10.1090/S0894-0347-02-00396-X.

Masahisa Inoue. On surfaces of class $\rm VII_0$. Inventiones Mathematicae, 24(4):269–310, 1974.

Jean-Pierre Demailly and Mihai Păun. Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. of Math. (2), 159(3):1247–1274, 2004. Published Theorem 0.7 and Conjecture 0.8 correspond to Theorem 0.6 and Conjecture 0.7 in arXiv:math/0105176v2, 24 September 2001; arXiv locators elsewhere are marked explicitly. URL: https://annals.math.princeton.edu/2004/159-3/p05, doi:10.4007/annals.2004.159.1247.

Tamás Darvas. Weak geodesic rays in the space of Kähler potentials and the class $\mathcal E(X,\omega )$. J. Inst. Math. Jussieu, 16(4):837–858, 2017. URL: https://doi.org/10.1017/S1474748015000316, doi:10.1017/S1474748015000316.

Giuseppe Barbaro. Pluriclosed deformations of Bismut flat metrics. 2026. URL: https://arxiv.org/abs/2609.35313v1, arXiv:2609.35313v1.

Rolf Schneider. Convex bodies: the Brunn-Minkowski theory. Volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014. ISBN 978-1-107-60101-7.

Tamás Darvas. The Mabuchi geometry of finite energy classes. Adv. Math., 285:182–219, 2015. URL: https://doi.org/10.1016/j.aim.2015.08.005, doi:10.1016/j.aim.2015.08.005.

S. S. Chern, Harold I. Levine, and Louis Nirenberg. Intrinsic norms on a complex manifold. In Global Analysis (Papers in Honor of K. Kodaira), pages 119–139. Univ. Tokyo Press, Tokyo, 1969.

Terence Tao. 245c, notes 2: the fourier transform. 2009. Online lecture notes, 6 April 2009. URL: https://terrytao.wordpress.com/2009/04/06/the-fourier-transform/.

Mingchen Xia. Integration by parts formula for non-pluripolar product. 2019. arXiv:1907.06359v1, posted 15 July 2019. URL: https://arxiv.org/abs/1907.06359v1.

Sébastien Boucksom. Singularities of plurisubharmonic functions and multiplier ideals. \href http://sebastien.boucksom.perso.math.cnrs.fr/notes/L2.pdflink, 2017.

Tien-Cuong Dinh, Xiaonan Ma, and Viêt-Anh Nguyên. Equidistribution speed for Fekete points associated with an ample line bundle. Annales scientifiques de l'École Normale Supérieure, 50(3):545–578, 2017. URL: https://numdam.org/item/10.24033/asens.2327/, doi:10.24033/asens.2327.

Omar Alehyane, Chinh H. Lu, and Mohammed Salouf. Monge–Ampère equations with prescribed singularities on compact Hermitian manifolds. 2025. arXiv:2511.02339v1, 4 November 2025; preprint version. URL: https://arxiv.org/abs/2511.02339v1.

Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider. Compact complex manifolds with numerically effective tangent bundles. Journal of Algebraic Geometry, 3(2):295–345, 1994. URL: https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/dps1.pdf.

Jean-Pierre Demailly. Analytic methods in algebraic geometry. Volume 1 of Surveys of Modern Mathematics. International Press, Somerville, MA; Higher Education Press, Beijing, 2012. ISBN 978-1-57146-234-3.

Jianchun Chu, Valentino Tosatti, and Ben Weinkove. On the $C^1,1$ regularity of geodesics in the space of Kähler metrics. Ann. PDE, 3(2):Paper No. 15, 12, 2017. arXiv version: arXiv:1611.02390v2, 20 April 2017. URL: https://doi.org/10.1007/s40818-017-0034-8, doi:10.1007/s40818-017-0034-8.

Tamás Darvas and Yanir A. Rubinstein. Kiselman's principle, the Dirichlet problem for the Monge–Ampère equation, and rooftop obstacle problems. 2014. Exact first arXiv version, 26 May 2014. URL: https://arxiv.org/abs/1405.6548v1, arXiv:1405.6548.

Edward Goldstein and Sergey Prokushkin. Geometric model for complex non-Kähler manifolds with $SU(3)$ structure. Communications in Mathematical Physics, 251:65–78, 2004. URL: https://arxiv.org/abs/hep-th/0212307v2, arXiv:hep-th/0212307, doi:10.1007/s00220-004-1167-7.

Akira Fujiki. Closedness of the Douady spaces of compact Kähler spaces. Publ. Res. Inst. Math. Sci., 14(1):1–52, 1978. doi:10.2977/PRIMS/1195189279.

Alexander Grothendieck. Techniques de construction en géométrie analytique. VI. Étude locale des morphismes: germes d'espaces analytiques, platitude, morphismes simples. Séminaire Henri Cartan, 13(1):1–13, 1960.

William Fulton. Introduction to toric varieties. Volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1993. ISBN 0-691-00049-2. The William H. Roever Lectures in Geometry. URL: https://doi.org/10.1515/9781400882526, doi:10.1515/9781400882526.

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, resultants and multidimensional determinants. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. ISBN 978-0-8176-4770-4. Reprint of the 1994 edition.

Papa Badiane, Chinh H. Lu, and Ahmed Zeriahi. Complex Monge–Ampère equations on Hermitian manifolds: from bounded to smooth solutions. 2026. arXiv:2610.06473v1, 5 October 2026; numbered statements refer to version 1. URL: https://arxiv.org/abs/2610.06473v1, arXiv:2610.06473.

Alexander Grothendieck and Jean Dieudonné. Éléments de géométrie algébrique IV.4. Number 32 in Publications Mathématiques de l'IHÉS. Institut des Hautes Études Scientifiques, 1967.

Vincent Guedj and Chinh H. Lu. Quasi-plurisubharmonic envelopes 2: bounds on Monge–Ampère volumes. Algebraic Geometry, 9(6):688–713, 2022. URL: https://content.algebraicgeometry.nl/2022-6/2022-6-021.pdf, doi:10.14231/AG-2022-021.

V. V. Shokurov. Three-fold log flips. Russian Acad. Sci. Izv. Math., 40(1):95–202, 1993. Appendix by Yujiro Kawamata. URL: https://doi.org/10.1070/IM1993v040n01ABEH001862, doi:10.1070/IM1993v040n01ABEH001862.

Vincent Guedj and Ahmed Zeriahi. Degenerate complex Monge-Ampère equations. Volume 26 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich, 2017. ISBN 978-3-03719-167-5. URL: https://doi.org/10.4171/167, doi:10.4171/167.

Mikael Passare and Hans Rullgård. Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope. Duke Math. J., 121(3):481–507, 2004. URL: https://doi.org/10.1215/S0012-7094-04-12134-7, doi:10.1215/S0012-7094-04-12134-7.

Mohamed El Kadiri. Remarks on weak convergence of complex Monge-Ampère measures. Indag. Math. (N.S.), 35(1):28–36, 2024. URL: https://doi.org/10.1016/j.indag.2023.08.001, doi:10.1016/j.indag.2023.08.001.

Ruadhaí Dervan and Rémi Reboulet. Ding stability and Kähler-Einstein metrics on manifolds with big anticanonical class. J. Reine Angew. Math., 816:201–239, 2024. URL: https://doi.org/10.1515/crelle-2024-0062, doi:10.1515/crelle-2024-0062.

Hyunsoo Ahn. Hölder regularity of Siciak–Zaharjuta extremal functions on compact Hermitian manifolds. arXiv:2604.01887v3, 2026. URL: https://arxiv.org/abs/2604.01887v3.

Robin Hartshorne. Algebraic geometry. Volume No. 52 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977. ISBN 0-387-90244-9.

Yun-Heng Du and Song-Yan Xie. A first-jet characterization of Griffiths positivity. 2026. arXiv:2609.26528v1. URL: https://arxiv.org/abs/2609.26528v1, arXiv:2609.26528.

Alan Michael Nadel. Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. of Math. (2), 132(3):549–596, 1990. URL: https://doi.org/10.2307/1971429, doi:10.2307/1971429.

Pierre Lelong. Fonctions plurisousharmoniques et formes différentielles positives. Gordon & Breach, Paris-London-New York; distributed by Dunod Éditeur, Paris, 1968.

Alexander Grothendieck and Jean Dieudonné. Éléments de géométrie algébrique IV.1. Number 20 in Publications Mathématiques de l'IHÉS. Institut des Hautes Études Scientifiques, 1964.

Mingchen Xia. Operations on transcendental non-Archimedean metrics. In Convex and complex: perspectives on positivity in geometry, volume 810 of Contemp. Math., pages 271–293. Amer. Math. Soc., Providence, RI, 2025. URL: https://doi.org/10.1090/conm/810/16216, doi:10.1090/conm/810/16216.

Tamás Darvas, Eleonora Di Nezza, and Chinh H. Lu. Monotonicity of nonpluripolar products and complex Monge-Ampère equations with prescribed singularity. Anal. PDE, 11(8):2049–2087, 2018. URL: https://doi.org/10.2140/apde.2018.11.2049, doi:10.2140/apde.2018.11.2049.

Robert J. Berman. Bergman kernels and equilibrium measures for line bundles over projective manifolds. Amer. J. Math., 131(5):1485–1524, 2009. URL: https://doi.org/10.1353/ajm.0.0077, doi:10.1353/ajm.0.0077.

Eleonora Di Nezza and Stefano Trapani. The regularity of envelopes. Ann. Sci. Éc. Norm. Supér. (4), 57(5):1347–1370, 2024. arXiv version: arXiv:2110.14314v3, 6 February 2023. URL: https://doi.org/10.24033/asens.2592, doi:10.24033/asens.2592.

Lars Hörmander. Notions of convexity. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2007. ISBN 978-0-8176-4584-7; 0-8176-4584-5. Reprint of the 1994 edition [of MR1301332].

Tien-Cuong Dinh and Viêt-Anh Nguyên. Characterization of Monge–Ampère measures with Hölder continuous potentials. Journal of Functional Analysis, 266(1):67–84, 2014. URL: https://arxiv.org/abs/1204.4883, doi:10.1016/j.jfa.2013.08.026.

Robert Berman and Sébastien Boucksom. Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math., 181(2):337–394, 2010. URL: https://doi.org/10.1007/s00222-010-0248-9, doi:10.1007/s00222-010-0248-9.

Marcel Brelot. Étude des fonctions sous-harmoniques au voisinage d'un point singulier. Ann. Inst. Fourier (Grenoble), 1:121–156, 1949. URL: http://www.numdam.org/item?id=AIF_1949__1__121_0.

Hans Grauert and Reinhold Remmert. Plurisubharmonische Funktionen in komplexen Räumen. Math. Z., 65:175–194, 1956. URL: https://doi.org/10.1007/BF01473877, doi:10.1007/BF01473877.

David Catlin. The Bergman kernel and a theorem of Tian. In Analysis and geometry in several complex variables (Katata, 1997), Trends Math., pages 1–23. Birkhäuser Boston, Boston, MA, 1999.

Jean-Pierre Demailly and János Kollár. Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup. (4), 34(4):525–556, 2001. URL: https://doi.org/10.1016/S0012-9593(01)01069-2, doi:10.1016/S0012-9593(01)01069-2.

Bo Berndtsson. Probability measures associated to geodesics in the space of Kähler metrics. In Algebraic and analytic microlocal analysis, volume 269 of Springer Proc. Math. Stat., pages 395–419. Springer, Cham, 2018. URL: https://doi.org/10.1007/978-3-030-01588-6_6, doi:10.1007/978-3-030-01588-6\_6.

Sébastien Boucksom, Charles Favre, and Mattias Jonsson. Differentiability of volumes of divisors and a problem of Teissier. J. Algebraic Geom., 18(2):279–308, 2009. URL: https://doi.org/10.1090/S1056-3911-08-00490-6, doi:10.1090/S1056-3911-08-00490-6.

Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi. Singular Kähler–Einstein metrics. Journal of the American Mathematical Society, 22(3):607–639, 2009. Numbered statements refer to arXiv:math/0603431v2, 24 September 2008. URL: https://arxiv.org/abs/math/0603431v2, doi:10.1090/S0894-0347-09-00629-8.

Constantin Bănică and Octavian Stănăşilă. Algebraic methods in the global theory of complex spaces. Editura Academiei, Bucharest; John Wiley & Sons, London-New York-Sydney, 1976. Translated from the Romanian.

Sébastien Boucksom and Mattias Jonsson. Global pluripotential theory over a trivially valued field. Ann. Fac. Sci. Toulouse Math. (6), 31(3):647–836, 2022. URL: https://doi.org/10.5802/afst.170, doi:10.5802/afst.170.

Tomoyuki Hisamoto. Restricted Bergman kernel asymptotics. 2012. arXiv:\allowbreak 1201.4233v2, 15 February 2012; author version. URL: https://arxiv.org/abs/1201.4233v2.

Haïm Brezis and Augusto C. Ponce. Kato's inequality when $\Delta u$ is a measure. C. R. Math. Acad. Sci. Paris, 338(8):599–604, 2004. doi:10.1016/j.crma.2003.12.032.

Shinichi Matsumura. Restricted volumes and divisorial Zariski decompositions. Amer. J. Math., 135(3):637–662, 2013.

Kuang-Ru Wu. Positively curved Finsler metrics on vector bundles III. 2023. arXiv:2312.16848v1, 28 December 2023. URL: https://arxiv.org/abs/2312.16848v1, arXiv:2312.16848.

Duc-Viet Vu. Relative non-pluripolar product of currents. Ann. Global Anal. Geom., 60(2):269–311, 2021. URL: https://doi.org/10.1007/s10455-021-09780-7, doi:10.1007/s10455-021-09780-7.

Quang-Tuan Dang. Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds. arXiv:2603.25520v2, 2026. Locators marked v1 refer to arXiv:2603.25520v1. URL: https://arxiv.org/abs/2603.25520v2.

Liviu Ornea and Misha Verbitsky. Oeljeklaus–Toma manifolds admitting no complex subvarieties. Mathematical Research Letters, 18(4):747–754, 2011. URL: https://arxiv.org/abs/1009.1101.

Pham Hoang Hiep. The weighted log canonical threshold. C. R. Math. Acad. Sci. Paris, 352(4):283–288, 2014. URL: https://doi.org/10.1016/j.crma.2014.02.010, doi:10.1016/j.crma.2014.02.010.

Pietro Mesquita-Piccione and David Witt Nyström. A transcendental non-Archimedean Calabi–Yau theorem with applications to the cscK problem. 2025. arXiv:2509.09442.

Edward Bierstone and Pierre D. Milman. Functoriality in Resolution of Singularities. Publ. Res. Inst. Math. Sci., 44(2):609–639, 2008. URL: https://ems.press/journals/prims/articles/2154, doi:10.2977/PRIMS/1210167338.

Charles H. Walter. Irreducibility of moduli spaces of vector bundles on birationally ruled surfaces. 1993. Numbered statements refer to arXiv:alg-geom/9311005v1, 16 November 1993. URL: https://arxiv.org/abs/alg-geom/9311005v1, arXiv:alg-geom/9311005.

Alexander Rashkovskii. Analytic approximations of plurisubharmonic singularities. Math. Z., 275(3-4):1217–1238, 2013. URL: https://doi.org/10.1007/s00209-013-1179-0, doi:10.1007/s00209-013-1179-0.

Tamás Darvas and Mingchen Xia. The volume of pseudoeffective line bundles and partial equilibrium. Geom. Topol., 28(4):1957–1993, 2024. URL: https://doi.org/10.2140/gt.2024.28.1957, doi:10.2140/gt.2024.28.1957.

Marcel Brelot. Éléments de la théorie classique du potentiel. “Les cours de Sorbonne”, 3e cycle. Centre de Documentation Universitaire, Paris, 1965. 3e édition.

Giuseppe Barbaro and Alexandra Otiman. Calabi–Yau locally conformally Kähler manifolds. arXiv:2509.18364v2, 2025. 7 December 2025. URL: https://arxiv.org/abs/2509.18364v2.

John L. Kelley. General topology. Volume No. 27 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.].

Ahmed Zeriahi. Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions. Indiana University Mathematics Journal, 50(1):671–703, 2001. Local Skoda integrability recalled in Section 1. URL: https://doi.org/10.1512/iumj.2001.50.2062, doi:10.1512/iumj.2001.50.2062.

Bo Guan. The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function. Communications in Analysis and Geometry, 6(4):687–703, 1998. URL: https://archive.intlpress.com/site/pub/files/_fulltext/journals/cag/1998/0006/0004/CAG-1998-0006-0004-a003.pdf, doi:10.4310/CAG.1998.v6.n4.a3.

Miguel Abreu. Kähler geometry of toric varieties and extremal metrics. Internat. J. Math., 9(6):641–651, 1998. URL: https://doi.org/10.1142/S0129167X98000282, doi:10.1142/S0129167X98000282.

Hans Grauert and Reinhold Remmert. Coherent analytic sheaves. Volume 265 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1984. ISBN 3-540-13178-7. URL: https://doi.org/10.1007/978-3-642-69582-7, doi:10.1007/978-3-642-69582-7.

Yum Tong Siu. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math., 27:53–156, 1974. URL: https://doi.org/10.1007/BF01389965, doi:10.1007/BF01389965.

Sébastien Boucksom, Tomoyuki Hisamoto, and Mattias Jonsson. Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble), 67(2):743–841, 2017. URL: http://aif.cedram.org/item?id=AIF_2017__67_2_743_0.

Vincent Guedj and Ahmed Zeriahi. Intrinsic capacities on compact Kähler manifolds. Journal of Geometric Analysis, 15(4):607–639, 2005. URL: https://www.math.univ-toulouse.fr/~guedj/fichierspdf/JGA05.pdf.

Robert Lazarsfeld and Mircea Mustaţă. Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. (4), 42(5):783–835, 2009. URL: https://doi.org/10.24033/asens.2109, doi:10.24033/asens.2109.

Qi'an Guan and Xiangyu Zhou. A proof of Demailly's strong openness conjecture. Ann. of Math. (2), 182(2):605–616, 2015. URL: https://annals.math.princeton.edu/2015/182-2/p05, doi:10.4007/annals.2015.182.2.5.

Michael E. Taylor. Function space and operator theory for nonlinear analysis. Author-hosted Chapter 13, Sections 5–6 and 8; accessed 4 and 8 October 2026. URL: https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/chap13.pdf.

Jean-Pierre Demailly. Regularization of closed positive currents and intersection theory. Author-hosted manuscript, 42 pages; retained PDF created 31 May 2007. The published paper is Journal of Algebraic Geometry 1 (1992), 361–409. URL: https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization.pdf.

Eric Bedford and B. A. Taylor. The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math., 37(1):1–44, 1976. URL: https://doi.org/10.1007/BF01418826, doi:10.1007/BF01418826.

Terence Tao. 245B, notes 12: continuous functions on locally compact Hausdorff spaces. 2009. Online lecture notes, 2 March 2009. URL: https://terrytao.wordpress.com/2009/03/02/245b-notes-12-continuous-functions-on-locally-compact-hausdorff-spaces/.

Dongwei Gu and Ngoc Cuong Nguyen. The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 18(4):1189–1248, 2018. URL: https://journals.sns.it/index.php/annaliscienze/article/view/611, doi:10.2422/2036-2145.201606_013.

David Witt Nyström. Two calabi-yau theorems for degenerations of compact kähler manifolds. 2025. arXiv:2510.03044.

Vincent Guedj and Ahmed Zeriahi. Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal., 15(4):607–639, 2005. URL: https://doi.org/10.1007/BF02922247, doi:10.1007/BF02922247.

Urban Cegrell. Pluricomplex energy. Acta Math., 180(2):187–217, 1998. URL: https://doi.org/10.1007/BF02392899, doi:10.1007/BF02392899.

Shin-ichi Matsumura. Injectivity theorems with multiplier ideal sheaves for higher direct images under Kähler morphisms. Algebr. Geom., 9(2):122–158, 2022. URL: https://doi.org/10.14231/ag-2022-005, doi:10.14231/ag-2022-005.

Zbigniew Błocki and Sławomir Kołodziej. On regularization of plurisubharmonic functions on manifolds. Proc. Amer. Math. Soc., 135(7):2089–2093, 2007. URL: https://gamma.im.uj.edu.pl/~blocki/publ/appr.pdf, doi:10.1090/S0002-9939-07-08858-2.

Takayuki Koike. Pluripotential geometry on semi-positive effective divisors of numerical dimension one. 2026. arXiv:2511.13903v2, 21 January 2026. URL: https://arxiv.org/abs/2511.13903v2, arXiv:2511.13903.

Hossein Raufi. Singular hermitian metrics on holomorphic vector bundles. Ark. Mat., 53(2):359–382, 2015. URL: https://doi.org/10.1007/s11512-015-0212-4, doi:10.1007/s11512-015-0212-4.

Robert J. Berman. Determinantal point processes and fermions on complex manifolds: large deviations and bosonization. Comm. Math. Phys., 327(1):1–47, 2014. URL: https://doi.org/10.1007/s00220-014-1891-6, doi:10.1007/s00220-014-1891-6.

Terence Tao. 245A, notes 6: outer measures, pre-measures, and product measures. 2010. Online lecture notes, 30 October 2010. URL: https://terrytao.wordpress.com/2010/10/30/245a-notes-6-outer-measures-pre-measures-and-product-measures/.

Tristan C. Collins and Valentino Tosatti. Kähler currents and null loci. Inventiones Mathematicae, 202(3):1167–1198, 2015. URL: https://arxiv.org/abs/1304.5216.

Neil S. Trudinger and Xu-Jia Wang. Hessian measures. III. J. Funct. Anal., 193(1):1–23, 2002. URL: https://doi.org/10.1006/jfan.2001.3925, doi:10.1006/jfan.2001.3925.

R. Tyrrell Rockafellar. Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., 1970.

Sébastien Boucksom. On the volume of a line bundle. Internat. J. Math., 13(10):1043–1063, 2002. URL: https://doi.org/10.1142/S0129167X02001575, doi:10.1142/S0129167X02001575.

Mingchen Xia and Kewei Zhang. A counterexample to the bounded mass property. 2026. arXiv:2608.21053v1, 21 August 2026. URL: https://arxiv.org/abs/2608.21053v1, arXiv:2608.21053.

Patrick Graf and Tim Kirschner. Finite quotients of three-dimensional complex tori. Ann. Inst. Fourier (Grenoble), 70(2):881–914, 2020. URL: http://aif.cedram.org/item?id=AIF_2020__70_2_881_0.

Hanwen Liu. Non-Kähler critical Hermitian metrics of the Dinew–Popovici functional. 2026. arXiv:2609.08113v1, 8 September 2026; numbered statements refer to version 1. URL: https://arxiv.org/abs/2609.08113v1, arXiv:2609.08113v1.

Jean-Pierre Demailly. Regularization of closed positive currents of type $(1,1)$ by the flow of a Chern connection. In Contributions to Complex Analysis and Analytic Geometry, volume E26 of Aspects of Mathematics, pages 105–126. Vieweg, 1994. URL: https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization_flow.pdf.

Terence Tao. Lecture notes 3 for 247A. UCLA Math 247A lecture notes, 2006. URL: https://www.math.ucla.edu/~tao/247a.1.06f/notes3.pdf.

Xiankui Meng and Xiangyu Zhou. On the restriction formula. J. Geom. Anal., 33(12):Paper No. 369, 30, 2023. URL: https://doi.org/10.1007/s12220-023-01425-4, doi:10.1007/s12220-023-01425-4.

Dan Popovici. An observation relative to a paper by J. Xiao. 2014. Preprint, exact arXiv:1405.2518v1, 11 May 2014. URL: https://arxiv.org/abs/1405.2518v1.

Mohamed El Kadiri, Bent Fuglede, and Jan Wiegerinck. Plurisubharmonic and holomorphic functions relative to the plurifine topology. J. Math. Anal. Appl., 381(2):706–723, 2011. URL: https://doi.org/10.1016/j.jmaa.2011.03.041, doi:10.1016/j.jmaa.2011.03.041.

Chinh H. Lu. Comparison of Monge-Ampère capacities. Ann. Polon. Math., 126(1):31–53, 2021. arXiv version: arXiv:2005.04264v1, 8 May 2020. URL: https://doi.org/10.4064/ap200513-2-1, doi:10.4064/ap200513-2-1.

Mingchen Xia. Pluripotential-theoretic stability thresholds. Int. Math. Res. Not. IMRN, pages 12324–12382, 2023. URL: https://doi.org/10.1093/imrn/rnac186, doi:10.1093/imrn/rnac186.

Bin Guo and Jian Song. Stability and gradient estimates for fully nonlinear elliptic equations on Hermitian manifolds. 2026. arXiv:2609.39726v1, 30 September 2026. URL: https://arxiv.org/abs/2609.39726v1, arXiv:2609.39726v1.

Sławomir Kołodziej and Ngoc Cuong Nguyen. Complex Hessian measures with respect to a background Hermitian form. Analysis and PDE, 19(1):107–166, 2026. URL: https://msp.org/apde/2026/19-1/apde-v19-n1-p02-p.pdf, doi:10.2140/apde.2026.19.107.

Tristan C. Collins and Valentino Tosatti. Restricted volumes on Kähler manifolds. Annales de la Faculté des Sciences de Toulouse, 31(3):907–947, 2022. URL: https://www.numdam.org/item/10.5802/afst.1708.pdf, doi:10.5802/afst.1708.

Dan Popovici. Stability of strongly Gauduchon manifolds under modifications. J. Geom. Anal., 23(2):653–659, 2013. Proposition 1.2 refers to arXiv:1009.5408v1, 27 September 2010. doi:10.1007/s12220-011-9257-1.

Valentino Tosatti and Ben Weinkove. On the evolution of a Hermitian metric by its Chern–Ricci form. J. Differential Geom., 99(1):125–163, 2015. Trace calculation and elliptic estimate in Sections 3 and 9; the maximal-time statement is Theorem 1.2 in arXiv:1201.0312v3. URL: https://arxiv.org/abs/1201.0312, doi:10.4310/JDG/1418345539.

Xiangsen Qin. Failure of semipositive metric extension in a smooth projective family. 2026. arXiv:2609.36410v1, 29 September 2026. URL: https://arxiv.org/abs/2609.36410v1, arXiv:2609.36410.

Stephen Semmes. Complex Monge-Ampère and symplectic manifolds. Amer. J. Math., 114(3):495–550, 1992. URL: https://doi.org/10.2307/2374768, doi:10.2307/2374768.

Robert J. Berman and Bo Berndtsson. Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties. Ann. Fac. Sci. Toulouse Math. (6), 22(4):649–711, 2013. URL: https://doi.org/10.5802/afst.1386, doi:10.5802/afst.1386.

Duong H. Phong and Jacob Sturm. Test configurations for K-stability and geodesic rays. J. Symplectic Geom., 5(2):221–247, 2007. URL: http://projecteuclid.org/euclid.jsg/1202004456.

Boris G. Moishezon. Resolution theorems for compact complex spaces with a sufficiently large field of meromorphic functions. Mathematics of the USSR-Izvestiya, 1(6):1331–1356, 1967. URL: https://www.mathnet.ru/eng/im2594, doi:10.1070/IM1967v001n06ABEH000624.

Takeo Ohsawa and Kensho Takegoshi. On the extension of $L^2$ holomorphic functions. Math. Z., 195(2):197–204, 1987. URL: https://doi.org/10.1007/BF01166457, doi:10.1007/BF01166457.

Dano Kim. Equivalence of plurisubharmonic singularities and Siu-type metrics. Monatsh. Math., 178(1):85–95, 2015. URL: https://doi.org/10.1007/s00605-015-0742-7, doi:10.1007/s00605-015-0742-7.

Tien-Cuong Dinh and Nessim Sibony. Distribution des valeurs de transformations méromorphes et applications. Comment. Math. Helv., 81(1):221–258, 2006. URL: https://doi.org/10.4171/CMH/50, doi:10.4171/CMH/50.

Bengt Josefson. On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on $\bf C\sp n$. Ark. Mat., 16(1):109–115, 1978. URL: https://doi.org/10.1007/BF02385986, doi:10.1007/BF02385986.

Bent Fuglede. Connexion en topologie fine et balayage des mesures. Annales de l'Institut Fourier, 21(3):227–244, 1971. URL: https://aif.centre-mersenne.org/item/10.5802/aif.388.pdf, doi:10.5802/aif.388.

Said El Marzguioui and Jan Wiegerinck. The pluri-fine topology is locally connected. Potential Anal., 25(3):283–288, 2006. arXiv version: arXiv:math/0512241v1, 12 December 2005. URL: https://doi.org/10.1007/s11118-006-9017-1, doi:10.1007/s11118-006-9017-1.

Mingchen Xia. Partial Okounkov bodies and Duistermaat-Heckman measures of non-Archimedean metrics. Geom. Topol., 29(3):1283–1344, 2025. URL: https://doi.org/10.2140/gt.2025.29.1283, doi:10.2140/gt.2025.29.1283.

Eleonora Di Nezza and Stefano Trapani. Monge-Ampère measures on contact sets. Math. Res. Lett., 28(5):1337–1352, 2021. URL: https://doi.org/10.4310/mrl.2021.v28.n5.a3, doi:10.4310/mrl.2021.v28.n5.a3.

Kefeng Liu and Xueyuan Wan. Stable ample vector bundles without semipositive Kobayashi curvature. 2026. arXiv:2609.37309v1, 29 September 2026. URL: https://arxiv.org/abs/2609.37309v1, arXiv:2609.37309.

Ana María Botero, José Ignacio Burgos Gil, David Holmes, and Robin de Jong. Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated. Duke Math. J., 173(12):2315–2396, 2024. URL: https://doi.org/10.1215/00127094-2023-0059, doi:10.1215/00127094-2023-0059.

Tamás Darvas, Eleonora Di Nezza, and Chinh H. Lu. Relative pluripotential theory on compact Kähler manifolds. Pure Appl. Math. Q., 21(3):1037–1118, 2025. arXiv version: arXiv:2303.11584v2, 3 October 2023. URL: https://doi.org/10.4310/pamq.250116004957, doi:10.4310/pamq.250116004957.

Sébastien Boucksom. Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4), 37(1):45–76, 2004. URL: https://doi.org/10.1016/j.ansens.2003.04.002, doi:10.1016/j.ansens.2003.04.002.

Osamu Fujino and Shin-ichi Matsumura. Injectivity theorem for pseudo-effective line bundles and its applications. Trans. Amer. Math. Soc. Ser. B, 8:849–884, 2021. URL: https://doi.org/10.1090/btran/86, doi:10.1090/btran/86.

Thomas Delzant. Hamiltoniens périodiques et images convexes de l'application moment. Bull. Soc. Math. France, 116(3):315–339, 1988. URL: https://doi.org/10.24033/bsmf.2100, doi:10.24033/bsmf.2100.

Ionuţ Chiose. Obstructions to the existence of Kähler structures on compact complex manifolds. Proc. Amer. Math. Soc., 142(10):3561–3568, 2014. Theorem 0.2 in the ten-page author manuscript. URL: https://www.imar.ro/~cjoita/MC-SCV/Chiose-closedobstruction.pdf, doi:10.1090/S0002-9939-2014-12128-9.

Jean-Pierre Demailly, Lawrence Ein, and Robert Lazarsfeld. A subadditivity property of multiplier ideals. Michigan Math. J., 48:137–156, 2000. Dedicated to William Fulton on the occasion of his 60th birthday. URL: https://doi.org/10.1307/mmj/1030132712, doi:10.1307/mmj/1030132712.

Shuwen Chen and Fangyang Zheng. Hermitian connections with parallel torsion and the Fino–Vezzoni conjecture. 2026. Version 1; numbered statements refer to this version. URL: https://arxiv.org/abs/2609.27809v1, arXiv:2609.27809.

Terence Tao. 245A, notes 5: differentiation theorems. 2010. Online lecture notes, 16 October 2010. URL: https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/.

Kiumars Kaveh and A. G. Khovanskii. Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. of Math. (2), 176(2):925–978, 2012. URL: https://doi.org/10.4007/annals.2012.176.2.5, doi:10.4007/annals.2012.176.2.5.

Sławomir Kołodziej and Ngoc Cuong Nguyen. Weak convergence of Monge–Ampère measures on compact Hermitian manifolds. In Complex Analysis and Geometry: KSCV14, volume 481 of Springer Proceedings in Mathematics & Statistics, pages 113–123. Springer, 2025. URL: https://arxiv.org/abs/2212.11550v1, doi:10.1007/978-981-96-0447-0_10.

Martino Bardi and Alessandro Goffi. New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs. Calculus of Variations and Partial Differential Equations, 58(6):184, 2019. Article 184. Checked proof locators refer to arXiv:1812.09589v1. URL: https://doi.org/10.1007/s00526-019-1620-2, doi:10.1007/s00526-019-1620-2.

Xiuxiong Chen. The space of Kähler metrics. J. Differential Geom., 56(2):189–234, 2000. URL: http://projecteuclid.org/euclid.jdg/1090347643.

Mohamed El Kadiri and Bent Fuglede. Classical fine potential theory. Springer Monographs in Mathematics. Springer, Singapore, [2025] ©2025. ISBN 978-981-96-0429-6; 978-981-96-0430-2. With a paper by Christian Berg. URL: https://doi.org/10.1007/978-981-96-0430-2, doi:10.1007/978-981-96-0430-2.

Yuetong Fang. Smoothness of bounded solutions to complex Hessian equations on compact Hermitian manifolds. 2026. Numbered statements refer to arXiv:2610.06719v1. URL: https://arxiv.org/abs/2610.06719v1, arXiv:2610.06719.

Mohamed El Kadiri and Jan Wiegerinck. Plurifinely plurisubharmonic functions and the Monge Ampère operator. Potential Anal., 41(2):469–485, 2014. URL: https://doi.org/10.1007/s11118-013-9378-1, doi:10.1007/s11118-013-9378-1.

Sławomir Kołodziej and Ngoc Cuong Nguyen. Stability and regularity of solutions of the Monge–Ampère equation on Hermitian manifolds. Advances in Mathematics, 346:264–304, 2019. Numbered statements refer to arXiv:1501.05749v4. URL: https://arxiv.org/abs/1501.05749v4, doi:10.1016/j.aim.2019.02.004.

Sébastien Boucksom, Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi. Monge-Ampère equations in big cohomology classes. Acta Math., 205(2):199–262, 2010. URL: https://doi.org/10.1007/s11511-010-0054-7, doi:10.1007/s11511-010-0054-7.

Huy Hoang Dao, Quang Dieu Nguyen, and Duc Hieu Tran. Continuity of pluricomplex Green functions with hypersurface poles on certain B-regular domains. 2026. URL: https://arxiv.org/abs/2610.00909v1, arXiv:2610.00909v1.

Bin Guo and Duong H. Phong. Uniform $L^\infty $ estimates: subsolutions to fully nonlinear partial differential equations. 2024. arXiv:2401.11572v1, 21 January 2024. URL: https://arxiv.org/abs/2401.11572v1, arXiv:2401.11572v1.

Daniele Angella, Vincent Guedj, and Chinh H. Lu. Plurisigned Hermitian metrics. Transactions of the American Mathematical Society, 376(7):4631–4659, 2023. URL: https://www.math.univ-toulouse.fr/~guedj/fichierspdf/TAMS22.pdf, doi:10.1090/tran/8916.

David A. Cox, John B. Little, and Henry K. Schenck. Toric varieties. Volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011. ISBN 978-0-8218-4819-7. URL: https://doi.org/10.1090/gsm/124, doi:10.1090/gsm/124.

Tien-Cuong Dinh and Nessim Sibony. Regularization of currents and entropy. Ann. Sci. École Norm. Sup. (4), 37(6):959–971, 2004. URL: https://doi.org/10.1016/j.ansens.2004.09.002, doi:10.1016/j.ansens.2004.09.002.

Tamás Darvas, Rémi Reboulet, David Witt Nyström, Mingchen Xia, and Kewei Zhang. Transcendental Okounkov bodies. J. Differential Geom., 132(1):135–178, 2026. arXiv version: arXiv:2309.07584v3, 5 January 2026. URL: https://doi.org/10.4310/jdg/1766433802, doi:10.4310/jdg/1766433802.

Yueh-Lin Chiang. Bergman kernel asymptotics for semipositive line bundles near curvature-degenerate points. 2026. arXiv:2609.24869v1, 21 September 2026; Example I.15 refers to this version. URL: https://arxiv.org/abs/2609.24869v1, arXiv:2609.24869.

Vincent Guedj and Chinh H. Lu. Degenerate complex Hessian equations on compact Hermitian manifolds. Pure and Applied Mathematics Quarterly, 21(3):1171–1194, 2025. Numbered statements refer to arXiv:2302.03354v1. URL: https://arxiv.org/abs/2302.03354v1, doi:10.4310/PAMQ.250116014513.

Tamás Darvas and Mingchen Xia. The trace operator of quasi-plurisubharmonic functions on compact Kähler manifolds. 2024. arXiv:2403.08259.

Tamás Darvas, Eleonora Di Nezza, and Hoang-Chinh Lu. The metric geometry of singularity types. J. Reine Angew. Math., 771:137–170, 2021. URL: https://doi.org/10.1515/crelle-2020-0019, doi:10.1515/crelle-2020-0019.

Mingchen Xia. Singularities in global pluripotential theory. Author-supplied manuscript. Supplied Final-2 version; source files retained 3 October 2026.

Tamás Darvas and Yanir A. Rubinstein. Kiselman's principle, the Dirichlet problem for the Monge-Ampère equation, and rooftop obstacle problems. J. Math. Soc. Japan, 68(2):773–796, 2016. URL: https://doi.org/10.2969/jmsj/06820773, doi:10.2969/jmsj/06820773.

Sławomir Kołodziej and Ngoc Cuong Nguyen. Weak solutions of complex Hessian equations on compact Hermitian manifolds. Compositio Mathematica, 152(11):2221–2248, 2016. Numbered statements refer to arXiv:1507.06755v2. URL: https://arxiv.org/abs/1507.06755v2, doi:10.1112/S0010437X16007417.

Henri Cartan. Sur les fondements de la théorie du potentiel. Bull. Soc. Math. France, 69:71–96, 1941. URL: http://www.numdam.org/item?id=BSMF_1941__69__71_0.

Sławomir Kołodziej and Ngoc Cuong Nguyen. Hölder continuous solutions of the Monge–Ampère equation on compact Hermitian manifolds. Annales de l'Institut Fourier, 68(7):2951–2964, 2018. URL: https://doi.org/10.5802/aif.3232, doi:10.5802/aif.3232.

Chung-Ming Pan. Gauduchon metrics and Hermite–Einstein metrics on non-Kähler varieties. arXiv:2503.02759v1, 2025. URL: https://arxiv.org/abs/2503.02759v1.

Tamás Darvas, Mingchen Xia, and Kewei Zhang. A transcendental approach to non-Archimedean metrics of pseudoeffective classes. Comment. Math. Helv., 100(2):269–322, 2025. URL: https://doi.org/10.4171/cmh/586, doi:10.4171/cmh/586.

Tien-Cuong Dinh, Viêt-Anh Nguyên, and Nessim Sibony. Exponential estimates for plurisubharmonic functions and stochastic dynamics. Journal of Differential Geometry, 84(3):465–488, 2010. URL: https://arxiv.org/abs/0801.1983.

Christer O. Kiselman. The partial Legendre transformation for plurisubharmonic functions. Invent. Math., 49(2):137–148, 1978. URL: https://doi.org/10.1007/BF01403083, doi:10.1007/BF01403083.

Valentino Tosatti. Regularity of envelopes in Kähler classes. Math. Res. Lett., 25(1):281–289, 2018. URL: https://doi.org/10.4310/mrl.2018.v25.n1.a12, doi:10.4310/mrl.2018.v25.n1.a12.

Eric Bedford and B. A. Taylor. A new capacity for plurisubharmonic functions. Acta Math., 149(1-2):1–40, 1982. URL: https://doi.org/10.1007/BF02392348, doi:10.1007/BF02392348.

Sébastien Boucksom, Vincent Guedj, and Chinh H. Lu. Volumes of Bott–Chern classes. Peking Mathematical Journal, 2025. Theorem locations refer to the 43-page publisher version, published online 16 June 2025. URL: https://doi.org/10.1007/s42543-025-00105-2, doi:10.1007/s42543-025-00105-2.

Tamás Darvas, Eleonora Di Nezza, and Chinh H. Lu. Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity. Math. Ann., 379(1-2):95–132, 2021. arXiv version: arXiv:1807.00276v3, 12 March 2020. URL: https://doi.org/10.1007/s00208-019-01936-y, doi:10.1007/s00208-019-01936-y.

Michael F. Atiyah. Convexity and commuting Hamiltonians. Bull. London Math. Soc., 14(1):1–15, 1982. URL: https://doi.org/10.1112/blms/14.1.1, doi:10.1112/blms/14.1.1.

Vincent Guedj and Ahmed Zeriahi. The weighted Monge-Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal., 250(2):442–482, 2007. URL: https://doi.org/10.1016/j.jfa.2007.04.018, doi:10.1016/j.jfa.2007.04.018.

George Marinescu and Xu Xing. Parameter dependence of weighted Bergman kernels beyond smoothness. 2026. arXiv:2609.24258v1. URL: https://arxiv.org/abs/2609.24258v1, arXiv:2609.24258.