The space of finite-energy metrics over a degeneration of complex manifolds
Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 659-701.

Given a degeneration of projective complex manifolds X𝔻 * with meromorphic singularities, and a relatively ample line bundle L on X, we study spaces of plurisubharmonic metrics on L, with particular focus on (relative) finite-energy conditions. We endow the space ^ 1 (L) of relatively maximal, relative finite-energy metrics with a d 1 -type distance given by the Lelong number at zero of the collection of fiberwise Darvas d 1 -distances. We show that this metric structure is complete and geodesic. Seeing X and L as schemes X K , L K over the discretely-valued field K=((t)) of complex Laurent series, we show that the space 1 (L K an ) of non-Archimedean finite-energy metrics over L K an embeds isometrically and geodesically into ^ 1 (L), and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case.

Étant donné une dégénérescence de variétés projectives complexes X𝔻 * avec des singularités méromorphes, et un fibré en droites relativement ample L sur X, nous étudions des espaces de métriques plurisousharmoniques sur L, avec une attention particulière aux conditions (relatives) d’énergie finie. Nous munissons l’espace ^ 1 (L) des métriques relativement maximales d’énergie finie d’une distance de type d 1 donnée par le nombre de Lelong en 0 de la famille des distances de Darvas d 1 fibre à fibre. Nous montrons que cette structure métrique est complète et géodésique. En considérant X et L comme des schémas X K , L K sur le champ discrètement valué K=((t)) des séries de Laurent complexes, nous montrons que l’espace 1 (L K an ) des métriques non archimédiennes d’énergie finie sur L K an s’immerge isométriquement et géodésiquement dans ^ 1 (L), et nous caractérisons son image. Ceci généralise un travail précédent de Berman-Boucksom-Jonsson, traitant le cas trivialement valué.

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Accepted:
Published online:
DOI: 10.5802/jep.229
Classification: 32U05, 32Q15, 14E99
Keywords: Berkovich spaces, complex manifolds, pluripotential theory, degenerations
Mot clés : Espaces de Berkovich, variétés complexes, théorie du pluripotentiel, dégénérescences

Rémi Reboulet 1

1 Department of mathematical sciences, Chalmers University of Technology 412 96 Gothenburg, Sweden
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Rémi Reboulet. The space of finite-energy metrics over a degeneration of complex manifolds. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 659-701. doi : 10.5802/jep.229. https://jep.centre-mersenne.org/articles/10.5802/jep.229/

[BBGZ13] R. J. Berman, S. Boucksom, V. Guedj & A. Zeriahi - “A variational approach to complex Monge-Ampère equations”, Publ. Math. Inst. Hautes Études Sci. 117 (2013), p. 179-245 | DOI | Numdam | Zbl

[BBJ21] R. J. Berman, S. Boucksom & M. Jonsson - “A variational approach to the Yau-Tian-Donaldson conjecture”, J. Amer. Math. Soc. 34 (3) (2021), p. 605-652 | DOI | MR | Zbl

[BDL17] R. J. Berman, T. Darvas & C. H. Lu - “Convexity of the extended K-energy and the large time behavior of the weak Calabi flow”, Geom. Topol. 21 (2017) no. 5, p. 2945-2988 | DOI | MR | Zbl

[BE21] S. Boucksom & D. Eriksson - “Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry”, Adv. Math. 378 (2021), article ID 107501, 124 pages | DOI | MR | Zbl

[Ber90] V. G. Berkovich - Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990 | DOI | MR

[Ber09] V. G. Berkovich - “A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures”, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Boston, MA, 2009, p. 49-67 | DOI | MR | Zbl

[BFJ08] S. Boucksom, C. Favre & M. Jonsson - “Valuations and plurisubharmonic singularities”, Publ. RIMS, Kyoto Univ. 44 (2008) no. 2, p. 449-494 | DOI | MR | Zbl

[BFJ16] S. Boucksom, C. Favre & M. Jonsson - “Singular semipositive metrics in non-Archimedean geometry”, J. Algebraic Geom. 25 (2016) no. 1, p. 77-139 | DOI | MR | Zbl

[BHJ19] S. Boucksom, T. Hisamoto & M. Jonsson - “Uniform K-stability and asymptotics of energy functionals in Kähler geometry”, J. Eur. Math. Soc. (JEMS) 21 (2019) no. 9, p. 2905-2944, Erratum: Ibid. 24 (2022), no. 2, p. 735–736 | DOI | Zbl

[BJ17] S. Boucksom & M. Jonsson - “Tropical and non-Archimedean limits of degenerating families of volume forms”, J. Éc. polytech. Math. 4 (2017), p. 87-139 | DOI | Numdam | MR | Zbl

[BJ22] S. Boucksom & M. Jonsson - “Global pluripotential theory over a trivially valued field”, Ann. Fac. Sci. Toulouse Math. (6) 31 (2022) no. 3, p. 647-836 | DOI | MR

[BK07] Z. Błocki & S. Kołodziej - “On regularization of plurisubharmonic functions on manifolds”, Proc. Amer. Math. Soc. 135 (2007) no. 7, p. 2089-2093 | DOI | MR | Zbl

[Bou18a] S. Boucksom - “Singularities of plurisubharmonic functions and multiplier ideals” (2018), Online lecture notes, http://sebastien.boucksom.perso.math.cnrs.fr/notes/L2.pdf

[Bou18b] S. Boucksom - “Variational and non-archimedean aspects of the Yau-Tian-Donaldson conjecture”, in Proceedings ICM—Rio de Janeiro 2018. Vol. II, World Sci. Publ., Hackensack, NJ, 2018, p. 591-617 | Zbl

[BT76] E. Bedford & B. A. Taylor - “The Dirichlet problem for a complex Monge-Ampère equation”, Invent. Math. 37 (1976) no. 1, p. 1-44 | DOI

[CGPT23] J. Cao, H. Guenancia, M. Paun & V. Tosatti - “Variation of singular Kähler–Einstein metrics: Kodaira dimension zero”, J. Eur. Math. Soc. (JEMS) 25 (2023) no. 2, p. 633-679 | DOI | Zbl

[CLD12] A. Chambert-Loir & A. Ducros - “Formes différentielles réelles et courants sur les espaces de Berkovich”, 2012 | arXiv

[Dar15] T. Darvas - “The Mabuchi geometry of finite energy classes”, Adv. Math. 285 (2015), p. 182-219 | DOI | MR | Zbl

[Dar19] T. Darvas - “Geometric pluripotential theory on Kähler manifolds”, 2019 | arXiv

[Del87] P. Deligne - “Le déterminant de la cohomologie”, in Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, American Mathematical Society, Providence, RI, 1987, p. 93-177 | DOI | Zbl

[DEL00] J.-P. Demailly, L. Ein & R. Lazarsfeld - “A subadditivity property of multiplier ideals”, Michigan Math. J. 48 (2000), p. 137-156 | DOI | MR | Zbl

[Dem92] J.-P. Demailly - “Regularization of closed positive currents and intersection theory”, J. Algebraic Geom. 1 (1992) no. 3, p. 361-409 | MR | Zbl

[Dem12] J.-P. Demailly - Analytic methods in algebraic geometry, Surveys of Modern Math., vol. 1, International Press & Higher Education Press, Somerville, MA & Beijing, 2012

[DL20] T. Darvas & C. H. Lu - “Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry”, Geom. Topol. 24 (2020) no. 4, p. 1907-1967 | DOI | MR | Zbl

[Don02] S. K. Donaldson - “Holomorphic discs and the complex Monge-Ampère equation”, J. Symplectic Geom. 1 (2002) no. 2, p. 171-196 | DOI | Zbl

[Don12] S. K. Donaldson - “Stability, birational transformations and the Kähler-Einstein problem”, in Surveys in differential geometry. Vol. XVII, Surv. Differ. Geom., vol. 17, Int. Press, Boston, MA, 2012, p. 203-228 | DOI | Zbl

[Elk89] R. Elkik - “Fibrés d’intersections et intégrales de classes de Chern”, Ann. Sci. École Norm. Sup. (4) 22 (1989) no. 2, p. 195-226 | DOI | Numdam | MR | Zbl

[Elk90] R. Elkik - “Métriques sur les fibrés d’intersection”, Duke Math. J. 61 (1990) no. 1, p. 303-328 | DOI | MR | Zbl

[Fav20] C. Favre - “Degeneration of endomorphisms of the complex projective space in the hybrid space”, J. Inst. Math. Jussieu 19 (2020) no. 4, p. 1141-1183 | DOI | MR | Zbl

[Gub07] W. Gubler - “Tropical varieties for non-Archimedean analytic spaces”, Invent. Math. 169 (2007) no. 2, p. 321-376 | DOI | MR | Zbl

[GZ12] V. Guedj & A. Zeriahi - “Dirichlet problem in domains of n , in Complex Monge-Ampère equations and geodesics in the space of Kähler metrics, Lect. Notes in Math., vol. 2038, Springer, Heidelberg, 2012, p. 13-32 | DOI | Zbl

[Kli91] M. Klimek - Pluripotential theory, London Math. Society Monographs, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991

[Lag12] A. Lagerberg - “Super currents and tropical geometry”, Math. Z. 270 (2012) no. 3-4, p. 1011-1050 | DOI | MR | Zbl

[Li22] C. Li - “Geodesic rays and stability in the cscK problem”, Ann. Sci. École Norm. Sup. (4) 55 (2022) no. 6, p. 1529-1574 | MR | Zbl

[MM07] X. Ma & G. Marinescu - Holomorphic Morse inequalities and Bergman kernels, Progress in Math., vol. 254, Birkhäuser Verlag, Basel, 2007 | DOI

[Mor99] A. Moriwaki - “The continuity of Deligne’s pairing”, Internat. Math. Res. Notices (1999) no. 19, p. 1057-1066 | DOI | MR | Zbl

[PRS08] D. H. Phong, J. Ross & J. Sturm - “Deligne pairings and the Knudsen-Mumford expansion”, J. Differential Geom. 78 (2008) no. 3, p. 475-496 | MR | Zbl

[PS10] D. H. Phong & J. Sturm - “The Dirichlet problem for degenerate complex Monge-Ampere equations”, Comm. Anal. Geom. 18 (2010) no. 1, p. 145-170 | DOI | MR | Zbl

[PS22] L. Pille-Schneider - “Hybrid convergence of Kähler-Einstein measures”, Ann. Inst. Fourier (Grenoble) 72 (2022) no. 2, p. 587-615 | DOI | Zbl

[Reb22] R. Reboulet - “Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy”, J. reine angew. Math. 793 (2022), p. 59-103 | DOI | MR | Zbl

[RN15] J. Ross & D. W. Nyström - “Harmonic discs of solutions to the complex homogeneous Monge-Ampère equation”, Publ. Math. Inst. Hautes Études Sci. 122 (2015), p. 315-335 | DOI | Zbl

[Sch12] G. Schumacher - “Positivity of relative canonical bundles and applications”, Invent. Math. 190 (2012) no. 1, p. 1-56, Erratum: Ibid. 192 (2013), no. 1, p. 253–255 | DOI | MR | Zbl

[Tsu10] H. Tsuji - “Dynamical construction of Kähler-Einstein metrics”, Nagoya Math. J. 199 (2010), p. 107-122 | DOI | MR | Zbl

[Xia19] M. Xia - “Mabuchi geometry of big cohomology classes with prescribed singularities”, 2019 | arXiv

[YZ21] X. Yuan & S.-W. Zhang - “Adelic line bundles over quasi-projective varieties”, 2021 | arXiv

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