Skoda–Zeriahi type integrability and entropy compactness for some measure with $L^1$-density
[Intégrabilité de type Skoda–Zeriahi et compacité d’entropie pour une certaine mesure avec densité $L^1$]
Aoi, Takahiro1
1National Institute of Technology (KOSEN), Wakayama College, 77, Nojima, Nada-chou, Gobo-shi Wakayama, 644-0023 (Japan)
Annales de l'Institut Fourier, Online first, 27 p.

In this article, we prove the Skoda–Zeriahi type integrability theorem with respect to some measure with $L^1$-density. In addition, we introduce the log-log threshold in order to detect singularities of Kähler potentials. We prove the positivity of the integrability threshold for such a measure and Kähler potentials with uniform log-log threshold. As an application, we prove the entropy compactness theorem for a family of potential functions of Poincaré type Kähler metrics with uniform log-log threshold. The Ohsawa–Takegoshi $L^2$-extension theorem and Skoda–Zeriahi’s integrability theorem play a very important role in this article.

Dans cet article, nous prouvons le théorème d’intégrabilité de type Skoda–Zeriahi pour une certaine mesure avec une fonction de densité $L^1$ . De plus, nous introduisons le seuil log-log afin de détecter les singularités des potentiels kählériennes. Nous prouvons la positivité de seuil d’intégrabilité pour une telle mesure et des potentiels kählériennes à seuil log-log uniforme. Pour l’application, nous prouvons le théorème de compacité d’entropie pour une famille de fonctions potentielles de métriques kählériennes de type Poincaré à seuil log-log uniforme. Le théorème d’extension $L^2$ d’Ohsawa–Takegoshi et le théorème d’intégrabilité de Skoda–Zeriahi jouent un rôle très important dans cet article.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3785
Classification : 31U05
Keywords: Plurisubharmonic functions, Kähler metrics of Poincaré type, relative entropy
Mots-clés : Fonctions plurisousharmonique, métriques kählériennes de type Poincaré, entropie relative

Aoi, Takahiro  1

1 National Institute of Technology (KOSEN), Wakayama College, 77, Nojima, Nada-chou, Gobo-shi Wakayama, 644-0023 (Japan) 
@unpublished{AIF_0__0_0_A74_0,
     author = {Aoi, Takahiro},
     title = {Skoda{\textendash}Zeriahi type integrability and entropy compactness for some measure with $L^1$-density},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3785},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Aoi, Takahiro
TI  - Skoda–Zeriahi type integrability and entropy compactness for some measure with $L^1$-density
JO  - Annales de l'Institut Fourier
PY  - 2026
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3785
LA  - en
ID  - AIF_0__0_0_A74_0
ER  - 
%0 Unpublished Work
%A Aoi, Takahiro
%T Skoda–Zeriahi type integrability and entropy compactness for some measure with $L^1$-density
%J Annales de l'Institut Fourier
%D 2026
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3785
%G en
%F AIF_0__0_0_A74_0
Aoi, Takahiro. Skoda–Zeriahi type integrability and entropy compactness for some measure with $L^1$-density. Annales de l'Institut Fourier, Online first, 27 p.

[1] Aoi, Takahiro A conical approximation of constant scalar curvature Kähler metrics of Poincaré type (2022) (https://arxiv.org/abs/2210.11862) | Zbl

[2] Auvray, Hugues The space of Poincaré type Kähler metrics on the complement of a divisor, J. Reine Angew. Math., Volume 722 (2017), pp. 1-64 | DOI | MR | Zbl

[3] Bedford, Eric; Taylor, B. Alan Fine topology, Šilov boundary, and (dd c ) n , J. Funct. Anal., Volume 72 (1987) no. 2, pp. 225-251 | DOI | MR | Zbl

[4] Berman, Robert J. A thermodynamical formalism for Monge–Ampère equations, Moser-Trudinger inequalities and Kähler–Einstein metrics, Adv. Math., Volume 248 (2013), pp. 1254-1297 | MR | DOI | Zbl

[5] Berman, Robert J.; Boucksom, Sebastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties, J. Reine Angew. Math., Volume 751 (2019), pp. 27-89 | DOI | MR | Zbl

[6] Berman, Robert J.; Boucksom, Sébastien; Guedj, Vincent; Zeriahi, Ahmed A variational approach to complex Monge–Ampère equations, Publ. Math., Inst. Hautes Étud. Sci., Volume 117 (2013), pp. 179-245 | Numdam | DOI | MR | Zbl

[7] Berman, Robert J.; Darvas, Tamás; Lu, Chinh H. Convexity of the extended K-energy and the large time behavior of the weak Calabi flow, Geom. Topol., Volume 21 (2017) no. 5, pp. 2945-2988 | DOI | MR | Zbl

[8] Bł ocki, Zbigniew Suita conjecture and the Ohsawa-Takegoshi extension theorem, Invent. Math., Volume 193 (2013) no. 1, pp. 149-158 | DOI | MR | Zbl

[9] Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | DOI | MR | Zbl

[10] Chen, Bo-Yong A simple proof of the Ohsawa-Takegoshi extension theorem (2011) (https://arxiv.org/abs/1105.2430) | Zbl

[11] Chen, Xiuxiong On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Not., Volume 2000 (2000) no. 12, pp. 607-623 | DOI | MR | Zbl

[12] Chen, Xiuxiong The space of Kähler metrics, J. Differ. Geom., Volume 56 (2000) no. 2, pp. 189-234 | DOI | MR | Zbl

[13] Chen, Xiuxiong; Cheng, Jingrui On the constant scalar curvature Kähler metrics (II)—Existence results, J. Am. Math. Soc., Volume 34 (2021) no. 4, pp. 937-1009 | DOI | MR | Zbl

[14] Darvas, Tamás The Mabuchi geometry of finite energy classes, Adv. Math., Volume 285 (2015), pp. 182-219 | DOI | MR | Zbl

[15] Darvas, Tamás Geometric pluripotential theory on Kähler manifolds, Advances in complex geometry (Contemporary Mathematics), Volume 735, American Mathematical Society, 2019, pp. 1-104 | MR | DOI | Zbl

[16] Demailly, Jean-Pierre Complex analytic and differential geometry (https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)

[17] Demailly, Jean-Pierre; Kollár, János Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér. (4), Volume 34 (2001) no. 4, pp. 525-556 | Numdam | DOI | MR | Zbl

[18] Dervan, Ruadhaí Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds, Ann. Fac. Sci. Toulouse, Math. (6), Volume 25 (2016) no. 4, pp. 919-934 | Numdam | DOI | MR | Zbl

[19] Di Nezza, Eleonora; Guedj, Vincent; Lu, Chinh H. Finite entropy vs finite energy, Comment. Math. Helv., Volume 96 (2021) no. 2, pp. 389-419 | DOI | MR | Zbl

[20] Di Nezza, Eleonora; Lu, Chinh H. Complex Monge–Ampère equations on quasi-projective varieties, J. Reine Angew. Math., Volume 727 (2017), pp. 145-167 | DOI | MR | Zbl

[21] Donaldson, Simon K. Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar (American Mathematical Society Translations, Series 2), Volume 196, American Mathematical Society, 1999, pp. 13-33 | DOI | MR | Zbl

[22] Guan, Qi’an; Zhou, Xiangyu A solution of an L 2 extension problem with an optimal estimate and applications, Ann. Math. (2), Volume 181 (2015) no. 3, pp. 1139-1208 | DOI | MR | Zbl

[23] Guedj, Vincent; Zeriahi, Ahmed Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal., Volume 15 (2005) no. 4, pp. 607-639 | DOI | MR | Zbl

[24] Guedj, Vincent; Zeriahi, Ahmed The weighted Monge–Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal., Volume 250 (2007) no. 2, pp. 442-482 | DOI | MR | Zbl

[25] Guedj, Vincent; Zeriahi, Ahmed Degenerate complex Monge–Ampère equations, EMS Tracts in Mathematics, 26, European Mathematical Society, 2017, xxiv+472 pages | DOI | MR | Zbl

[26] Guenancia, Henri Kähler–Einstein metrics: from cones to cusps, J. Reine Angew. Math., Volume 759 (2020), pp. 1-27 | DOI | MR | Zbl

[27] Hörmander, Lars L 2 estimates and existence theorems for the ∂ ¯ operator, Acta Math., Volume 113 (1965), pp. 89-152 | DOI | MR | Zbl

[28] Hörmander, Lars An introduction to complex analysis in several variables, North-Holland Mathematical Library, 7, North-Holland, 1990, xii+254 pages | MR | Zbl

[29] Kaufmann, Lucas A Skoda-type integrability theorem for singular Monge-Ampère measures, Mich. Math. J., Volume 66 (2017) no. 3, pp. 581-594 | DOI | MR | Zbl

[30] Lempert, László; Vivas, Liz Geodesics in the space of Kähler metrics, Duke Math. J., Volume 162 (2013) no. 7, pp. 1369-1381 | DOI | MR | Zbl

[31] Ma, Xiaonan; Marinescu, George Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, 254, Birkhäuser, 2007, xiv+422 pages | DOI | MR | Zbl

[32] Mabuchi, Toshiki Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math., Volume 24 (1987) no. 2, pp. 227-252 | MR | Zbl

[33] McNeal, Jeffery D.; Varolin, Dror Analytic inversion of adjunction: L 2 extension theorems with gain, Ann. Inst. Fourier, Volume 57 (2007) no. 3, pp. 703-718 | Numdam | DOI | MR | Zbl

[34] Ohsawa, Takeo; Takegoshi, Kenshō On the extension of L 2 holomorphic functions, Math. Z., Volume 195 (1987) no. 2, pp. 197-204 | DOI | MR | Zbl

[35] Semmes, Stephen Complex Monge–Ampère and symplectic manifolds, Am. J. Math., Volume 114 (1992) no. 3, pp. 495-550 | DOI | MR | Zbl

[36] Skoda, Henri Sous-ensembles analytiques d’ordre fini ou infini dans C n , Bull. Soc. Math. Fr., Volume 100 (1972), pp. 353-408 | Numdam | DOI | MR | Zbl

[37] Székelyhidi, Gábor Extremal metrics and K-stability, Ph. D. Thesis, Imperial College London (2006)

[38] Tian, Gang On Kähler–Einstein metrics on certain Kähler manifolds with C 1 (M)>0, Invent. Math., Volume 89 (1987) no. 2, pp. 225-246 | DOI | MR | Zbl

[39] Tian, Gang Canonical metrics in Kähler geometry. Notes taken by Meike Akveld, Birkhäuser, 2000 | DOI | MR | Zbl

[40] Zeriahi, Ahmed Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions, Indiana Univ. Math. J., Volume 50 (2001) no. 1, pp. 671-703 | DOI | MR | Zbl

[41] Zheng, Kai Existence of constant scalar curvature Kähler cone metrics, properness and geodesic stability, Math. Ann., Volume 393 (2025) no. 3-4, pp. 3639-3711 | DOI | MR | Zbl

Cité par Sources :