This note grew out of a series of lectures held in Cortona in 2019 and whose aim was to understand the recent breakthrough obtained by Chen and Cheng on the existence of constant scalar curvature Kähler metrics. We present a detailed version of the and a priori estimates within the realm of pluripotential theory.
Cette note est le fruit d’une série d’exposés donnés à Cortona en 2019 et dont le but visait à comprendre les avancées majeures due à Chen et Cheng sur l’existence de métriques kähleriennes à courbure scalaire constante. Nous donnons ici une preuve alternative et détaillée des estimées a priori dites et dans le cadre de la théorie du pluripotentiel.
Alix Deruelle 1; Eleonora Di Nezza 2, 3
@article{AFST_2022_6_31_3_975_0, author = {Alix Deruelle and Eleonora Di Nezza}, title = {Uniform estimates for {cscK} metrics}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {975--993}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 31}, number = {3}, year = {2022}, doi = {10.5802/afst.1710}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1710/} }
TY - JOUR AU - Alix Deruelle AU - Eleonora Di Nezza TI - Uniform estimates for cscK metrics JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2022 SP - 975 EP - 993 VL - 31 IS - 3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1710/ DO - 10.5802/afst.1710 LA - en ID - AFST_2022_6_31_3_975_0 ER -
%0 Journal Article %A Alix Deruelle %A Eleonora Di Nezza %T Uniform estimates for cscK metrics %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2022 %P 975-993 %V 31 %N 3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1710/ %R 10.5802/afst.1710 %G en %F AFST_2022_6_31_3_975_0
Alix Deruelle; Eleonora Di Nezza. Uniform estimates for cscK metrics. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, AMAZER, Volume 31 (2022) no. 3, pp. 975-993. doi : 10.5802/afst.1710. https://afst.centre-mersenne.org/articles/10.5802/afst.1710/
[1] Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math., Volume 102 (1978) no. 1, pp. 63-95 | MR | Zbl
[2] Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer, 1998, xviii+395 pages | DOI | MR
[3] Regularity of weak minimizers of the K-energy and applications to properness and K-stability, Ann. Sci. Éc. Norm. Supér., Volume 53 (2020) no. 2, pp. 267-289 | DOI | MR | Zbl
[4] Monge-Ampère equations in big cohomology classes., Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | DOI | Zbl
[5] Extremal Kähler metrics, Seminar on Differential Geometry (Annals of Mathematics Studies), Volume 102, Princeton University Press, 1982, pp. 259-290 | MR | Zbl
[6] Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013), pp. 879-916 | DOI | Numdam | MR | Zbl
[7] On the constant scalar curvature Kähler metrics, general automorphism group (2018) | arXiv
[8] On the constant scalar curvature Kähler metrics, a priori estimates, J. Am. Math. Soc., Volume 34 (2021), pp. 909-936 | DOI | Zbl
[9] On the constant scalar curvature Kähler metrics, existence results, J. Am. Math. Soc., Volume 34 (2021), pp. 937-1009 | DOI | Zbl
[10] Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 183-197 | DOI | MR | Zbl
[11] Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than , J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 199-234 | DOI | MR | Zbl
[12] Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches and completion of the main proof, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 235-278 | DOI | MR | Zbl
[13] Monotonicity of nonpluripolar products and complex Monge-Ampère equations with prescribed singularity, Anal. PDE, Volume 11 (2018) no. 8, pp. 2049-2087 | DOI | MR | Zbl
[14] Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity, Math. Ann. (2019), pp. 1-38
[15] Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics, J. Am. Math. Soc., Volume 30 (2017) no. 2, pp. 347-387 | DOI | MR | Zbl
[16] Scalar curvature and stability of toric varieties, J. Differ. Geom., Volume 62 (2002) no. 2, pp. 289-349 | MR | Zbl
[17] An obstruction to the existence of Einstein Kähler metrics, Invent. Math., Volume 73 (1983) no. 3, pp. 437-443 | DOI | MR
[18] The weighted Monge–Ampère energy of quasi plurisubharmonic functions, J. Funct. Anal., Volume 250 (2007), pp. 442-482 | DOI | Zbl
[19] Degenerate complex Monge-Ampère equations, EMS Tracts in Mathematics, 26, European Mathematical Society, 2017, xxiv+472 pages | DOI | MR
[20] The complex Monge-Ampère equation, Acta Math., Volume 180 (1998) no. 1, pp. 69-117 | DOI | MR | Zbl
[21] An introduction to extremal Kähler metrics, Graduate Studies in Mathematics, 152, American Mathematical Society, 2014, xvi+192 pages | DOI | MR
[22] K-stability and Kähler-Einstein metrics, Commun. Pure Appl. Math., Volume 68 (2015) no. 7, pp. 1085-1156 | DOI | MR
[23] On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I., Commun. Pure Appl. Math., Volume 31 (1978), pp. 339-411 | Zbl
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