[L'itération de Ricci et ses applications]
Dans cette Note nous introduisons et étudions des systèmes dynamiques reliées à l'opérateur de Ricci sur l'espace des métriques kählériennes comme discrétisations des certains flots géométriques. Nous posons une conjecture concernant leurs convergence vers des métriques kählériennes canoniques and nous étudions le cas où la première classe de Chern est négative, zéro ou positive. Cette construction a plusieurs applications en géométrie kählérienne, parmi elles une réponse à une question de Nadel et une construction des faisceaux d'idéaux multiplicateurs.
In this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kähler metrics and study the case where the first Chern class is negative, zero or positive. This construction has several applications in Kähler geometry, among them an answer to a question of Nadel and a construction of multiplier ideal sheaves.
Accepté le :
Publié le :
Yanir A. Rubinstein 1
@article{CRMATH_2007__345_8_445_0, author = {Yanir A. Rubinstein}, title = {The {Ricci} iteration and its applications}, journal = {Comptes Rendus. Math\'ematique}, pages = {445--448}, publisher = {Elsevier}, volume = {345}, number = {8}, year = {2007}, doi = {10.1016/j.crma.2007.09.020}, language = {en}, }
Yanir A. Rubinstein. The Ricci iteration and its applications. Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 445-448. doi : 10.1016/j.crma.2007.09.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.09.020/
[1] Équations du type Monge–Ampère sur les variétés kählériennes compactes, Bull. Sci. Math., Volume 102 (1978), pp. 63-95
[2] Uniqueness of Kähler–Einstein metrics modulo connected group actions, Sendai, 1985, Kinokuniya (1987), pp. 11-40
[3] Deformations of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds, Invent. Math., Volume 81 (1985), pp. 359-372
[4] Ricci flow on Kähler–Einstein surfaces, Invent. Math., Volume 147 (2002), pp. 487-544
[5] D.Z.-D. Guan, Extremal-solitons and convergence of the modified Calabi flow on certain bundles, preprint, December 22, 2006
[6] Three-manifolds with positive Ricci curvature, J. Diff. Geom., Volume 17 (1982), pp. 255-306
[7] Multiplier ideal sheaves and Kähler–Einstein metrics of positive scalar curvature, Ann. Math., Volume 132 (1990), pp. 549-596
[8] On the absence of periodic points for the Ricci curvature operator acting on the space of Kähler metrics, Modern Methods in Complex Analysis, Princeton University Press, 1995, pp. 277-281
[9] Multiplier ideal sheaves and the Kähler–Ricci flow (preprint, arxiv) | arXiv
[10] Y.A. Rubinstein, On iteration of the Ricci operator on the space of Kähler metrics, I, manuscript, August 14, 2005, unpublished
[11] On energy functionals, Kähler–Einstein metrics and the Moser–Trudinger–Onofri neighborhood (preprint, arxiv) | arXiv
[12] Y.A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics I, preprint, 2007; II, preprint, in preparation
[13] Kähler–Einstein metrics with positive scalar curvature, Invent. Math., Volume 130 (1997), pp. 1-37
[14] Convergence of Kähler–Ricci flow, J. Amer. Math. Soc., Volume 20 (2007), pp. 675-699
[15] On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampére equation, I, Comm. Pure Appl. Math., Volume 31 (1978), pp. 339-411
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