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{\bf Approche variationnelle pour les \'equations de Monge-Amp\`ere complexes et applications g\'eom\'etriques [d'apr\`es Berman, Boucksom, Eyssidieux, Guedj, Zeriahi ...]}

{\bf R\'esum\'e}. Les \'equations de Monge-Amp\`ere sur les vari\'et\'es  k\"ahleriennes compactes peuvent \^etre r\'esolues par une m\'ethode variationnelle ind\'ependante du th\'eor\`eme de Yau. La technique repose sur l'\'etude de certaines fonctionnelles (Ding-Tian, Mabuchi) sur l'espace des m\'etriques de K\"ahler, et sur leur convexit\'e g\'eod\'esique, due \`a Berndtsson-Berman dans sa forme g\'en\'erale. Les applications incluent l'existence et l'unicit\'e de m\'etriques de K\"ahler-Einstein sur les vari\'et\'e ${\mathbb Q}$-Fano \`a singularit\'es terminales, et une nouvelle preuve d'une version uniforme de la conjecture de Yau-Tian-Donaldson.

{\bf Mots-cl\'e}. Vari\'et\'e k\"ahlerienne compacte, \'equation de Monge-Amp\`ere, m\'ethode variationnelle, flot de Ricci, potentiel plurisousharmonique, courant positif ferm\'e, fonctionnelle de Ding-Tian, fonctionnelle de Mabuchi, convexit\'e g\'eod\'esique, vari\'et\'e de Fano, K-stabilit\'e, conjecture de Yau-Tian-Donaldson

{\bf Classification MSC.} 32Q15, 32Q20, 53C44  

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{\bf Variational approach for complex Monge-Amp\`ere equations and geometric applications [after Berman, Boucksom, Eyssidieux, Guedj, Zeriahi ...]}

{\bf Abstract.}
Solutions of Monge-Amp\`ere equations on compact K\"ahler manifolds can be obtained by a variational method independent of Yau’s theorem. The technique relies on the study of certain functionals (Ding-Tian, Mabuchi) on the space of K\"ahler metrics, and on their geodesic convexity, due to Berndtsson-Berman in its general form. Applications include the existence and uniqueness of K\"ahler-Einstein metrics on ${\mathbb Q}$-Fano varieties with log terminal singularities and a new proof of a uniform version of the Yau-Tian-Donaldson conjecture.

{\bf Keywords}. Compact K\"ahler variety, Monge-Amp\`ere equation, variational method, Ricci flow, plurisubharmonic potential, closed positive current, Ding-Tian functional, Mabuchi functional, geodesic convexity, Fano variety, K-stability, Yau-Tian-Donaldson conjecture

{\bf Classification MSC.} 32Q15, 32Q20, 53C44  

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