Classes de cohomologie positives dans les variétés kählériennes compactes

Classes de cohomologie positives dans les variétés kählériennes compactes [ Positive cohomology classes in compact Kähler varieties ]

Debarre, Olivier

Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque no. 307 (2006), Talk no. 943, p. 199-228

Abstract
Résumé

Let $X$ be a compact Kähler manifold. In the real vector space $H^{1, 1} (X, 𝐑) \subset H^{2} (X, 𝐑)$ of Dolbeault cohomology classes of type $(1, 1)$ , we study the convex cone of Kähler classes and the larger cone of classes of positive closed currents of type $(1, 1)$ . When $X$ is projective, theses cones cut out, on the Néron-Severi subspace $NS {(X)}_{𝐑} \subset H^{1, 1} (X, 𝐑)$ generated by integral classes, the cone of classes of ample divisors and the closure of the cone of classes of effective divisors.

Étant donnée une variété kählérienne compacte $X$ , on étudie dans l’espace vectoriel réel de cohomologie de Dolbeault $H^{1, 1} (X, 𝐑) \subset H^{2} (X, 𝐑)$ le cône convexe des classes de Kähler ainsi que celui, plus grand, des classes de courants positifs fermés de type $(1, 1)$ . Lorsque $X$ est projective, les traces de ces cônes sur l’espace de Néron-Severi $NS {(X)}_{𝐑} \subset H^{1, 1} (X, 𝐑)$ engendré par les classes entières sont respectivement le cône des classes de diviseurs amples et l’adhérence de celui des classes de diviseurs effectifs.

MR 2296419 | Zbl 1125.32009 | 1 citation in Numdam

Classification: 32J27, 14M20, 14E30, 14C20, 14C17, 14C30, 32C30
Keywords: Kähler manifold, hyperkähler manifold, ample cone, nef cone, pseudo-effective cone, big cone, Kähler cone, current, singular metric, Zariski decomposition, volume of a line bundle, uniruled variety, mobile curve

@incollection{SB_2004-2005__47__199_0,
     author = {Debarre, Olivier},
     title = {Classes de cohomologie positives dans les vari\'et\'es k\"ahl\'eriennes compactes},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     note = {talk:943},
     pages = {199-228},
     zbl = {1125.32009},
     mrnumber = {2296419},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2004-2005__47__199_0}
}

Debarre, Olivier. Classes de cohomologie positives dans les variétés kählériennes compactes, in Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 943, pp. 199-228. http://www.numdam.org/item/SB_2004-2005__47__199_0/

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