Classes de cohomologie positives dans les variétés kählériennes compactes

Debarre, Olivier

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

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 $\mathrm{NS}{\left(X\right)}_{𝐑}\subset H^{1,1}(X,𝐑)$ engendré par les classes entières sont respectivement le cône des classes de diviseurs amples et l’adhérence de celui des classes de diviseurs effectifs.

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 $\mathrm{NS}{\left(X\right)}_{𝐑}\subset H^{1,1}(X,𝐑)$ generated by integral classes, the cone of classes of ample divisors and the closure of the cone of classes of effective divisors.

Classification : 32J27, 14M20, 14E30, 14C20, 14C17, 14C30, 32C30
Mot clés : variété kählérienne, variété hyperkählérienne, cône ample, cône nef, cône pseudo-effectif, classes grandes, cône de Kähler, courant, métrique singulière, décomposition de Zariski, volume d'un fibré en droites, variété uniréglée, courbe mobile
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

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