Positive cohomology classes in compact Kähler varieties

Olivier Debarre

Séminaire Bourbaki (2004-2005)

  • Volume: 47, page 199-228
  • ISSN: 0303-1179

Abstract

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Let X be a compact Kähler manifold. In the real vector space H 1 , 1 ( X , 𝐑 ) ⊂ 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 ⊂ 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.

How to cite

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Debarre, Olivier. "Classes de cohomologie positives dans les variétés kählériennes compactes." Séminaire Bourbaki 47 (2004-2005): 199-228. <http://eudml.org/doc/252164>.

@article{Debarre2004-2005,
abstract = {Étant donnée une variété kählérienne compacte $X$, on étudie dans l’espace vectoriel réel de cohomologie de Dolbeault $H^\{1,1\}(X,\{\bf R\})\subset H^2(X,\{\bf R\})$ 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 $\mathop \{\rm NS\}\nolimits (X)_\{\bf R\}\subset H^\{1,1\}(X,\{\bf R\})$ engendré par les classes entières sont respectivement le cône des classes de diviseurs amples et l’adhérence de celui des classes de diviseurs effectifs.},
author = {Debarre, Olivier},
journal = {Séminaire Bourbaki},
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},
language = {fre},
pages = {199-228},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Classes de cohomologie positives dans les variétés kählériennes compactes},
url = {http://eudml.org/doc/252164},
volume = {47},
year = {2004-2005},
}

TY - JOUR
AU - Debarre, Olivier
TI - Classes de cohomologie positives dans les variétés kählériennes compactes
JO - Séminaire Bourbaki
PY - 2004-2005
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 47
SP - 199
EP - 228
AB - Étant donnée une variété kählérienne compacte $X$, on étudie dans l’espace vectoriel réel de cohomologie de Dolbeault $H^{1,1}(X,{\bf R})\subset H^2(X,{\bf R})$ 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 $\mathop {\rm NS}\nolimits (X)_{\bf R}\subset H^{1,1}(X,{\bf R})$ engendré par les classes entières sont respectivement le cône des classes de diviseurs amples et l’adhérence de celui des classes de diviseurs effectifs.
LA - fre
KW - 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
UR - http://eudml.org/doc/252164
ER -

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