Comptes Rendus
no. 1-2
Analytic Geometry
A Note on the cone of mobile curves
[Une Note sur le cône des courbes mobiles]

Présenté par : Jean-Pierre Demailly

Matei Toma1, 2 
1 Institut Élie-Cartan, Nancy-Université, CNRS, INRIA, B.P. 239, 54506 Vandoeuvre-lès-Nancy cedex, France
2 Institute of Mathematics of the Romanian Academy, Romania
Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 71-73.

S. Boucksom, J.-P. Demailly, M. Păun and Th. Peternell proved that the cone of mobile curves ME(X)¯ of a projective complex manifold X is dual to the cone generated by classes of effective divisors and conjectured an extension of this duality in the Kähler set-up. We show that their conjecture implies that ME(X)¯ coincides with the cone of integer classes represented by closed positive smooth (n−1,n−1)-forms. Without assuming the validity of the conjecture we prove that this equality of cones still holds at the level of degree functions.

S. Boucksom, J.-P. Demailly, M. Păun et Thomas Peternell ont montré que le cône des courbes mobiles ME(X)¯ d'une variété projective complexe X est le dual du cône engendré par les classes de diviseurs effectifs, et ils ont conjecturé que cette dualité pouvait s'étendre dans le contexte kählerien. Nous montrons que cette conjecture implique que ME(X)¯ coïncide avec le cône des classes entières représentées par des formes positives fermées de type (n−1,n−1) et de classe C∞. Sans supposer que cette conjecture soit vraie, nous montrons que cette égalité de cônes a lieu en tout cas au niveau des fonctions degré associées.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.11.003

Matei Toma 1, 2

1 Institut Élie-Cartan, Nancy-Université, CNRS, INRIA, B.P. 239, 54506 Vandoeuvre-lès-Nancy cedex, France
2 Institute of Mathematics of the Romanian Academy, Romania 
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Matei Toma. A Note on the cone of mobile curves. Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 71-73. doi : 10.1016/j.crma.2009.11.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.11.003/

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