Recent progress in the study of hyperbolic algebraic varieties

Jean-Pierre Demailly

Description

Recent progress in the study of hyperbolic algebraic varieties Jean-Pierre Demailly Institut Fourier, Universite de Grenoble I, France December 18, 2009 / Colloquium CAS, Beijing Jean-Pierre Demailly (Grenoble I), Beijing, 18/12/2009 Recent progress in the study of hyperbolic algebraic varieties

• constant holomorphic map


• simply connected


• algebraic varieties


• has no


• dimensional manifold


• liouville's theorem


• entire curves

Sujets

• Demailly
• Simply connected space
• Algebraic variety
• Liouville's theorem

Informations

• Publié par: pefav
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

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December 18, 2009 / Colloquium CAS, Beijing

Jean-Pierre Demailly

Recent progress in the study of hyperbolic algebraic varieties

InstitutFourier,Universite´deGrenobleI,France

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Definition.By anentire curvewe mean a non constant holomorphic mapf:C→Xinto a complex n-dimensional manifold.

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Definition.By anentire curvewe mean a non constant holomorphic mapf:C→Xinto a complex n-dimensional manifold. IfXis aboundedopen subset Ω⊂C n , then there are no entire curvesf:C→Ω (Liouville’s theorem)

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Definition.By anentire curvewe mean a non constant holomorphic mapf:C→Xinto a complex n-dimensional manifold. IfXis aboundedopen subset Ω⊂C n , then there are no entire curvesf:C→Ω (Liouville’s theorem) X=C r{01∞}=C r{01}has no entire curves (Picard’s theorem)

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˜ ˜ ˜ f(t) = (f 1 (t)    f n (t))

˜ andf j :C→Ccan be arbitrary entire functions.

Definition.By anentire curvewe mean a non constant holomorphic mapf:C→Xinto a complex n-dimensional manifold. IfXis aboundedopen subset Ω⊂C n , then there are no entire curvesf:C→Ω (Liouville s theorem) ’ X=C r{01∞}=C r{01}has no entire curves (Picard’s theorem) A complex torusX=C n lattice) has a lot of entireΛ (Λ curves. AsCsimply connected, everyf:C→X=C n Λ ˜ lifts asf:C→C n ,

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Consider now the complex projectiven-space

P n =P n C = (C n+1 r{0})C ∗ 

r{0})C

C = (C

[z] = [z 0 :z 1 :  :z n ]

[z] = [z 0 :z 1 :  :z n ]

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