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Recent progress in the study of hyperbolic algebraic varieties

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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


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Language English
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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:CXinto a complex n-dimensional manifold.
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Definition.By anentire curvewe mean a non constant holomorphic mapf:CXinto a complex n-dimensional manifold. IfXis aboundedopen subset ΩCn, then there are no entire curvesf:CΩ (Liouville’s theorem)
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Definition.By anentire curvewe mean a non constant holomorphic mapf:CXinto a complex n-dimensional manifold. IfXis aboundedopen subset ΩCn, then there are no entire curvesf:CΩ (Liouville’s theorem) X=C r{01∞}=C r{01}has no entire curves (Picard’s theorem)
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˜ ˜ ˜ f(t) = (f1(t)    fn(t))
˜ andfj:CCcan be arbitrary entire functions.
Definition.By anentire curvewe mean a non constant holomorphic mapf:CXinto a complex n-dimensional manifold. IfXis aboundedopen subset ΩCn, then there are no entire curvesf:CΩ (Liouville s theorem) X=C r{01∞}=C r{01}has no entire curves (Picard’s theorem) A complex torusX=Cnlattice) has a lot of entireΛ (Λ curves. AsCsimply connected, everyf:CX=CnΛ ˜ lifts asf:CCn,
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Consider now the complex projectiven-space
Pn=PnC= (Cn+1r{0})C
r{0})C
C= (C
[z] = [z0:z1:  :zn]
[z] = [z0:z1:  :zn]
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