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
Voir
- constant holomorphic map
- simply connected
- algebraic varieties
- has no
- dimensional manifold
- liouville's theorem
- entire curves
December 18, 2009 / Colloquium CAS, Beijing
Jean-Pierre Demailly
Recent progress in the study of hyperbolic algebraic varieties
InstitutFourier,Universite´deGrenobleI,France
Definition.By anentire curvewe mean a non constant holomorphic mapf:C→Xinto a complex n-dimensional manifold.
itseraeiiavcegrb
Definition.By anentire curvewe mean a non constant holomorphic mapf:C→Xinto a complex n-dimensional manifold. IfXis aboundedopen subset Ω⊂Cn, then there are no entire curvesf:C→Ω (Liouville’s theorem)
Definition.By anentire curvewe mean a non constant holomorphic mapf:C→Xinto a complex n-dimensional manifold. IfXis aboundedopen subset Ω⊂Cn, then there are no entire curvesf:C→Ω (Liouville’s theorem) X=C r{01∞}=C r{01}has no entire curves (Picard’s theorem)
Jean-PierreDemayllierG(lbon,)IeijBeg,in/118202/sruevriceEtn
˜ ˜ ˜ f(t) = (f1(t) fn(t))
˜ andfj: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⊂Cn, then there are no entire curvesf:C→Ω (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:C→X=CnΛ ˜ lifts asf:C→Cn,
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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]
rietieseseiP-naeJiameDerrGrenlly(I),Boblegn1,ieij2/00/821
