Entire curves and algebraic differential equations
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66
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Entire curves and algebraic differential equations Jean-Pierre Demailly Institut Fourier, Universite de Grenoble I, France April 16 2009, Saint-Martin d'Heres IF - IMPA Conference Jean-Pierre Demailly (Grenoble I), 16/04/2009 Entire curves and algebraic differential equations / IF - IMPA
Voir
- constant holomorphic map
- simply connected
- open subset
- has no
- dimensional manifold
- liouville's theorem
- entire curves
Entire curves and aic differential equ
algebr
Jean-Pierre
ations
Demailly
InstitutFourier,Universit´edeGrenobleI,France
April
16 2009, Saint-Martin d’Heres ` IF - IMPA Conference
Jean-Pierre Demailly (Grenoble I), 16/04/2009
Entire curves and algebraic differential equations / IF - IMPA
Entire curves
Definition.By anentire curvewe mean a non constant holomorphic mapf:C→Xinto a complex n-dimensional manifold.
Jean-Pierre Demailly (Grenoble I), 16/04/2009
Entire curves and algebraic differential equations / IF - IMPA
Entire curves
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)
Jean-Pierre Demailly (Grenoble I), 16/04/2009
Entire curves and algebraic differential equations / IF - IMPA
Entire curves
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-Pierre Demailly (Grenoble I), 16/04/2009
Entire curves and algebraic differential equations / IF - IMPA
Entire curves
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=CnΛ (Λ lattice) has a lot of entire curves. AsCsimply connected, everyf:C→X=CnΛ n ˜ lifts asf:C→C,
˜ ˜ ˜ f(t) = (f1(t) fn(t))
˜ andfj:C→Ccan be arbitrary entire functions.
Jean-Pierre Demailly (Grenoble I), 16/04/2009
Entire curves and algebraic differential equations / IF - IMPA
Projective
algebraic
Consider
now
varieties
the complex
projective
Pn=Pn(Cn+1r{0})C∗ C=
Jean-Pierre Demailly (Grenoble I), 16/04/2009
n-space
[z] = [z0
:z1
: :zn]
Entire curves and algebraic differential equations / IF - IMPA
