Entire curves and algebraic differential equations

Jean-Pierre Demailly

Description

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

• constant holomorphic map


• simply connected


• open subset


• has no


• dimensional manifold


• liouville's theorem


• entire curves

Sujets

• Demailly
• Simply connected space
• Open set
• Liouville's theorem

Informations

• Publié par: pefav
• Nombre de lectures: 99
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

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⊂C n , 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⊂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)

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⊂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 Λ (Λ lattice) has a lot of entire curves. AsCsimply connected, everyf:C→X=C n Λ n ˜ lifts asf:C→C,

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

˜ andf j :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

P n =P n (C n+1 r{0})C ∗  C =

Jean-Pierre Demailly (Grenoble I), 16/04/2009

n-space

[z] = [z 0

:z 1

:  :z n ]

Entire curves and algebraic differential equations / IF - IMPA