Algebraic structure of the ring of jet differential operators and hyperbolic

Jean-Pierre Demailly

Description

Algebraic structure of the ring of jet differential operators and hyperbolic varieties Jean-Pierre Demailly Institut Fourier, Universite de Grenoble I, France December 21 / ICCM 2007, Hangzhou Jean-Pierre Demailly (Grenoble I), 21/12/2007 Jet differential operators and hyperbolic varieties / Hangzhou

• kobayashi metric


• jet differential


• holomorphic curves


• hyperbolic varieties


• brody has


• compact riemann


• kobayashi pseudo-metric

Sujets

• Demailly

Informations

• Publié par: pefav
• Nombre de lectures: 17
• Langue: English

Extrait

Algebraic differenti

structure of the al operators and varieties

Jean-Pierre

Demailly

ring of jet hyperbolic

InstitutFourier,Universit´edeGrenobleI,France

December 21

Jean-Pierre Demailly (Grenoble I), 21/12/2007

/ ICCM

2007,

Hangzhou

Jet differential operators and hyperbolic varieties / Hangzhou

Kobayashi

metric /

hyperbolic

manifolds

LetXbe a complex manifold,n= dim C X. Xis said to behyperbolic in the sense of Brodyif there are no non-constant entire holomorphic curves f:C→X.

Jean-Pierre Demailly (Grenoble I), 21/12/2007

Jet differential operators and hyperbolic varieties / Hangzhou

Kobayashi metric / hyperbolic manifolds

LetXbe a complex manifold,n= dim C X. Xis said to behyperbolic in the sense of Brodyif there are no non-constant entire holomorphic curves f:C→X. Brody has shown that forXcompact, hyperbolicity is equivalent to thenon degeneracy of the Kobayashi pseudo-metric:x∈X,ξ∈T X

k x (ξ) = inf{λ >0 ;∃f:D→X

Jean-Pierre Demailly (Grenoble I), 21/12/2007

f(0) =x λf ∗ (0) =ξ}

Jet differential operators and hyperbolic varieties / Hangzhou

Kobayashi metric / hyperbolic manifolds

LetXbe a complex manifold,n= dim C X. Xis said to behyperbolic in the sense of Brodyif there are no non-constant entire holomorphic curves f:C→X. Brody has shown that forXcompact, hyperbolicity is equivalent to thenon degeneracy of the Kobayashi pseudo-metric:x∈X,ξ∈T X

k x (ξ) = inf{λ >0 ;∃f:D→Xf(0) =x λf ∗ (0) =ξ}

Hyperbolic varieties are especially interesting for their expected diophantine properties : Conjecture(S. Lang)If a projective variety X defined overQis hyperbolic, then X(Q)is finite.

Jean-Pierre Demailly (Grenoble I), 21/12/2007

Jet differential operators and hyperbolic varieties / Hangzhou

Hyperbolicity

Case

n= 1

and

curvature

(compact

Riemann

X=P 1 X=C(Z+Zτ)

obviously non hyperbolic :∃f

Jean-Pierre Demailly (Grenoble I), 21/12/2007

surfaces):

(g= 0 (g= 1

:C→X.

T X T X

>0) = 0)

Jet differential operators and hyperbolic varieties / Hangzhou