Algebraic structure of the ring of jet differential operators and hyperbolic

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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

Voir

Publié par

pefav

Langue

English

Demailly

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= dimCX. 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= dimCX. 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∈TX

kx(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

kx(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=P1 X=C(Z+Zτ)

obviously non hyperbolic :∃f

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

surfaces):

(g= 0 (g= 1

:C→X.

TX TX

>0) = 0)

Jet diﬀerential operators and hyperbolic varieties / Hangzhou

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