Kahler manifolds and transcendental techniques in algebraic geometry

Jean-Pierre Demailly

Description

Kahler manifolds and transcendental techniques in algebraic geometry Jean-Pierre Demailly Institut Fourier, Universite de Grenoble I, France August 29 / ICM 2006, Madrid Jean-Pierre Demailly (Grenoble I), 29/08/2006 Transcendental techniques in algebraic geometry

• holomorphic coordinates


• ipq ∑


• cohomological properties


• charts equipped


• transcendental techniques


• dzj1 ?


• distribution coefficient


• complex manifolds

Sujets

• Demailly
• Partition coefficient
• Complex manifold

Informations

• Publié par: pefav
• Nombre de lectures: 28
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

K¨ahler tech

manifolds and transcendental niques in algebraic geometry

Jean-Pierre

Demailly

InstitutFourier,Universite´deGrenobleI,France

August

Jean-Pierre Demailly (Grenoble I), 29/08/2006

29

/

ICM

2006,

Madrid

Transcendental techniques in algebraic geometry

Complex

manifolds

/

(pq)-forms

Goal : study thegeometric / topological / cohomological propertiesofcompactK¨ahlermanifolds

Jean-Pierre Demailly (Grenoble I), 29/08/2006

Transcendental techniques in algebraic geometry

Complex manifolds / (pq)-forms

Goal : study thegeometric / topological / cohomological propertiesofcompactK¨ahlermanifolds A complexn-dimensional manifold is given by coordinate charts equipped with local holomorphic coordinates (z 1 z 2     z n ).

Jean-Pierre Demailly (Grenoble I), 29/08/2006

Transcendental techniques in algebraic geometry

Complex manifolds / (pq)-forms

Goal : study thegeometric / topological / cohomological propertiesofcompactKa¨hlermanifolds A complexn-dimensional manifold is given by coordinate charts equipped with local holomorphic coordinates (z 1 z 2     z n ). A differential formuof type (pq) can be written as a sum u(z) = X u JK (z)dz J ∧d z K |J|=p|K|=q

whereJ= (j 1     j p ),K= (k 1     k q ),

dz J =dz j 1 ∧  ∧dz j p d z K =d z k 1 ∧  ∧d z k q 

Jean-Pierre Demailly (Grenoble I), 29/08/2006

Transcendental techniques in algebraic geometry

Complex

manifolds

/

Currents

A current is a differential form withdistribution coefficients T(z) =i pq X T JK (z)dz J ∧d z K |J|=p|K|=q

Jean-Pierre Demailly (Grenoble I), 29/08/2006

Transcendental techniques in algebraic geometry

Complex manifolds / Currents

A current is a differential form withdistribution coefficients T(z) =i pq X T JK (z)dz J ∧d z K

|J|=p|K|=q

The currentTis said to bepositiveif the distribution P λ j λ k T JK is a positive real measure for all (λ J )∈C N (so thatT KJ =T JK , henceT=T). The coefficientsT JK are thencomplex measures– and the diagonal onesT JJ arepositive real measures.

Jean-Pierre Demailly (Grenoble I), 29/08/2006

Transcendental techniques in algebraic geometry

Complex manifolds / Currents

A current is a differential form withdistribution coefficients T(z) =i pq X T JK (z)dz J ∧d z K

|J|=p|K|=q

The currentTis said to bepositiveif the distribution P λ j λ k T JK is a positive real measure for all (λ J )∈C N (so thatT KJ =T JK , henceT=T). The coefficientsT JK are thencomplex measures– and the diagonal onesT JJ arepositive real measures. Tis said to beclosedifdT= 0 in the sense of distributions.

Jean-Pierre Demailly (Grenoble I), 29/08/2006

Transcendental techniques in algebraic geometry