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Kahler manifolds and transcendental techniques in algebraic geometry

De
72 pages
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


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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
/
(pq)-forms
Goal : study thegeometric / topological / cohomological propertiesofcompactK¨ahlermanifolds
Jean-Pierre Demailly (Grenoble I), 29/08/2006
Transcendental techniques in algebraic geometry
Complex manifolds / (pq)-forms
Goal : study thegeometric / topological / cohomological propertiesofcompactK¨ahlermanifolds A complexn-dimensional manifold is given by coordinate charts equipped with local holomorphic coordinates (z1z2    zn).
Jean-Pierre Demailly (Grenoble I), 29/08/2006
Transcendental techniques in algebraic geometry
Complex manifolds / (pq)-forms
Goal : study thegeometric / topological / cohomological propertiesofcompactKa¨hlermanifolds A complexn-dimensional manifold is given by coordinate charts equipped with local holomorphic coordinates (z1z2    zn). A differential formuof type (pq) can be written as a sum u(z) =XuJK(z)dzJd zK |J|=p|K|=q
whereJ= (j1    jp),K= (k1    kq),
dzJ=dzj1  dzjpd zK=d zk1  d zkq
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) =ipqXTJK(z)dzJd zK |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) =ipqXTJK(z)dzJd zK
|J|=p|K|=q
The currentTis said to bepositiveif the distribution PλjλkTJKis a positive real measure for all (λJ)CN (so thatTKJ=TJK, henceT=T). The coefficientsTJKare thencomplex measures– and the diagonal onesTJJarepositive 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) =ipqXTJK(z)dzJd zK
|J|=p|K|=q
The currentTis said to bepositiveif the distribution PλjλkTJKis a positive real measure for all (λJ)CN (so thatTKJ=TJK, henceT=T). The coefficientsTJKare thencomplex measures– and the diagonal onesTJJarepositive 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
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