Introduction a la theorie de Hodge by Jose Bertin, Jean-Pierre Demailly, Luc Illusie, Chris Peters

By Jose Bertin, Jean-Pierre Demailly, Luc Illusie, Chris Peters

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By Jose Bertin, Jean-Pierre Demailly, Luc Illusie, Chris Peters

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Il suffit donc de prouver que d) implique c). Comme du = 0, nous avons d′ u = d′′ u = 0, et comme u est suppos´ee orthogonale `a Hp,q (X, C), le th. 1 ⋆ ). Le th´eor`eme analogue au th. 1 pour implique u = d′′ s, s ∈ C ∞ (X, Λp,q−1 TX ′ d (qui s’en d´eduit d’ailleurs par conjugaison) montre qu’on a s = h + d′ v + d′⋆ w, ⋆ ⋆ avec h ∈ Hp,q−1 (X, C), v ∈ C ∞ (X, Λp−1,q−1 TX ) et w ∈ C ∞ (X, Λp+1,q−1 TX ). 16. Comme d′ u = 0, la composante d′⋆ d′′ w orthogonale `a Ker d′ doit ˆetre nulle. 6 nous d´eduisons le corollaire suivant, qui `a son tour implique que la d´ecomposition de Hodge ne d´epend pas de la m´etrique k¨ahl´erienne choisie.

On associe d’abord `a K •,• le complexe total (K • , d) tel que K l = p+q=l K p,q , muni de la diff´erentielle totale d = d′ + d′′ . 1) F pK l = K j,l−j . 2) F p H l (K • ) := Im H l (F p K • ) → H l (K • ) , et on note Gp H l (K • ) = F p H l (K • )/F p+1 H l (K • ) le module gradu´e associ´e. La th´eorie des suites spectrales (voir par exemple [God57]) nous dit qu’il existe une suite de complexes doubles Er•,• , r 1, munis de diff´erentielles dr : Erp,q → Erp+r,q−r+1 de bidegr´e (r, −r + 1), telle que Er+1 = H • (Er ) se calcule par r´ecurrence comme la cohomologie du complexe (Er•,• , dr ), et telle que la limite p,q E∞ = limr→+∞ Erp,q s’identifie avec le module gradu´e G• H • (K • ), de fa¸con pr´ecise p,q E∞ = Gp H p+q (K • ).

N /ζ0 ) les coordonn´ees non homog`enes de la carte Cn ⊂ Pn . Un calcul montre que ω= i i ′ ′′ d d log(1 + |z|2 ) = Θ(O(1)), 2π 2π ω n = 1. Pn Comme les seuls groupes de cohomologie enti`ere non nuls de Pn sont H 2p (P, Z) ≃ Z pour 0 p n, nous voyons que h = {ω} ∈ H 2 (Pn , Z) est un g´en´erateur de l’anneau de cohomologie H • (Pn , Z). En d’autres termes, H • (Pn , Z) ≃ Z[h]/(hn+1 ) comme anneau. 5. Exemple. Un tore complexe est un quotient X = Cn /Γ de Cn par un r´eseau Γ de rang 2n. C’est donc une vari´et´e complexe compacte.

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