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A vanishing theorem on arithmetic surfaces. (English) Zbl 0834.14013
Soient \(\pi : X \to S\) une surface complexe, normale, plate sur une courbe projective \(S\), \(L\) un fibré de rang 1, génériquement de degré positif sur \(X\). Alors \(R^0 \pi_* (L^{-1}) = 0\) et l’annulation de \(H^1 (X, L^{- 1})\) est équivalente à l’annulation de \(H^0 (S,R^1 \pi_* (L^{- 1}))\). Un théorème d’annulation de Kodaira-Ramanujan nous dit alors que ces groupes s’annulent quant \(L\) est “big” et “n.e.f.” (i.e. numériquement effectif). L’objet du présent papier est de prouver un analogue de ce théorème d’annulation pour les surfaces arithmétiques.
Soit \(X\) un schéma intègre de dimension 2 plat et projectif sur le spectre \(S\) de l’anneau des entiers d’un corps de nombres \(K\) et tel que la fibre générique \(X_K\) soit lisse et géométriquement irréductible, l’A. montre que si \(\overline L= (L,h^L)\) est un fibré linéaire hermitien tel que la courbure de \(\overline L\) soit un multiple positif de la forme de Kähler \(\mu\) normalisée et le degré de la restriction de \(L\) a une composante d’une fibre de \(X\) sur \(S\) non négatif, alors le logarithme de la \(L^2\)-norme d’un élément \(e\) de torsion dans \(H^1 (X, L^{-1)}\) est borné inferieurement par un multiple de la selfintersection de \(I\). Plus précisement, si \(m\) est défini par \(c_1(I) = m \cdot \mu\), \(e(I) = \inf_P h_I (P)\) (pour une certaine hauteur normalisée \(h_I\) borneé inférieurement), \(P =\) point algébrique sur \(X_K\), \(\widehat c_1 (L)^2 = \widehat c_1 (f^* (L))^2\), \(f : \widetilde X \to X\) avec \(\widetilde X\) régulier t.q. \(f\) induise un isomorphisme sur les fibres génériques \((\widehat c_1 (L)\) est indépendant de \(f)\), alors \[ \widehat c_1 (I)^2 + (m^2 - 2m) e(I) \leq [K:\mathbb{Q}]m^2 \bigl( \log |e |+ 1 \bigr) \] \((\widehat c_1 =\) première classe de Chern arithmetique).
Reviewer: J.C.Douai (Lille)

MSC:
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14J25 Special surfaces
14F17 Vanishing theorems in algebraic geometry
32L20 Vanishing theorems
14C20 Divisors, linear systems, invertible sheaves
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