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By Jean-Pierre Demailly

This quantity is a ramification of lectures given via the writer on the Park urban arithmetic Institute (Utah) in 2008, and on different events. the aim of this quantity is to explain analytic thoughts important within the examine of questions referring to linear sequence, multiplier beliefs, and vanishing theorems for algebraic vector bundles. the writer goals to be concise in his exposition, assuming that the reader is already a bit of conversant in the fundamental innovations of sheaf concept, homological algebra, and intricate differential geometry. within the ultimate chapters, a few very fresh questions and open difficulties are addressed--such as effects relating to the finiteness of the canonical ring and the abundance conjecture, and effects describing the geometric constitution of Kahler types and their optimistic cones.

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1. — Supposons que tp : Xp → Yp soit un isomorphisme pour p ≤ n. Alors, pour tout Faisceau F de SF , et at induit un isomorphisme ∼ τ

5. — Supposons que pour tout n, les fl`eches Xn+1 → cosqn (X)n+1 soient de S-descente cohomologique universelle. Alors, X → S est de descente cohomologique universelle. DESCENTE COHOMOLOGIQUE 33 D´emonstration. 2, les fl`eches cosqn+1 (X)p → cosqn (X)p sont des ´equivalences de descente cohomologique pour tout p. 1.

On peut supposer p > n + 1. On ´ecrit alors cosqn+1 (X)p = lim Xq ←− [q]→[p] q≤n+1 comme le noyau de la double fl`eche ΠX = d´ ef Xq ⇒ Xi = ΞX α [q]→[p] q≤n+1 [i] →[j] [p] j≤n+1 o` u la composante αX d’indice α ∈ Hom[p] ([i], [j]) de la double fl`eche est la double fl`eche form´ee d’une part du morphisme ΠX → Xi de projection d’indice [i] → [p] et, d’autre part, du morphisme ΠX → Xj → Xi , compos´e de la projection d’indice [j] → [p] et de α ∈ Hom(Xj , Xi ). 3. — Soit p un entier naturel. Le diagramme / ΠX cosqn+1 (X)p  ˜ p cosqn+1 (X) /  ˜ ΠX est cart´esien.

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