By James D. Lewis

ISBN-10: 0821805681

ISBN-13: 9780821805688

This ebook offers an advent to a subject matter of imperative curiosity in transcendental algebraic geometry: the Hodge conjecture. which include 15 lectures plus addenda and appendices, the quantity relies on a chain of lectures brought by means of Professor Lewis on the Centre de Recherches Mathematiques (CRM). The booklet is a self-contained presentation, thoroughly dedicated to the Hodge conjecture and similar themes. It contains many examples, and such a lot effects are thoroughly confirmed or sketched. the incentive in the back of a few of the effects and heritage fabric is supplied. This entire method of the ebook supplies it a ``user-friendly'' type. Readers needn't seek in other places for numerous effects. The ebook is appropriate to be used as a textual content for a subject matters path in algebraic geometry; comprises an appendix by way of B. Brent Gordon.

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Il est repr´esent´e par un espace rigide ´etale fini au-dessus de M. Soit U un espace rigide quasicompact. On notera (I, ρ, η) pour une section de MΛ,K sur U . Cela signifie que l’on se donne un mod`ele entier U de U ∼ U rig −−→ U puis une section (I, ρ) ∈ M(U) 32 Chapitre I. D´ecomposition cellulaire de la tour de Lubin-Tate qui d´efinit un ´el´ement s ∈ M(U ) et enfin une section η du produit fibr´e du diagramme suivant GM MΛ,K y s U que l’on notera parfois en ∼ η : Λ −−→ Tp (I) [K] une structure de niveau K sur la fibre g´en´erique de I.

3. 2. Si G est un groupe de Lubin-Tate et K ⊂ Id + π k End(Λ), alors sur XK le groupe G est muni d’une structure de niveau de Drinfeld, de niveau k. C’est une cons´equence du fait que l’espace classifiant des structures de Drinfeld de niveau k est fini au-dessus de X a mˆeme fibre g´en´erique que XK (en fibre g´en´erique toutes les d´efinitions des structures de niveau co¨ıncident) et est donc en-dessous du normalis´e. 3. , la donn´ee pour i entre 1 et n − 1 de nombres rationnels αi , 0 < αi < 1, tels que le polygone commen¸cant en (0, 1), passant par les (q i , αi ) et finissant en (q n , 0) soit convexe.

4). Les valuations des ´el´ements de [π]−1 Fx (ker f \ λ1 λi λ1 1 {0}) sont donc ( n , . . , n ). Or, n < λi , puisque λ1 ≤ , et l’´egalit´e q q q n(q − 1) 24 Chapitre I. D´ecomposition cellulaire de la tour de Lubin-Tate v(xi ) = 1 − i 1 implique que λi ≥ . Donc, i n n(q − q i−1 ) ∀β ∈ ker f \ {0}, ∀α ∈ [π]−1 Fx (ker f \ {0}), v(α) < v(β) On en d´eduit que v(f (α)) = q i v(α), qui sont donc les ( λ1 λi , . . , n−i ), lorsque α q n−i q varie. Or, q i λi+1 ≥ · · · ≥ q i λn > λ1 λi ≥ · · · ≥ n−i q n−i q On en d´eduit aussitˆot que x ∈ ∂n−i D.

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A survey of the Hodge conjecture by James D. Lewis


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