Algebraic geometry V. Fano varieties by A.N. Parshin, I.R. Shafarevich, Yu.G. Prokhorov, S. Tregub,

By A.N. Parshin, I.R. Shafarevich, Yu.G. Prokhorov, S. Tregub, V.A. Iskovskikh

The purpose of this survey, written by means of V.A. Iskovskikh and Yu.G. Prokhorov, is to supply an exposition of the constitution conception of Fano kinds, i.e. algebraic vareties with an plentiful anticanonical divisor. Such forms obviously seem within the birational class of types of damaging Kodaira measurement, and they're very just about rational ones. This EMS quantity covers various techniques to the category of Fano forms equivalent to the classical Fano-Iskovskikh "double projection" process and its ameliorations, the vector bundles process because of S. Mukai, and the strategy of extremal rays. The authors speak about uniruledness and rational connectedness in addition to contemporary development in rationality difficulties of Fano kinds. The appendix includes tables of a few periods of Fano types. This publication may be very priceless as a reference and examine advisor for researchers and graduate scholars in algebraic geometry.

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Il est ´egalement construit sans recours `a la structure de F-cristal mais en utilisant la rigidit´e des quasi-isog´enies dans [22] (cf. 15 de [22]). 2) est un isomorphisme de F ⊗Qp F˘ -modules, via l’action de OF sur H et H. Consid´erons la d´ecomposition isotypique D(H)Q = D(H)Q,τ τ :F 0 →W (F q )Q Le morphisme ι : W (Fq )Q → F˘ induit un plongement τ0 : F 0 → W (Fq )Q . Celui-ci ∼ induit un isomorphisme O ⊗OF 0 ,τ W (Fq ) −−→ WO (Fq ) (WO d´esigne les vecteurs 0 de Witt ramifi´es, cf. [9]). Notons alors DO (H)Q = D(H)Q,τ0 o` u O est l`a pour OF .

1 Hypoth`eses et notations Soit F |Qp une extension de degr´e fini, d’uniformisante π et de corps r´esiduel k = Fq = OF /πOF . On note F˘ = F nr le compl´et´e de l’extension maximale nonramifi´ee de F dans une clˆ oture alg´ebrique de celui-ci et F 0 l’extension maximale ˘ pour O ˘ . On fixe non-ramifi´ee de Qp dans F . On note parfois O pour OF et O F un isomorphisme entre le corps r´esiduel de OF˘ et Fq une clˆoture alg´ebrique de Fq . Si Z est un sch´ema formel d’id´eal de d´efinition I, par d´efinition, la cat´egorie des groupes p-divisibles sur Z est la cat´egorie limite projective lim ←− k≥1 Groupes p-divisibles sur Spec(OZ /Ik ) de la cat´egorie fibr´ee des groupes p-divisibles sur le syst`eme de sch´emas (Z mod Ik )k≥1 .

N−1}. On en d´eduit les formules de r´ecurrence suivantes: (k+1) Si b = 0 f0 (k+1) ∀i > 0 fi (k) = f0 k (k) (k) = fi + xqi f0 Si b = 0, en posant x0 = 1 et ∀i ∈ Z, xi = xj , o` u j ≡ i mod n, j ∈ {0, . . , n − 1} (k+1) fb (k+1) ∀i = b fi o` u ⎧ ⎨ −1 0 α(b, i) = ⎩ −1 (k) = fb k (k) (k) = fi + π α(b,i) xqn−b+i fb si si si i=0 1≤i≤b−1 b+1≤i≤n−1 On peut alors retrouver les formules donn´ees dans [26] pour l’application des p´eriodes. 1. Posons D = {x ∈ Xrig | ∀1 ≤ i ≤ n − 1, v(xi ) ≥ 1 − i } n un ouvert admissible quasi-compact dans la boule unit´e ouverte de dimension n−1.

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