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Algebraic Surfaces by Arnaud Beauville (auth.), G. Tomassini (eds.)

By Arnaud Beauville (auth.), G. Tomassini (eds.)

Lectures: A. Beauville: Surfaces algébriques complexes.- F.A. Bogomolov: the idea of invariants and its functions to a few difficulties within the algebraic geometry.- E. Bombieri: equipment of algebraic geometry in Char. P and their applications.- Seminars: F. Catanese: Pluricanonical mappings of surfaces with okay² =1,2, q=pg=0.- F. Catanese: On a category of surfaces of basic type.- I. Dolgacev: Algebraic surfaces with p=pg =0.- A. Tognoli: a few feedback in regards to the "Nullstellensatz".

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7) m already when -- It is, however, the f a i l e r e of 2 which is responsible for the fzilzlre of Kodaira's vanishing theorem i n char 3 0. Now we analyze i n inore d e t a i l t h e s i t u a t i o n i n case t h e c r u c i a l cohomology group t o study i s D E l ~ f s, i s a s e c t i o n of has 6een used and t h e second t i n e , allowing d i v i s i o n t o define the splitting. dim X char = 0 dim X = 2. Here I f there i s a divisor H'(x,L-I). 12) 1 dim8k e r ( ~(X, LOX) We a r e l e d J t o Problem 1.

Hodgepolarisee de poids 2 sur r, dl08 une application T + G O ( r C ) . de Hodge permet de prouver que cette application est analyticye; on dit que c'est~19"applicationdes pdriodes" de la faille de surfaces X -+ T. est justifide par la raison suivante. La structure:de. H2,~5 HO(X,~~)et H2(x,e),au anal de B~(x,F), la flsche i : XO(X,Q~) -t H (x,c)* correspondant & 1' inclusion est donnde per 2 2 = w pour w E HO(X,S? ), Y E H~(X,Z). Si lton fixe m e base (wJ ' . de,Ii0(~,$) Y 4 et une base (yi) de H2 (x,z), la struktuke de Hodge est ddtemiinbe par l a "matrice des pdriodes" (jl;%) - .

Le cas des surfaces K3 8 'et'e r s s o l n r&emment par Chafarevitch e t Piatechki/ Chapiro ( [ ~ h - P I ) ; on va donner i c i une indication de l e u r d6monstration. s) & l ' a r t i c l e CCh-PI. me de T o r e l l i pour l e s surfaces ab'eliennes. La s t r u c t u r e de Hodge de poids un d'une t e l l e surface A dktermine sa variktk de Picard, c ' e s t a d i r e 1 8 vari6t'e abglienne duale l ' o n r6cupSre A come v a r i k t 6 abklienne duale de 2. x; il e s t bien connu m e On a kgalement un th6orS- me de T o r e l l i pour l e s s t r u c t u r e s de Hodge de poids 2 : ?

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