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Quelques Points D'int Rt ATELIER INTERNATIONAL SUR LA THORIE ALGBRIQUE et ANALYTIQUE DES RSIDUS ET SES APPLICATIONS PARIS IHP du au MAI La thorie des rsidus est un domaine qui connat une rcente et intressante activit Plusieurs group es de travail se sont p enchs sur ce thme durant cette anne Cet atelier est donc lo ccasion dune prsentation confrontation et synthse des ides sur le sujet aussi bien dun p oint de vue thorique que pratique Quelques p oints dintrt Unication des direntes appro ches algbriques sur les rsidus Connexion entre les appro ches analytiques et algbriques Complexes rsiduels et leur constructions Thormes de BrianonSkoda Calculs eectifs de sommes de rsidus Applications des rsidus la thorie de llimination aux formules de reprsentations aux rsolutions de systmes p olynomiaux Orateurs invits Reinhold HBL Craig HUNEKE Ernst KUNZ Joseph LIPMAN Organisateurs Marc CHARDIN Centre de Mathmatiques Ecole p olytechnique F Palaiseau France Mohamed ELKADI Dept de Mathmatiques Univ de Nice SophiaAntipolis Parc Valrose Nice France Bernard MOURRAIN INRIA Pro jet SAFIR route des Lucioles F Valbonne France Les organisateurs ont t aids dans leur travail par les participants de deux sminaires informels sur ce sujet lun Nice Formes Formules et lautre Paris et aussi par David EISENBUD et JeanPierre JOUANOLOU Cet atelier a reu le soutien du Lab oratoire GAGE cole p olytechnique Palaiseau et du pro jet SAFIR Nice SophiaAntipolis travers le programme HCM SAC INTERNATIONAL WORKSHOP ON ALGEBRAIC AND ANALYTIC THEORY OF RESIDUES AND ITS APPLICATIONS PARIS IHP MAY Residues theory is a eld which has known recent and interesting activities Several working groups have fo cus on this theme during the last years The workshop is the o ccasion of a presentation confrontation and synthesis of ideas on this sub jects as well from a theoretical p oint of view as for its applications Some p oints of interest Unication of the dierent algebraic approaches on residues Residual complexes and their construction Connection b etween the analytic and algebraic approaches BrianonSkoda theorems Eective computation of sums of residues Application of Residue Theory to elimination theory representation formulas and p oly nomial systems solving Invited sp eakers Reinhold HBL Craig HUNEKE Ernst KUNZ Joseph LIPMAN Organizers Marc CHARDIN Centre de Mathmatiques Ecole p olytechnique F Palaiseau France Mohamed ELKADI Dept de Mathmatiques Univ de Nice SophiaAntipolis Parc Valrose Nice France Bernard MOURRAIN INRIA Pro jet SAFIR route des Lucioles F Valbonne France The organizers were and are help ed in their task by the participants of two informal seminars on this sub ject one in Nice and one in Paris and by David EISENBUD and Jean Pierre JOUANOLOU This workshop is supp orted by the Lab oratoire GAGE Ecole p olytechnique Palaiseau the pro ject SAFIR Nice SophiaAntipolis via the HCM programm SAC PROGRAMME PROGRAM JEUDI THURSDAY MATIN MORNING h h Amnon YEKUTIELI Adeles and the De RhamResidue Complex h h Uwe STORCH Resultants for homogeneous regular sequences APRESMIDI AFTERNOON h h Carlos BERENSTEIN On the residue formula of Jacobi and its applications h h Alain YGER Integral representation formulas and multidimensional residues h h Alicia DICKENSTEIN Residues in Toric Varieties h h MarieFrancoise ROY Quadratic forms and Bezoutians VENDREDI FRIDAY MATIN MORNING h h Craig HUNEKE Tight closure and theorems of BrianconSkoda type h h Joseph LIPMAN Formal Duality Fundamental Class and the Residue Theorem APRESMIDI AFTERNOON h h IChiau HUANG An explicit construction of residual complexes h h Leovigildo ALONSOTARRIO Formal Completion and Duality h h Mikael PASSARE Courants rsiduels et faisceaux dualisants h h Teresa KRICK Elimination by arithmetic circuits The duality to ol SAMEDI SATURDAY MATIN MORNING h h Ernst KUNZ Generalization of a theorem of Chasles h h Reinhold HBL Generalization of a theorem of Waring APRESMIDI AFTERNOON h h Ab dellah AL AMRANI Fibrs pro jectifs tordus et classes de Chern h h Salomon OFMAN Application de lalgorithme de GelfandLerayShilov aux courantsrsidus h h Djilali BOUDIAF Interprtation des rsidus comp oss laide des courants Interprtation des rsidus comp oss laide des courants rsiduels Djilali Boudiaf Rsum Soient X une varit analytique complexe de dimension n et F fY Y g p n p e une famille ordonne dhypersurfaces complexes de X de runion Y et dintersection Y telle que chaque sousfamille F fY Y g i p soit en p osition dintersection complte i i e cestdire dim F n i o F Y Y On dsigne par E Y resp D C i i i X X D le complexe des faisceaux des germes des formes semimromorphes sur X p les Y e sur Y resp des courants sur X supp orts sur Y et par X D lensemble des sections Y X globales de D supp ort dans Y Suivant HerreraLieb erman p our le cas p on X montre que le diagramme de cohomologie suivant est commutatif e I Y q q e e H X E Y H X nY C X p p r es R F F y y q p q p H X C H X D Y X Y Y e o I Y est lisomorphisme de Grothendieck est un isomorphisme comp os de Y lisomorphisme de Poly q p H X D H X D H Y C Y nq p Y X nq p X q p X C et de H Y C H nq p Y inverse de lisomorphisme de dualit de Poincar X obtenu par cappro duit par la classe p fondamentale de X est lhomomorphisme rsiduel dni dans partir de lop rateur R F p q p q p e est lhomomorphisme rsidu et enn r es Y D E de ColeHerrera R F Y X F compos dni par Poly partir de travaux de Sorani Cest la gnralisation de lhomomorphisme de Leray et Norguet en eet si les hypersurfaces Y sont lisses et j p e en position gnrale alors lhomomorphisme r es asso cie toute classe de cohomologie de F p e e X n Y la classe de cohomologie prsidu compos de LerayNorguet note r es Si est F p e une forme semimromorphe p les simples sur Y alors la forme prsidu compos r es F est dnie et on a la relation p p p I Y r es i R F F o le premier membre est le courant rsiduel asso ci et I Y est le courant dintgration de Lelong sur Y Rfrences Boudiaf D Sur les courants rsiduels Thse Universit Paris VI Coleff N et Herrera M Les courants rsiduels asso cis une forme mromorphe Lecture Notes in Math Springer Verlag Coleff N Herrera M et Lieberman D Algebraic cycles as residues of meromorphic forms Math Ann Grothendieck A On the de Rham cohomology of algebraic varieties Inst Hautes tudes Sci Publ Math Herrera M et Lieberman D Residues and principal values on complex spaces Math Ann Leray J Le calcul direntiel et intgral sur une varit analytique complexe Problme de Cauchy I I I Bull So c Math France Norguet F Introduction la thorie cohomologique des rsidus Sm P Lelong Anal yse Anne Lecture Notes in Math SpringerVerlag Poly JB Formules des rsidus et intersection des chanes sousanalytiques Thses Universit de Poitiers Poly JB Morphismes de MayerVietoris et rsidus comp oss paratre Sorani G Sui residui delle forme diferenziali di una variet analitica complessa Rend Mat Appl V Ser Authors Address Djilali Boudiaf Universit de Poitiers Mathmatiques Avenue du Recteur Pineau POITIERS Residues in Toric Varieties Eduardo Cattani David Cox and Alicia Dickenstein Toric residues provide a to ol for the study of certain homogeneous ideals of the homogeneous co ordinate ring of a toric variety such as those app earing in the description of the Ho dge structure of their hypersurfaces BC They were introduced in C where some of their prop erties were describ ed in the sp ecial case when all of the divisors involved were linearly equivalent The main results of the present work are an extension of the Isomorphism Theorem of C to the case of nonequivalent ample divisors a global transformation law for toric residues and a theorem expressing the toric residue as a sum of lo cal Grothendieck residues Assume that X is a complete simplicial toric variety of dimension n S denotes the homogeneous co ordinate ring of X which is the p olynomial ring S Cx x Here nr the variables x corresp ond to the generators of the dimensional cones in the fan which i determines X and hence to torusinvariant irreducible divisors D of X S is graded by i P d nr nr i declaring that the monomial x has degree d D in the Chow group A X i i n i i i C P nr We let degx A X denote the anticanonical class on X Then given i n i for i n we dene their critical degree to b e homogeneous p olynomials F S i i P n A X Each H S determines a meromorphic nform H i n F i H on X BC F F n If in addition the F dont vanish simultaneously on X then relative to the op en cover i n n U fx X F x g of X this gives a ech class H H X The toric i i F X residue Res S hF F i C F n is given by the formula Res H Tr H F F n n where Tr H X C is the trace map X Our rst main result is the following Global Transformation Law for i n Suppose and G S Theorem Let F S i i i i n X G A F j ij i i where A is homogeneous of degree and assume the G dont vanish simultaneously ij j i i on X Let be the critical degree for F F Then for each H S H detA is of n ij the critical degree for G G and n Res H Res H detA F G ij The pro of uses a ech co chain argument One application of this transformation law is that in certain cases we can describ e explicit elements of S with nonzero residue For this purp ose assume X is complete and its fan contains a ndimensional simplicial cone Then denote the variables of
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