Géométrie algébrique et géométrie analytique

Serre, Jean-Pierre

doi:10.5802/aif.59

Serre, Jean-Pierre

Annales de l'Institut Fourier, Tome 6 (1956), pp. 1-42.

Résumé

Toute variété algébrique $X$ sur le corps des nombres complexes peut être munie, de façon canonique, d’une structure d’espace analytique ; tout faisceau algébrique cohérent sur $X$ détermine un faisceau analytique cohérent. Lorsque $X$ est une variété projective, nous montrons que, réciproquement, tout faisceau analytique cohérent sur $X$ peut être obtenu ainsi, et de façon unique ; de plus, cette correspondance préserve les groupes de cohomologie. Ces résultats contiennent comme cas particuliers des théorèmes classiques de Chow et Lefschetz, et permettent d’aborder la comparaison entre espaces fibrés algébriques et espaces fibrés analytiques de base une variété algébrique projective.

MR Zbl | 82 citations dans Numdam

DOI : 10.5802/aif.59

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     year = {1956},
     doi = {10.5802/aif.59},
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Serre, Jean-Pierre. Géométrie algébrique et géométrie analytique. Annales de l'Institut Fourier, Tome 6 (1956), pp. 1-42. doi : 10.5802/aif.59. https://www.numdam.org/articles/10.5802/aif.59/

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Cattaneo, Andrea; Fu, Lie Finiteness of Klein actions and real structures on compact hyperkähler manifolds, Mathematische Annalen, Volume 375 (2019) no. 3-4, p. 1783 | DOI:10.1007/s00208-019-01876-7
Taskinen, Jari; Vilonen, Kari Cartan Theorems for Stein Manifolds Over a Discrete Valuation Base, The Journal of Geometric Analysis, Volume 29 (2019) no. 1, p. 577 | DOI:10.1007/s12220-018-0012-8
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Harris, Michael; Zucker, Steven Boundary cohomology of Shimura varieties, Inventiones Mathematicae, Volume 116 (1994) no. 1, p. 243 | DOI:10.1007/bf01231562
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Beretta, Lucia Quelques propriétés des applications analytiques près d'un homéomorphisme, ANNALI DELL UNIVERSITA DI FERRARA, Volume 30 (1984) no. 1, p. 41 | DOI:10.1007/bf02853268
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Sasakura, Nobuo A type of comparison theorem in polynomial growth cohomology, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 59 (1983) no. 7 | DOI:10.3792/pjaa.59.297
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Borel, Armand Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, Journal of Differential Geometry, Volume 6 (1972) no. 4 | DOI:10.4310/jdg/1214430642
Griffiths, Phillip A. Function theory of finite order on algebraic varieties. I(B), Journal of Differential Geometry, Volume 7 (1972) no. 1-2 | DOI:10.4310/jdg/1214430819
Hartshorne, Robin Algebraic De Rham cohomology, Manuscripta Mathematica, Volume 7 (1972) no. 2, p. 125 | DOI:10.1007/bf01679709
Gerritzen, L. On non-archimedean representations of abelian varieties, Mathematische Annalen, Volume 196 (1972) no. 4, p. 323 | DOI:10.1007/bf01428221
Herrera, M.; Lieberman, D. Duality and the deRham cohomology of infinitesimal neighborhoods, Inventiones Mathematicae, Volume 13 (1971) no. 1-2, p. 97 | DOI:10.1007/bf01390096
Mather, John N. Stable map-germs and algebraic geometry, Manifolds — Amsterdam 1970, Volume 197 (1971), p. 176 | DOI:10.1007/bfb0068617
Golovin, V. D. Cohomologies and analytic differential forms, Mathematical Notes of the Academy of Sciences of the USSR, Volume 9 (1971) no. 5, p. 330 | DOI:10.1007/bf01094361
Hurt, Norman E. A classification theory for quantizable dynamical systems, Reports on Mathematical Physics, Volume 2 (1971) no. 3, p. 211 | DOI:10.1016/0034-4877(71)90005-x
Raynaud, Michèle Geometrie Algebrique Et Geometrie Analytique, Revêtements Etales et Groupe Fondamental, Volume 224 (1971), p. 311 | DOI:10.1007/bfb0058667
Brieskorn, Egbert Die monodromie der isolierten singularit�ten von hyperfl�chen, Manuscripta Mathematica, Volume 2 (1970) no. 2, p. 103 | DOI:10.1007/bf01155695
Siu, Yum-tong Analytic sheaves of local cohomology, Transactions of the American Mathematical Society, Volume 148 (1970) no. 2, p. 347 | DOI:10.1090/s0002-9947-1970-0257403-3
Wiegmann, Klaus-Werner Strukturen auf Quotienten komplexer Räume, Commentarii Mathematici Helvetici, Volume 44 (1969) no. 1, p. 93 | DOI:10.1007/bf02564516
Palamodov, V. P. Differential operators on the class of convergent power series and the Weierstrass auxiliary lemma, Functional Analysis and Its Applications, Volume 2 (1969) no. 3, p. 235 | DOI:10.1007/bf01076125
Bloom, Thomas; Herrera, Miguel De Rham cohomology of an analytic space, Inventiones Mathematicae, Volume 7 (1969) no. 4, p. 275 | DOI:10.1007/bf01425536
Richardson, Jr., R. W. Deformations of subalgebras of Lie algebras, Journal of Differential Geometry, Volume 3 (1969) no. 3-4 | DOI:10.4310/jdg/1214429055
Brieskorn, Egbert Rationale Singularit�ten komplexer Fl�chen, Inventiones Mathematicae, Volume 4 (1968) no. 5, p. 336 | DOI:10.1007/bf01425318
Forster, Otto; Ramspott, Karl Josef Homotopieklassen von Idealbasen in Steinschen Algebren, Inventiones Mathematicae, Volume 5 (1968) no. 4, p. 255 | DOI:10.1007/bf01389776
Deligne, P. Théorème de Lefschetz et Critères de Dégénérescence de Suites Spectrales, Publications mathématiques de l'IHÉS, Volume 35 (1968) no. 1, p. 107 | DOI:10.1007/bf02698925
Newstead, P.E. Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology, Volume 7 (1968) no. 3, p. 205 | DOI:10.1016/0040-9383(68)90001-3
Fischer, Gerd Lineare Faserräume und kohärente Modulgarben über komplexen Räumen, Archiv der Mathematik, Volume 18 (1967) no. 6, p. 609 | DOI:10.1007/bf01898870
Mumford, David Abelian quotients of the Teichmüller modular group, Journal d'Analyse Mathématique, Volume 18 (1967) no. 1, p. 227 | DOI:10.1007/bf02798046
Igusa, Jun-Ichi A desingularization problem in the theory of Siegel modular functions, Mathematische Annalen, Volume 168 (1967) no. 1, p. 228 | DOI:10.1007/bf01361555
Gottschling, Erhard Die Uniformisierbarkeit der Fixpunkte eigentlich diskontinuierlicher Gruppen von biholomorphen Abbildungen, Mathematische Annalen, Volume 169 (1967) no. 1, p. 26 | DOI:10.1007/bf01399530
Wiegmann, Klaus-Werner �ber Quotienten holomorph-konvexer komplexer R�ume, Mathematische Zeitschrift, Volume 97 (1967) no. 4, p. 251 | DOI:10.1007/bf01112168
Forster, Otto Zur Theorie der Steinschen Algebren und Moduln, Mathematische Zeitschrift, Volume 97 (1967) no. 5, p. 376 | DOI:10.1007/bf01112815
Wiegmann, Klaus-Werner Einbettungen komplexer R�ume in Zahlenr�ume, Inventiones Mathematicae, Volume 1 (1966) no. 3, p. 229 | DOI:10.1007/bf01452243
Forster, Otto; Ramspott, Karl Josef Okasche Paare von Garben nicht-abelscher Gruppen, Inventiones Mathematicae, Volume 1 (1966) no. 3, p. 260 | DOI:10.1007/bf01452245
Grothendieck, A. Un theoreme sur les homomorphismes de schemas abeliens, Inventiones Mathematicae, Volume 2 (1966) no. 1, p. 59 | DOI:10.1007/bf01403390
Forster, Otto; Ramspott, Karl Josef Analytische Modulgarben und Endromisb�ndel, Inventiones Mathematicae, Volume 2 (1966) no. 2, p. 145 | DOI:10.1007/bf01404550
Grothendieck, A. On the de rham cohomology of algebraic varieties, Publications mathématiques de l'IHÉS, Volume 29 (1966) no. 1, p. 95 | DOI:10.1007/bf02684807
Ramspott, Karl Josef Stetige und holomorphe Schnitte in Bündeln mit homogener Faser, Mathematische Zeitschrift, Volume 89 (1965) no. 3, p. 234 | DOI:10.1007/bf02116867
Bibliography, Volume 14 (1964), p. 471 | DOI:10.1016/s0079-8169(08)61566-4
Forster, Otto Prim�rzerlegung in Steinschen Algebren, Mathematische Annalen, Volume 154 (1964) no. 4, p. 307 | DOI:10.1007/bf01362567
Wolffhardt, Klaus Existenzs�tze f�r maximale holomorphe und meromorphe Abbildungen, Mathematische Zeitschrift, Volume 85 (1964) no. 4, p. 328 | DOI:10.1007/bf01110679
Forster, Otto Funktionswerte als Randintegrale in komplexen R�umen, Mathematische Annalen, Volume 150 (1963) no. 4, p. 317 | DOI:10.1007/bf01470760
Kuhlmann, Norbert �ber die Aufl�sung der Singularit�ten 3-dimensionaler komplexer R�ume I, Mathematische Annalen, Volume 151 (1963) no. 4, p. 304 | DOI:10.1007/bf01470823
Forster, Otto Banach-Algebren stetiger Funktionen auf kompakten R�umen, Mathematische Zeitschrift, Volume 81 (1963) no. 1, p. 1 | DOI:10.1007/bf01111516
Röhrl, Helmut Holomorphic fiber bundles over Riemann surfaces, Bulletin of the American Mathematical Society, Volume 68 (1962) no. 3, p. 125 | DOI:10.1090/s0002-9904-1962-10715-0
Kuhlmann, Norbert �ber die normalen Punkte eines komplexen Raumes, Mathematische Annalen, Volume 146 (1962) no. 5, p. 397 | DOI:10.1007/bf01470653
Atiyah, M.F.; Hirzebruch, F. The Riemann–Roch theorem for analytic embeddings, Topology, Volume 1 (1962) no. 2, p. 151 | DOI:10.1016/0040-9383(65)90023-6
Kuhlmann, Norbert Die Normalisierung komplexer R�ume, Mathematische Annalen, Volume 144 (1961) no. 2, p. 110 | DOI:10.1007/bf01451331
Lech, Christer Note on multiplicities of ideals, Arkiv för Matematik, Volume 4 (1960) no. 1, p. 63 | DOI:10.1007/bf02591323
Teleman, Costake Sur les structures fibrées osculatrices d'une surface deRiemann, Commentarii Mathematici Helvetici, Volume 34 (1960) no. 1, p. 175 | DOI:10.1007/bf02565934
Kuhlmann, Norbert Projektive Modifikationen komplexer R�ume, Mathematische Annalen, Volume 139 (1960) no. 3, p. 217 | DOI:10.1007/bf01352913
Remmert, Reinhold; Stein, Karl Eigentliche holomorphe Abbildungen, Mathematische Zeitschrift, Volume 73 (1960) no. 2, p. 159 | DOI:10.1007/bf01162476
Ishikawa, Takeshi A theorem on flat couples, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 36 (1960) no. 7 | DOI:10.3792/pja/1195523970
Grauert, Hans; Remmert, Reinhold Komplexe R�ume, Mathematische Annalen, Volume 136 (1958) no. 3, p. 245 | DOI:10.1007/bf01362011
Nastold, Hans-Joachim Über meromorphe Schnitte komplex-analytischer Vektorraumbündel und Anwendungen auf Riemannsche Klassen. II, Mathematische Zeitschrift, Volume 70 (1958) no. 1, p. 55 | DOI:10.1007/bf01558577
Grothendieck, Alexander Sur quelques points d'algèbre homologique, II, Tohoku Mathematical Journal, Volume 9 (1957) no. 3 | DOI:10.2748/tmj/1178244774

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