The DOI matches its Crossref record; 5 of the 5 references it cites were found in a registry or an index; 1 of the 27 sources its claims cite were checked for support.
Géométrie algébrique et géométrie analytique
Open accessAbstract
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.
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
Serre proves that for a projective complex algebraic variety, coherent algebraic sheaves and coherent analytic sheaves correspond bijectively and have isomorphic cohomology. This GAGA equivalence yields algebraic-versus-analytic comparison results, including Chow’s theorem and algebraicity criteria for certain principal bundles.
Key findings
- For projective \(X\) over \(\mathbb C\) and any coherent algebraic sheaf \(\mathcal F\), the natural map \(H^q(X,\mathcal F)\to H^q(X^{\mathrm{an}},\mathcal F^{\mathrm{an}})\) is bijective for every \(q\ge 0\) (Theorem 1). Theorems 2 and 3 add that analytic morphisms between associated sheaves come from algebraic morphisms, and every coherent analytic sheaf comes from a coherent algebraic sheaf, uniquely up to isomorphism.
- Applications include invariance of Betti numbers of nonsingular varieties under automorphisms of \(\mathbb C\) (Proposition 12), algebraicity of compact analytic subsets of algebraic varieties (Proposition 14), regularity of holomorphic maps from compact algebraic varieties to algebraic varieties (Proposition 15), and uniqueness of an algebraic structure on a compact analytic space (corollary).
- For principal bundles, the algebraic-to-analytic comparison map is injective for compact \(X\) (Proposition 16) and bijective for the additive group \(\mathbb C\) (Proposition 17), for \(\mathrm{GL}_n(\mathbb C)\) (Proposition 18), and for groups satisfying a rational-section condition (R), including solvable groups, \(\mathrm{SL}_n(\mathbb C)\), and \(\mathrm{Sp}_{2m}(\mathbb C)\) (Proposition 20 examples).
- Locally, the algebraic local ring and the ring of holomorphic germs at a point have the same completion and form a flat couple; consequently they have the same dimension (Proposition 3, Corollaries 1–2).
Methods
- The paper is a mathematical proof study: it defines analytic spaces following H. Cartan, constructs the analytic space \(X^{\mathrm{an}}\) associated to a complex algebraic variety, and compares local rings and sheaves on \(X\) and \(X^{\mathrm{an}}\).
- The main arguments use coherent algebraic sheaf theory, Cartan’s Theorems A and B, exact sequences of sheaves, the five lemma, induction on cohomological degree and dimension, and Dolbeault theory for Betti-number applications.
- There is no empirical sample or dataset; the analysis is local and global sheaf cohomology together with commutative algebra of local rings, completions, and flat couples.
Limitations
- The author states that the fiber-space applications are incomplete: among semisimple groups, only the unimodular linear group and the symplectic group are treated, and applications to automorphic functions are left aside.
- The author notes uncertainty about whether \(X\) and its transform \(X^\sigma\) are always homeomorphic; a genus-1 curve example shows they are not always analytically isomorphic. The rational-section condition (R) fails for \(\mathrm{O}_n(\mathbb C)\) when \(n\ge 3\), and bijectivity in that case is unknown.
- Evident from the text: the main equivalence theorems require projectivity and fail even for affine varieties; several applications also assume nonsingularity or compactness.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
1 of 27 cited sources checked · 0 supported · 0 flagged · accuracy not yet measured
In the paper
“COROLLAIRE 1. -Le couple £ anneaux (Q^ 3€^) est un couple plat”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
What the sources say
- Not matchedcites Annexe, déf.4
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
In the paper
“LEMME 3. -Soit A un anneau commutatif, dans lequel Vidéal 0 soit intersection Sun nombre fini d^idéaux premiers minimaux distincts p;; soit K^ le corps des fractions de A/p^, et soit S V ensemble des éléments de A qui ^appartiennent à aucun des pc U anneau de fractions As est alors isomorphe au composé direct des K,.”
§ ANNALES DE L'INSTITUT FOURIER > As c As.
What the sources say
- Not matchedcites [15] [15] P. Samuel. Commutative Algebra (Notes by D. Herzig). Cornell Univ., 1953.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“La topologie de X^ est plus fine que la topologie de X”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesNone of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“b) 5i Y est un sous-espace analytique fermé de X, le faisceau .^)(Y)est un faisceau analytique cohérent”
§ ANNALES DE L'INSTITUT FOURIER > 3. Faisceaux analytiques.
What the sources say
- Not matchedcites [1] [1] H. Cartan. Idéaux et modules de fonctions analytiques de variables complexes. Bull. Soc. Math. France, 78, 1950, pp. 29-64.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [2] [2] H. Cartan. Séminaire E. N. S., 1951-1952.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“COROLLAIRE 3. -Si X est une variété algébrique irréductible de dimension r, l'espace analytique X^ est de dimension analytique r en chacun de ses points”
§ ANNALES DE L'INSTITUT FOURIER > COROLLAIRE 2. -Les anneaux Oy; et Mp ont même dimension.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“COROLLAIRE 2. -Les anneaux Oy; et Mp ont même dimension.”
§ ANNALES DE L'INSTITUT FOURIER > COROLLAIRE 2. -Les anneaux Oy; et Mp ont même dimension.
What the sources say
- Not matchedcites [16] [16] P. Samuel. Algèbre locale. Mém. Sci. Math., 123, Paris, 1953.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“la correspondance entre ces deux catégories de faisceaux laisse invariants les groupes de cohomologie”
§ ANNALES DE L'INSTITUT FOURIER > INTRODUCTION
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“II existe sur X une structure d'espace analytique et une seule telle que, pour toute carte ç : V ->-U, Fensemble Z-owert V soit ouvert, et y soit un isomorphisme analytique de V”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Rests on an assumption: A polynomial is continuous for the usual topology and holomorphic.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“PROPOSITION 5. -Soient X une variété algébrique^ et U une partie de X. Si U est ï-owerte et ï-dense dans X, alors U est dense dans X.”
§ ANNALES DE L'INSTITUT FOURIER > COROLLAIRE 2. -Les anneaux Oy; et Mp ont même dimension.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesNone of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“PROPOSITION 6. ^-Pour qu'une variété algébrique X soit complète, il faut et il suffit quelle soit compacte.”
§ ANNALES DE L'INSTITUT FOURIER > PROPOSITION 6. ^-Pour qu'une variété algébrique X soit complète, il faut et il suffit quelle soit compacte.
What the sources say
- Not matchedcites [7] [7] W-L. Chow. On the projective embedding of homogeneous varieties. Lefschetz's volume, Princeton, 1956.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [19] [19] J.-P. Serre. Sur la cohomologie des variétés algébriques. J. de Maths. Pures et Appl., 35, 1956.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Rests on an assumption: Every projective algebraic variety is compact in the usual topology.
- Not matched
- It may state two separate claims as one. 7
- The quote only partly states this claim. 4
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“Pour n = 1, GL^(C) coïncide avec le groupe multiplicatif C*; si l'on suppose que X est une variété normale, le groupe H 1 (X, fî) coïncide avec le groupe des classes de diviseurs localement linéairement équivalents à zéro (cf.[20], § 3) et la proposition 18 signifie que tout espace fibre analytique de base X et de groupe structural C* provient d'un tel diviseur.”
§ ANNALES DE L'INSTITUT FOURIER > 20. Espaces fibres algébriques et espaces fibres analytiques.
What the sources say
- Not matchedcites [20] [20] A. Weil. Fibre-spaces in algebraic geometry (Notes by A. Wallace). Chicago Univ., 1952.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Partly supportedcites [12] [12] K. Kodaira and D. C. Spencer. Divisor class groups on algebraic varieties. Proc. Nat. Acad. Sci. U. S. A., 39, 1953, pp. 872-877. sourceExperimental — accuracy not yet measured
In the opening discussion and Theorem 1 (pages not supplied), the paper identifies divisor classes with complex line bundles and proves that every such bundle comes from a divisor. Its stated setting is a non-singular projective algebraic variety, however; it does not establish the cited identification of H¹(X,O*) with those divisor classes for an arbitrary normal X. The statement about Proposition 18 is the citing author's own interpretation, not a result attributed to this source. (LLM, conf=0.84)
Tests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“X^ est un espace localement compact dénombrable à Vinfini.”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“faisceaux algébriques cohérents et faisceaux analytiques cohérents se correspondent biunivoquement”
§ ANNALES DE L'INSTITUT FOURIER > INTRODUCTION
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Le faisceau 3^ est un faisceau de 3-6-modules, c'est-à-dire un faisceau analytique', l'injection 0'-^ 3^ définit un homomorphisme canonique a : 9' ->-9^.”
§ ANNALES DE L'INSTITUT FOURIER > As c As.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“THÉORÈME 3 . -Pour tout faisceau analytique cohérent jll» sur X 71 , il existe un faisceau algébrique cohérent ^ sur X tel que ^ soit isomorphe à JIL.De plus, cette propriété détermine 9 de façon unique, à un isomorphisme près.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Pour tout a;çX, on a donc 3^ == 3^; les faisceaux 9 et y ne diffèrent que par leur topologie (9' n'est pas autre chose que V image réciproque de 9 par l'application continue X^X).”
§ ANNALES DE L'INSTITUT FOURIER > As c As.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“En vertu du lemme 8, il existe un entier n tel que M)(n) soit isomorphe à un faisceau quotient d'un faisceau 3^, et M) est donc isomorphe à un quotient de Wo (-n^.”
§ ANNALES DE L'INSTITUT FOURIER > 16. Démonstration du théorème 3. Les faisceaux ll(n).
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“Les deux faisceaux en question sont nuls en dehors de Y^; il nous suffira donc de montrer que leurs restrictions à Y 71 sont isomorphes.”
§ ANNALES DE L'INSTITUT FOURIER > 10. Prolongement d'un faisceau.
What the sources say
- Not matchedcites [18] [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe quote only partly states this claim.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Ces trois théorèmes signifient que la théorie des faisceaux analytiques cohérents sur X^ coïncide essentiellement avec celle des faisceaux algébriques cohérents sur X. Bien entendu, ils tiennent à ce que X est une variété projectile, et sont inexacts même pour une variété affine.”
§ ANNALES DE L'INSTITUT FOURIER > On a alors :
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“c) lorsque G == Sj^(C), n = 2m; dans ce cas, l'espace homogène GL^(C)/G est l'espace des formes alternées non dégénérées”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe quote only partly states this claim.
The paper relies on this without support.
Every projective algebraic variety is compact in the usual topology.
“comme toute variété projective est compacte pour la topologie usuelle”
The paper relies on this without support.
The fraction field of the local ring of an irreducible component of an algebraic variety at a point is the field of rational functions on that component.
“le corps des quotients de A;, soit K(, n'est donc pas autre chose que le corps des fonctions rationnelles sur la variété irréductible X^.”
The paper relies on this without support.
A polynomial is continuous for the usual topology and holomorphic.
“comme un polynôme est continu pour la topologie usuelle (resp.holomorphe)”
Citations
Cites
Some Basic Theorems on Algebraic Groups
Rosenlicht, M. (1956) · American Journal of Mathematics · doi:10.2307/2372523 · Cited by 495
On Compact Complex Analytic Varieties
Chow, W.-L. (1949) · American Journal of Mathematics · doi:10.2307/2372375 · Cited by 132
Divisor class groups on algebraic varieties
Kodaira, K., & Spencer, D. C. (1953) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.39.8.872 · Cited by 39
Zum Eliminationsproblem der Potenzreihenideale
Rückert, W. (1932) · Springer eBooks · doi:10.1007/978-3-662-41142-1_1 · Cited by 23
Fibre-spaces in algebraic geometry
WEIL, A. A. (1979) · Springer eBooks · doi:10.1007/978-1-4757-1705-1_51 · Cited by 9
Flagged first, then by citations
Research works printed but not linked: 7
- [1] H. Cartan. Idéaux et modules de fonctions analytiques de variables complexes. Bull. Soc. Math. France, 78, 1950, pp. 29-64.
- [5] H. Cartan et J.-P. Serre. Un théorème de finitude concernant les variétés analytiques compactes. C. R., 237, 1953, pp. 128-130.
- [8] P. Dolbeault. Sur la cohomologie des variétés analytiques complexes. C. R., 236, 1953, pp. 175-177.
- [9] J. Frenkel. Cohomologie à valeurs dans un faisceau non abélien. C. R., 240, 1955, pp. 2368-2370.
- [11] K. Kodaira. On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties). Ann. of Maths., 60, 1954, pp. 28-48.
- [17] P. Samuel. Méthodes d'algèbre abstraite en géométrie algébrique. Ergebn. der Math., Springer, 1955.
- [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Cited by
Sorted by citations
The Arithmetic of Elliptic Curves
Silverman, J. H. (1986) · Graduate Texts in Mathematics · doi:10.1007/978-1-4757-1920-8 · Cited by 4,044
Grothendieck, A. (1964) · Publications mathématiques de l IHÉS · doi:10.1007/bf02684747 · Cited by 2,994
Deligne, P. (1974) · Publications mathématiques de l IHÉS · doi:10.1007/bf02685881 · Cited by 1,201
Space of Unitary Vector Bundles on a Compact Riemann Surface
Seshadri, C. S. (1967) · Annals of Mathematics · doi:10.2307/1970444 · Cited by 872
Variation of hodge structure: The singularities of the period mapping
Schmid, W. (1973) · Inventiones Mathematicae · doi:10.1007/bf01389674 · Cited by 817
On the de Rham cohomology of algebraic varieties
Grothendieck, A. (1966) · Publications mathématiques de l IHÉS · doi:10.1007/bf02684807 · Cited by 615
Instantons and algebraic geometry
Atiyah, M. F., & Ward, R. S. (1977) · Communications in Mathematical Physics · doi:10.1007/bf01626514 · Cited by 478
Noncommutative Geometry, Quantum Fields and Motives
Connes, A., & Marcolli, M. (2007) · Colloquium Publications - American Mathematical Society/Colloquium Publications · doi:10.1090/coll/055 · Cited by 504
Serre, J.-P. (2011) · A K Peters/CRC Press eBooks · doi:10.1201/b10588 · Cited by 367
Self-citationA theory of generalized Donaldson–Thomas invariants
Joyce, D., & Song, Y. (2011) · Memoirs of the American Mathematical Society · doi:10.1090/s0065-9266-2011-00630-1 · Cited by 396
Rationale Singularit�ten komplexer Fl�chen
Brieskorn, E. (1968) · Inventiones mathematicae · doi:10.1007/bf01425318 · Cited by 325
Differential Galois Theory and Non-Integrability of Hamiltonian Systems
Ruiz, J. J. M. (1999) · Modern Birkhäuser classics · doi:10.1007/978-3-0348-0723-4 · Cited by 311
Die monodromie der isolierten singularit�ten von hyperfl�chen
Brieskorn, E. (1970) · manuscripta mathematica · doi:10.1007/bf01155695 · Cited by 302
Théorème de Lefschetz et critères de dégénérescence de suites spectrales
Deligne, P. (1968) · Publications mathématiques de l IHÉS · doi:10.1007/bf02698925 · Cited by 290
On the de Rham cohomology of algebraic varieties
Hartshorne, R. (1975) · Publications mathématiques de l IHÉS · doi:10.1007/bf02684298 · Cited by 274
Algebraic Cohomology and Deformation Theory
Gerstenhaber, M., & Schack, S. D. (1988) · Springer eBooks · doi:10.1007/978-94-009-3057-5_2 · Cited by 269
Lacunas for hyperbolic differential operators with constant coefficients I
Atiyah, M. F., Bott, R., & Gårding, L. (1970) · Acta Mathematica · doi:10.1007/bf02394570 · Cited by 259
Analytic Methods in Algebraic Geometry
Demailly, J.-P. (2007) · Cited by 251
Algebraization of Formal Moduli: II. Existence of Modifications
Artin, M. (1970) · Annals of Mathematics · doi:10.2307/1970602 · Cited by 242
Lifting smooth homotopies of orbit spaces
Schwarz, G. W. (1980) · Publications mathématiques de l IHÉS · doi:10.1007/bf02684776 · Cited by 224
Hodge Cycles on Abelian Varieties
Deligne, P. (1982) · Lecture notes in mathematics · doi:10.1007/978-3-540-38955-2_3 · Cited by 244
Introduction to Singularities and Deformations
Greuel, G., Lossen, C., & Shustin, E. (2025) · Springer monographs in mathematics · doi:10.1007/978-3-031-86043-0 · Cited by 213
Galoisian obstructions to integrability of Hamiltonian systems
Morales-Ruiz, J. J., & Ramis, J. P. (2001) · Methods and Applications of Analysis · doi:10.4310/maa.2001.v8.n1.a3 · Cited by 200
Grauert, H., & Remmert, R. (1958) · Mathematische Annalen · doi:10.1007/bf01362011 · Cited by 211
Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem
Borel, A. (1972) · Journal of Differential Geometry · doi:10.4310/jdg/1214430642 · Cited by 197
Algebraic solutions of differential equations (p-curvature and the Hodge filtration)
Katz, N. M. (1972) · Inventiones mathematicae · doi:10.1007/bf01389714 · Cited by 205
Poonen, B. (2017) · Graduate studies in mathematics · doi:10.1090/gsm/186 · Cited by 186
Montaldi, J. (2021) · Cambridge University Press eBooks · doi:10.1017/9781316585085.009 · Cited by 183
Local Properties of Analytic Varieties
Whitney, H. (1992) · Birkhäuser Boston eBooks · doi:10.1007/978-1-4612-2972-8_33 · Cited by 176
Lectures on Logarithmic Algebraic Geometry
Ogus, A. (2018) · Cambridge University Press eBooks · doi:10.1017/9781316941614 · Cited by 191
Integrability of Hamiltonian systems and differential Galois groups of higher variational equations
Morales-Ruiz, J. J., Ramis, J.-P., & Simó, C. (2007) · Annales Scientifiques de l École Normale Supérieure · doi:10.1016/j.ansens.2007.09.002 · Cited by 161
Differential operators on homogeneous spaces. III
Borho, W., & Brylinski, J.-L. (1985) · Inventiones mathematicae · doi:10.1007/bf01388547 · Cited by 156
Period Mappings and Period Domains
Carlson, J. A., Müller-Stach, S. J., & Peters, C. A. M. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316995846 · Cited by 154
The invariant holonomic system on a semisimple Lie algebra
Hotta, R., & Kashiwara, M. (1984) · Inventiones mathematicae · doi:10.1007/bf01388568 · Cited by 128
Arbarello, E., & Cornalba, M. (1981) · Commentarii Mathematici Helvetici · doi:10.1007/bf02566195 · Cited by 117
L2 vanishing theorems for positive line bundles and adjunction theory
Demailly, J.-P. (1996) · Lecture notes in mathematics · doi:10.1007/bfb0094302 · Cited by 125
The Tamagawa number conjecture of adjoint motives of modular forms
Diamond, F., Flach, M., & Guo, L. (2004) · Annales Scientifiques de l École Normale Supérieure · doi:10.1016/j.ansens.2004.09.001 · Cited by 125
Euler characteristics of Teichmüller curves in genus two
Bainbridge, M. (2007) · Geometry & Topology · doi:10.2140/gt.2007.11.1887 · Cited by 96
Analytic cycles and vector bundles on non-compact algebraic varieties
Cornalba, M., & Griffiths, P. (1975) · Inventiones mathematicae · doi:10.1007/bf01389905 · Cited by 102
A desingularization problem in the theory of Siegel modular functions
Igusa, J.-I. (1967) · Mathematische Annalen · doi:10.1007/bf01361555 · Cited by 103
Some aspects of the Hodge conjecture
Voisin, C. (2007) · Japanese journal of mathematics · doi:10.1007/s11537-007-0639-x · Cited by 90
Algebraicity of holomorphic mappings between real algebraic sets in Cn
Baouendi, M. S., Ebenfelt, P., & Rothschild, L. P. (1996) · Acta Mathematica · doi:10.1007/bf02392622 · Cited by 101
Zur Theorie der Steinschen Algebren und Moduln
Förster, O. (1967) · Mathematische Zeitschrift · doi:10.1007/bf01112815 · Cited by 97
Multiloop amplitudes in the theory of quantum strings and complex geometry
Knizhnik, V. G. (1989) · Soviet Physics Uspekhi · doi:10.1070/pu1989v032n11abeh002775 · Cited by 96
Stable bundles of rank 2 and odd degree over a curve of genus 2
Newstead, P. E. (1968) · Topology · doi:10.1016/0040-9383(68)90001-3 · Cited by 96
Galois theory on the line in nonzero characteristic
Abhyankar, S. S. (1992) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-1992-00270-7 · Cited by 98
Multiplier ideal sheaves and analytic methods in algebraic geometry
Demailly, J.-P. (2000) · Cited by 96
Quantizations of conical symplectic resolutions I: local and global structurePreprint
Braden, T., Proudfoot, N. J., & Webster, B. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1208.3863 · Cited by 96
Montaldi, J. A. (2021) · doi:10.1017/9781316585085.025 · Cited by 95
Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
Buczyński, J., Ginensky, A., & Landsberg, J. M. (2013) · Journal of the London Mathematical Society · doi:10.1112/jlms/jds073 · Cited by 88
Diophantine Approximation and Nevanlinna Theory
Vojta, P. (2010) · Lecture notes in mathematics · doi:10.1007/978-3-642-15945-9_3 · Cited by 87
The Poincaré Lemma for Variations of Polarized Hodge Structure
Kashiwara, M., & Kawai, T. (1987) · Publications of the Research Institute for Mathematical Sciences · doi:10.2977/prims/1195176545 · Cited by 85
Lacunas for hyperbolic differential operators with constant coefficients. II
Atiyah, M. F., Bott, R., & Gårding, L. (1973) · Acta Mathematica · doi:10.1007/bf02392039 · Cited by 84
Compact four-dimensional self-dual EINSTEIN manifolds with positive scalar curvature
Friedrich, T., & Kurke, H. (1982) · Mathematische Nachrichten · doi:10.1002/mana.19821060124 · Cited by 85
Factorisation of generalised theta functions. I
Narasimhan, M. S., & Ramadas, T. R. (1993) · Inventiones mathematicae · doi:10.1007/bf01232680 · Cited by 78
Motivic Eilenberg-MacLane spaces
Voevodsky, V. (2010) · Publications mathématiques de l IHÉS · doi:10.1007/s10240-010-0024-9 · Cited by 79
“— See [22].”
De Rham cohomology of an analytic space
Bloom, T., & Herrera, M. E. (1969) · Inventiones mathematicae · doi:10.1007/bf01425536 · Cited by 78
The Riemann–Roch theorem for analytic embeddings
Atiyah, M. F., & Hirzebruch, F. E. P. (1962) · Topology · doi:10.1016/0040-9383(65)90023-6 · Cited by 66
Stable vector bundles and instantons
Hartshorne, R. (1978) · Communications in Mathematical Physics · doi:10.1007/bf01614151 · Cited by 69
Noncommutative geometry and motives: The thermodynamics of endomotives
Alain Connes, A., Caterina Consani, B., & Marcolli, M. (2007) · Advances in Mathematics · doi:10.1016/j.aim.2007.03.006 · Cited by 62
The Topology of Quotient Varieties
Neeman, A. (1985) · Annals of Mathematics · doi:10.2307/1971309 · Cited by 64
Abelian quotients of the Teichmüller modular group
Mumford, D. B. (1967) · Journal d Analyse Mathématique · doi:10.1007/bf02798046 · Cited by 63
Generalized intermediate Jacobians and the theorem on normal functions
Zucker, S. W. (1976) · Inventiones mathematicae · doi:10.1007/bf01404203 · Cited by 63
Galois coverings of the arithmetic line
Harbater, D. (1987) · Lecture notes in mathematics · doi:10.1007/bfb0072980 · Cited by 59
Tame topology of arithmetic quotients and algebraicity of Hodge loci
Bakker, B., Klingler, B., & Tsimerman, J. (2020) · Journal of the American Mathematical Society · doi:10.1090/jams/952 · Cited by 58
Eigentliche holomorphe Abbildungen
Remmert, R., & Stein, K. (1960) · Mathematische Zeitschrift · doi:10.1007/bf01162476 · Cited by 57
Note on multiplicities of ideals
Lech, C. (1960) · Arkiv för matematik · doi:10.1007/bf02591323 · Cited by 54
Manin, Y. I. (1976) · Journal of Mathematical Sciences · doi:10.1007/bf01083779 · Cited by 54
Dualité de Spanier–Whitehead en géométrie algébrique
Riou, J. (2005) · Comptes Rendus Mathématique · doi:10.1016/j.crma.2005.02.002 · Cited by 53
“4, se ramène à vérifier que les deux théories coh giques considérées coïncident sur les variétés projectiveslisses, ce qui résulte ici de [15].”
Higher analytic stacks and GAGA theorems
Porta, M., & Yu, T. Y. (2016) · Advances in Mathematics · doi:10.1016/j.aim.2016.07.017 · Cited by 49
Deformations of subalgebras of Lie algebras
Richardson, R. W. (1969) · Journal of Differential Geometry · doi:10.4310/jdg/1214429055 · Cited by 45
Sur quelques points d'algèbre homologique, II
Grothendieck, A. (1957) · Tohoku Mathematical Journal · doi:10.2748/tmj/1178244774 · Cited by 51
Harbater, D., & Hartmann, J. (2010) · Israel Journal of Mathematics · doi:10.1007/s11856-010-0021-1 · Cited by 52
“1), Serre’s GAGA [29], and Cartan’s Lemma on factoring matrices [2].”
BackgroundCohomologie étale: les points de départ
Deligne, P., & Boutot, J.-F. (1977) · Lecture notes in mathematics · doi:10.1007/bfb0091518 · Cited by 51
Grothendieck’s existence theorem in formal geometry
Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S. L., Nitsure, N., & Vistoli, A. (2006) · Mathematical surveys and monographs · doi:10.1090/surv/123/08 · Cited by 49
Scala, L. (2009) · Duke Mathematical Journal · doi:10.1215/00127094-2009-050 · Cited by 46
o-minimal GAGA and a conjecture of Griffiths
Bakker, B., Brunebarbe, Y., & Tsimerman, J. (2022) · Inventiones mathematicae · doi:10.1007/s00222-022-01166-1 · Cited by 70
“Note that X is not required to be proper over C, but in contrast to Serre’s classical GAGA theorem [35] (as well as most other GAGA-type theorems for proper algebraic spaces), it is not true that every definable coherent sheaf is algebraic (see Example 3.”
BackgroundThe elliptic Weyl character formula
Ganter, N. (2014) · Compositio Mathematica · doi:10.1112/s0010437x1300777x · Cited by 45
Hazewinkel, M., & Martin, C. F. (1982) · Journal of Pure and Applied Algebra · doi:10.1016/0022-4049(82)90037-8 · Cited by 45
Okasche Paare von Garben nicht-abelscher Gruppen
Förster, O., & Ramspott, K. J. (1966) · Inventiones mathematicae · doi:10.1007/bf01452245 · Cited by 44
On the passage from local to global in number theory
Mazur, B. (1993) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-1993-00414-2 · Cited by 42
Analytische Modulgarben und Endromisb�ndel
Förster, O., & Ramspott, K. J. (1966) · Inventiones mathematicae · doi:10.1007/bf01404550 · Cited by 44
Holomorphic flexibility properties of complex manifolds
Forstnerič, F. (2006) · American Journal of Mathematics · doi:10.1353/ajm.2006.0005 · Cited by 49
Diophantine problems and p-adic period mappings
Lawrence, B., & Venkatesh, A. (2020) · Inventiones mathematicae · doi:10.1007/s00222-020-00966-7 · Cited by 45
Differentiable and Deformation Type of Algebraic Surfaces, Real and Symplectic Structures
Catanese, F. (2008) · Lecture notes in mathematics · doi:10.1007/978-3-540-78279-7_2 · Cited by 45
Equivariant elliptic cohomology and rigidity
Roşu, I. (2001) · American Journal of Mathematics · doi:10.1353/ajm.2001.0027 · Cited by 42
Seminormal rings and weakly normal varieties
Leahy, J. V., & Vitulli, M. A. (1981) · Nagoya Mathematical Journal · doi:10.1017/s0027763000019279 · Cited by 41
Complex Algebraic Varieties: Periods of Integrals and Hodge Structures
Kulikov, V. S., & Kurchanov, P. F. (1998) · Encyclopaedia of mathematical sciences · doi:10.1007/978-3-662-03662-4_1 · Cited by 37
p -adic uniformization of unitary Shimura varieties
Varshavsky, Y. (1998) · Publications mathématiques de l IHÉS · doi:10.1007/bf02698861 · Cited by 42
An Overview of Recent Advances in Hodge Theory
Brylinski, J.-L., & Zucker, S. W. (1998) · Complex Manifolds · doi:10.1007/978-3-642-61299-2_2 · Cited by 42
Arens–Michael envelopes, homological epimorphisms, and relatively quasi-free algebras
Pirkovskii, A. Y. (2008) · Transactions of the Moscow Mathematical Society · doi:10.1090/s0077-1554-08-00169-6 · Cited by 41
Mebkhout, Z. (1989) · Publications mathématiques de l IHÉS · doi:10.1007/bf02698840 · Cited by 40
Meredith, D. (1972) · Nagoya Mathematical Journal · doi:10.1017/s0027763000014641 · Cited by 40
Ramified primes in the field of moduli of branched coverings of curves
Beckmann, S. (1989) · Journal of Algebra · doi:10.1016/0021-8693(89)90303-7 · Cited by 40
L'algèbre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle, II
Ayoub, J. (2013) · Journal für die reine und angewandte Mathematik (Crelles Journal) · doi:10.1515/crelle-2012-0090 · Cited by 39
“[43] J.”
Non-commutative complex differential geometry
Beggs, E. J., & Smith, S. P. (2013) · Journal of Geometry and Physics · doi:10.1016/j.geomphys.2013.03.018 · Cited by 38
Ducros, A. (2018) · Astérisque · doi:10.24033/ast.1050 · Cited by 38
Un theoreme sur les homomorphismes de schemas abeliens
Grothendieck, A. (1966) · Inventiones mathematicae · doi:10.1007/bf01403390 · Cited by 36
On the algebraicity of the zero locus of an admissible normal function
Brosnan, P., & Pearlstein, G. (2013) · Compositio Mathematica · doi:10.1112/s0010437x1300729x · Cited by 38
Holomorphic fiber bundles over Riemann surfaces
Röhrl, H. (1962) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1962-10715-0 · Cited by 37
Metadata
- Authors
- Serre, J.-P.
- Corresponding author
- Serre, Jean-Pierre
- ORCID
- Serre, Jean-Pierre
- Venue
- Annales de l'Institut Fourier
- Publisher
- MathDoc/Centre Mersenne
- Published
- 1956
- Volume
- 6
- Pages
- 1-42
- ISSN
- 0373-0956 (ISSN-L)
- 1777-5310
- OpenAlex
- W2092940386
- Cited by
- 645
- References
- 4
- Citations by year
- 2026: 6 · 2025: 12 · 2024: 15 · 2023: 22 · 2022: 19 · 2021: 62 · 2020: 28 · 2019: 26 · 2018: 27 · 2017: 20 · 2016: 16 · 2015: 9 · 2014: 15 · 2013: 10 · 2012: 12
- FWCI
- 6.67
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Annales de l’institut Fourier (publishedVersion)
- Annals of the Fourier Institute (Institut Fourier) (submittedVersion)
- French digital mathematics library (Numdam) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Algebraic Geometry and Number Theory
- Physical Sciences › Mathematics › Geometry and Topology › Geometry and complex manifolds
- Physical Sciences › Mathematics › Algebra and Number Theory › Advanced Algebra and Geometry
- Keywords
- analytic spaces, coherent sheaves, projective varieties, cohomology groups, Lefschetz theorem
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-09-28, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-09-28 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
