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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
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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.
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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.
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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
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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.
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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
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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 :
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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
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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.
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- [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.
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Bemerkungen �ber eigentliche holomorphe Abbildungen
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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
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