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Cited by 651Algebraic Geometry and Number TheoryGeometry and complex manifoldsAdvanced Algebra and Geometry

Géométrie algébrique et géométrie analytique

Open access
Serre, Jean-Pierre·1956·Annales de l'Institut Fourier
DOI 10.5802/aif.59
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Abstract

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.

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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

Crossref · 10.5802/aif.59

Cites

5 of 5 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

1 of 27 cited sources checked · 0 supported · 0 flagged · accuracy not yet measured

  • Findingreasoning

    The pair consisting of the local ring of regular functions and the local ring of holomorphic functions at a point is a flat couple.

    “COROLLAIRE 1. -Le couple £ anneaux (Q^ 3€^) est un couple plat”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Annexe, déf.4

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Findingreasoning

    For a commutative ring whose zero ideal is the intersection of finitely many distinct minimal primes, the localization away from the union of those primes is isomorphic to the direct product of the fraction fields of the quotients by those primes.

    “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,.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [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.

  • Findingreasoning

    The topology of the associated analytic space is finer than the Zariski topology of the algebraic variety.

    “La topologie de X^ est plus fine que la topologie de X”
    Quote in the paperSubstantial claim
  • Findingcitation

    For a closed analytic subspace of an analytic space, the sheaf of ideals of functions vanishing on it is a coherent analytic sheaf.

    “b) 5i Y est un sous-espace analytique fermé de X, le faisceau .^)(Y)est un faisceau analytique cohérent”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [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.

    • cites [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.

  • Findingreasoning

    If an algebraic variety is irreducible of dimension r, its associated analytic space has analytic dimension r at every point.

    “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”
    Quote in the paperSubstantial claim
  • May combine two claims. 7
  • The quote only partly states this claim. 4

  • May combine two claims.

    Interpretationcitation

    For n=1, GL_1(C)=C*; if X is normal, H^1(X,O*) is the group of divisor classes locally linearly equivalent to zero, and Proposition 18 means every analytic fiber space over X with structure group C* comes from such a divisor.

    “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.”
    Quote in the paperNot a substantial claim1 of 2 sources checked
    • cites [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.

    • cites [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. source

      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)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Findingreasoning

    The associated analytic space is locally compact and countable at infinity.

    “X^ est un espace localement compact dénombrable à Vinfini.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingreasoning

    For a projective algebraic variety over the complex numbers, coherent algebraic sheaves and coherent analytic sheaves are in one-to-one correspondence.

    “faisceaux algébriques cohérents et faisceaux analytiques cohérents se correspondent biunivoquement”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingreasoning

    For any algebraic sheaf F on X, the associated sheaf F^ is a sheaf of modules over the analytic structure sheaf, hence an analytic sheaf, and the inclusion O' into F^ defines a canonical homomorphism a from F' to F^.

    “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^.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For every coherent analytic sheaf on X^h, there exists a coherent algebraic sheaf on X whose associated analytic sheaf is isomorphic to it; moreover, this algebraic sheaf is unique up to isomorphism.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For any sheaf F on an algebraic variety X, the construction gives a sheaf F' on the associated analytic space X^ with the same stalks at every point, differing from F only in topology, and F' is the inverse image of F under the continuous map X^ to X.

    “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).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    Lemma 8: For a coherent analytic sheaf M on X=P^r(C), there exists an integer n such that M(n) is isomorphic to a quotient of a sheaf of the form F^, and M is isomorphic to a quotient of F^(-n).

    “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^.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning

    For a closed algebraic subvariety Y of X and a coherent algebraic sheaf S on Y, extending S by zero to X and then passing to the analytic space gives the same coherent analytic sheaf on X^ as passing to the analytic space on Y and then extending by zero to X^.

    “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.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [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.

  • May combine two claims.

    Interpretationreasoning

    The three theorems imply that the theory of coherent analytic sheaves on X^ essentially coincides with that of coherent algebraic sheaves on X, and this depends on X being projective; the statements fail even for affine varieties.

    “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.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Premisereasoning

    Condition (R) holds when G is the symplectic group S_{2m}(C), because the generic nondegenerate alternating form can be reduced to canonical form by a linear change of variables over C(U_ij).

    “c) lorsque G == Sj^(C), n = 2m; dans ce cas, l'espace homogène GL^(C)/G est l'espace des formes alternées non dégénérées”
    Quote in the paperSubstantial claim
10 of 10

  • 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)”
3 of 3

Cites

Flagged first, then by citations

Research works printed but not linked: 7

  1. [1] H. Cartan. Idéaux et modules de fonctions analytiques de variables complexes. Bull. Soc. Math. France, 78, 1950, pp. 29-64.
  2. [5] H. Cartan et J.-P. Serre. Un théorème de finitude concernant les variétés analytiques compactes. C. R., 237, 1953, pp. 128-130.
  3. [8] P. Dolbeault. Sur la cohomologie des variétés analytiques complexes. C. R., 236, 1953, pp. 175-177.
  4. [9] J. Frenkel. Cohomologie à valeurs dans un faisceau non abélien. C. R., 240, 1955, pp. 2368-2370.
  5. [11] K. Kodaira. On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties). Ann. of Maths., 60, 1954, pp. 28-48.
  6. [17] P. Samuel. Méthodes d'algèbre abstraite en géométrie algébrique. Ergebn. der Math., Springer, 1955.
  7. [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.

Cited by

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Related

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
DOI
10.5802/aif.59
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
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 › Mathematical Physics › Advanced Algebra and Geometry
Keywords
analytic spaces, coherent sheaves, projective varieties, cohomology groups, Lefschetz theorem

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