# Algebraic geometry and analytic geometry

Contents 1 Main statement 2 Background 3 Important results 3.1 Riemann's existence theorem 3.2 The Lefschetz principle 3.3 Chow's theorem 3.4 GAGA 3.5 Formal statement of GAGA 4 Notes 5 References 6 External links Main statement Let X be a projective complex algebraic variety. Because X is a complex variety, its set of complex points X(C) can be given the structure of a compact complex analytic space. This analytic space is denoted Xan. Similarly, if {displaystyle {mathcal {F}}} is a sheaf on X, then there is a corresponding sheaf {displaystyle {mathcal {F}}^{text{an}}} on Xan. This association of an analytic object to an algebraic one is a functor. The prototypical theorem relating X and Xan says that for any two coherent sheaves {displaystyle {mathcal {F}}} and {displaystyle {mathcal {G}}} on X, the natural homomorphism: {displaystyle {text{Hom}}_{{mathcal {O}}_{X}}({mathcal {F}},{mathcal {G}})rightarrow {text{Hom}}_{{mathcal {O}}_{X}^{text{an}}}({mathcal {F}}^{text{an}},{mathcal {G}}^{text{an}})} is an isomorphism. Here {displaystyle {mathcal {O}}_{X}} is the structure sheaf of the algebraic variety X and {displaystyle {mathcal {O}}_{X}^{text{an}}} is the structure sheaf of the analytic variety Xan. In other words, the category of coherent sheaves on the algebraic variety X[1] is equivalent to the category of analytic coherent sheaves on the analytic variety Xan, and the equivalence is given on objects by mapping {displaystyle {mathcal {F}}} to {displaystyle {mathcal {F}}^{text{an}}} . (Note in particular that {displaystyle {mathcal {O}}_{X}^{text{an}}} itself is coherent, a result known as the Oka coherence theorem.)[2] Another important statement is as follows: For any coherent sheaf {displaystyle {mathcal {F}}} on an algebraic variety X the homomorphisms {displaystyle varepsilon _{q} : H^{q}(X,{mathcal {F}})rightarrow H^{q}(X^{an},{mathcal {F}}^{an})} are isomorphisms for all q's. This means that the q-th cohomology group on X is isomorphic to the cohomology group on Xan.

The theorem applies much more generally than stated above (see the formal statement below). It and its proof have many consequences, such as Chow's theorem, the Lefschetz principle and Kodaira vanishing theorem.

Background Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way.

For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem). For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant. Thus f is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere.

Important results There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century. Some of the more important advances are listed here in chronological order.

Riemann's existence theorem Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem[3][4] a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves—that is, such coverings all come from finite extensions of the function field.

The Lefschetz principle In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0, by treating K as if it were the complex number field. An elementary form of it asserts that true statements of the first order theory of fields about C are true for any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski and are based in mathematical logic.[5][6] This principle permits the carrying over of some results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0.

Chow's theorem Chow (1949), proved by Wei-Liang Chow, is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective space that is closed (in the ordinary topological sense) is an algebraic subvariety.[7] This can be rephrased as "any analytic subspace of complex projective space which is closed in the strong topology is closed in the Zariski topology." This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry.

GAGA Foundations for the many relations between the two theories were put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique Serre (1956) by Jean-Pierre Serre, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves.

Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings.

Formal statement of GAGA Let {displaystyle (X,{mathcal {O}}_{X})} be a scheme of finite type over C. Then there is a topological space Xan which as a set consists of the closed points of X with a continuous inclusion map λX: Xan → X. The topology on Xan is called the "complex topology" (and is very different from the subspace topology). Suppose φ: X → Y is a morphism of schemes of locally finite type over C. Then there exists a continuous map φan: Xan → Yan such λY ° φan = φ ° λX. There is a sheaf {displaystyle {mathcal {O}}_{X}^{mathrm {an} }} on Xan such that {displaystyle (X^{mathrm {an} },{mathcal {O}}_{X}^{mathrm {an} })} is a ringed space and λX: Xan → X becomes a map of ringed spaces. The space {displaystyle (X^{mathrm {an} },{mathcal {O}}_{X}^{mathrm {an} })} is called the "analytification" of {displaystyle (X,{mathcal {O}}_{X})} and is an analytic space. For every φ: X → Y the map φan defined above is a mapping of analytic spaces. Furthermore, the map φ ↦ φan maps open immersions into open immersions. If X = Spec(C[x1,...,xn]) then Xan = Cn and {displaystyle {mathcal {O}}_{X}^{mathrm {an} }(U)} for every polydisc U is a suitable quotient of the space of holomorphic functions on U. For every sheaf {displaystyle {mathcal {F}}} on X (called algebraic sheaf) there is a sheaf {displaystyle {mathcal {F}}^{mathrm {an} }} on Xan (called analytic sheaf) and a map of sheaves of {displaystyle {mathcal {O}}_{X}} -modules {displaystyle lambda _{X}^{*}:{mathcal {F}}rightarrow (lambda _{X})_{*}{mathcal {F}}^{mathrm {an} }} . The sheaf {displaystyle {mathcal {F}}^{mathrm {an} }} is defined as {displaystyle lambda _{X}^{-1}{mathcal {F}}otimes _{lambda _{X}^{-1}{mathcal {O}}_{X}}{mathcal {O}}_{X}^{mathrm {an} }} . The correspondence {displaystyle {mathcal {F}}mapsto {mathcal {F}}^{mathrm {an} }} defines an exact functor from the category of sheaves over {displaystyle (X,{mathcal {O}}_{X})} to the category of sheaves of {displaystyle (X^{mathrm {an} },{mathcal {O}}_{X}^{mathrm {an} })} . The following two statements are the heart of Serre's GAGA theorem (as extended by Alexander Grothendieck,[4] Amnon Neeman,[8] and others.) If f: X → Y is an arbitrary morphism of schemes of finite type over C and {displaystyle {mathcal {F}}} is coherent then the natural map {displaystyle (f_{*}{mathcal {F}})^{mathrm {an} }rightarrow f_{*}^{mathrm {an} }{mathcal {F}}^{mathrm {an} }} is injective. If f is proper then this map is an isomorphism. One also has isomorphisms of all higher direct image sheaves {displaystyle (R^{i}f_{*}{mathcal {F}})^{mathrm {an} }cong R^{i}f_{*}^{mathrm {an} }{mathcal {F}}^{mathrm {an} }} in this case. Now assume that Xan is Hausdorff and compact. If {displaystyle {mathcal {F}},{mathcal {G}}} are two coherent algebraic sheaves on {displaystyle (X,{mathcal {O}}_{X})} and if {displaystyle fcolon {mathcal {F}}^{mathrm {an} }rightarrow {mathcal {G}}^{mathrm {an} }} is a map of sheaves of {displaystyle {mathcal {O}}_{X}^{mathrm {an} }} -modules then there exists a unique map of sheaves of {displaystyle {mathcal {O}}_{X}} -modules {displaystyle varphi :{mathcal {F}}rightarrow {mathcal {G}}} with {displaystyle f=varphi ^{mathrm {an} }} . If {displaystyle {mathcal {R}}} is a coherent analytic sheaf of {displaystyle {mathcal {O}}_{X}^{mathrm {an} }} -modules over Xan then there exists a coherent algebraic sheaf {displaystyle {mathcal {F}}} of {displaystyle {mathcal {O}}_{X}} -modules and an isomorphism {displaystyle {mathcal {F}}^{mathrm {an} }cong {mathcal {R}}} .

In slightly lesser generality, the GAGA theorem asserts that the category of coherent algebraic sheaves on a complex projective variety X and the category of coherent analytic sheaves on the corresponding analytic space Xan are equivalent. The analytic space Xan is obtained roughly by pulling back to X the complex structure from Cn through the coordinate charts. Indeed, phrasing the theorem in this manner is closer in spirit to Serre's paper, seeing how the full scheme-theoretic language that the above formal statement uses heavily had not yet been invented by the time of GAGA's publication.

Notes ^ The notion of coherent was first studied in the complex manifolds and the complex analytic varieties that are the subject of analytic geometry, and then introduced by FAC (Serre (1955)) into the algebraic varieties that are the subject of algebraic geometry. So it could say that the method of analytic techniques are applied to algebraic varieties was already used in FAC. ^ Hall (2018) ^ Harbater (2003) ^ Jump up to: a b Grothendieck & Raynaud (2002) ^ For discussions see Seidenberg (1958), Comments on Lefschetz's Principle; Frey & Rück (1986), The strong Lefschetz principle in algebraic geometry. ^ F.-V., Kuhlmann (2001) [1994], "Transfer principle", Encyclopedia of Mathematics, EMS Press ^ Hartshorne (1970) ^ Neeman (2007) References Chow, Wei-Liang (1949). "On Compact Complex Analytic Varieties". American Journal of Mathematics. 71 (4): 893–914. doi:10.2307/2372375. JSTOR 2372375. Frey, Gerhard; Rück, Hans -Georg (1986). "The strong Lefschetz principle in algebraic geometry". Manuscripta Mathematica. 55 (3–4): 385–401. doi:10.1007/BF01186653. S2CID 122967192. Grothendieck, Alexander; Raynaud, Michele (2002). "Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique". Revêtements étales et groupe fondamental (SGA 1) (in French). arXiv:math/0206203. doi:10.1007/BFb0058656. ISBN 978-2-85629-141-2. Harbater, David (21 July 2003). "Galois Groups and Fundamental Groups§9.Patching and Galois theory (Dept. of Mathematics, University of Pennsylvania)" (PDF). In Schneps, Leila (ed.). Galois Groups and Fundamental Groups. Cambridge University Press. ISBN 9780521808316. Hall, Jack (2018). "GAGA theorems". arXiv:1804.01976. Neeman, Amnon (2007). Algebraic and Analytic Geometry. doi:10.1017/CBO9780511800443. ISBN 9780511800443. Seidenberg, A. (1958). "Comments on Lefschetz's Principle". The American Mathematical Monthly. 65 (9): 685–690. doi:10.1080/00029890.1958.11991979. JSTOR 2308709. Hartshorne, Robin (1970). Ample Subvarieties of Algebraic Varieties. Lecture Notes in Mathematics. Vol. 156. doi:10.1007/BFb0067839. ISBN 978-3-540-05184-8. Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4757-3849-0. ISBN 978-0-387-90244-9. MR 0463157. S2CID 197660097. Zbl 0367.14001. Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents" (PDF), Annals of Mathematics, 61 (2): 197–278, doi:10.2307/1969915, JSTOR 1969915, MR 0068874 Serre, Jean-Pierre (1956). "Géométrie algébrique et géométrie analytique". Annales de l'Institut Fourier (in French). 6: 1–42. doi:10.5802/aif.59. ISSN 0373-0956. MR 0082175. External links Kiran Kedlaya. 18.726 Algebraic Geometry (LEC # 30 - 33 GAGA)Spring 2009. Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons BY-NC-SA. hide vte Topics in algebraic curves Rational curves Five points determine a conicProjective lineRational normal curveRiemann sphereTwisted cubic Elliptic curves Analytic theory Elliptic functionElliptic integralFundamental pair of periodsModular form Arithmetic theory Counting points on elliptic curvesDivision polynomialsHasse's theorem on elliptic curvesMazur's torsion theoremModular elliptic curveModularity theoremMordell–Weil theoremNagell–Lutz theoremSupersingular elliptic curveSchoof's algorithmSchoof–Elkies–Atkin algorithm Applications Elliptic curve cryptographyElliptic curve primality Higher genus De Franchis theoremFaltings's theoremHurwitz's automorphisms theoremHurwitz surfaceHyperelliptic curve Plane curves AF+BG theoremBézout's theoremBitangentCayley–Bacharach theoremConic sectionCramer's paradoxCubic plane curveFermat curveGenus–degree formulaHilbert's sixteenth problemNagata's conjecture on curvesPlücker formulaQuartic plane curveReal plane curve Riemann surfaces Belyi's theoremBring's curveBolza surfaceCompact Riemann surfaceDessin d'enfantDifferential of the first kindKlein quarticRiemann's existence theoremRiemann–Roch theoremTeichmüller spaceTorelli theorem Constructions Dual curvePolar curveSmooth completion Structure of curves Divisors on curves Abel–Jacobi mapBrill–Noether theoryClifford's theorem on special divisorsGonality of an algebraic curveJacobian varietyRiemann–Roch theoremWeierstrass pointWeil reciprocity law Moduli ELSV formulaGromov–Witten invariantHodge bundleModuli of algebraic curvesStable curve Morphisms Hasse–Witt matrixRiemann–Hurwitz formulaPrym varietyWeber's theorem Singularities AcnodeCrunodeCuspDelta invariantTacnode Vector bundles Birkhoff–Grothendieck theoremStable vector bundleVector bundles on algebraic curves Categories: Algebraic geometryAnalytic geometry

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