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54-525: 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 X . Similarly, if F {\displaystyle {\mathcal {F}}} is a sheaf on X , then there is a corresponding sheaf F an {\displaystyle {\mathcal {F}}^{\text{an}}} on X . This association of an analytic object to an algebraic one

108-410: A S -scheme X is projective if and only if it is proper and there exists a very ample sheaf on X relative to S . Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, if X is projective, then the pullback of O ( 1 ) {\displaystyle {\mathcal {O}}(1)} under the closed immersion of X into

162-464: A complex projective variety X and the category of coherent analytic sheaves on the corresponding analytic space X are equivalent. The analytic space X is obtained roughly by pulling back to X the complex structure from C 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

216-450: A coordinate-free analog of the above. Namely, given a finite-dimensional vector space V over k , we let where k [ V ] = Sym ⁡ ( V ∗ ) {\displaystyle k[V]=\operatorname {Sym} (V^{*})} is the symmetric algebra of V ∗ {\displaystyle V^{*}} . It is the projectivization of V ; i.e., it parametrizes lines in V . There

270-432: A polynomial ring yields a projective scheme (required to be prime ideal to give a variety). In this sense, examples of projective varieties abound. The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely. The important class of complex projective varieties, i.e., the case k = C {\displaystyle k=\mathbb {C} } ,

324-456: A projective space is very ample. That "projective" implies "proper" is deeper: the main theorem of elimination theory . By definition, a variety is complete , if it is proper over k . The valuative criterion of properness expresses the intuition that in a proper variety, there are no points "missing". There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety

378-411: A projective variety. There are at least two equivalent ways to define the degree of X relative to its embedding. The first way is to define it as the cardinality of the finite set where d is the dimension of X and H i 's are hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, if X is a hypersurface, then the degree of X is the degree of

432-420: Is k [ x 0 , … , x n ] {\displaystyle k[x_{0},\ldots ,x_{n}]} and its Hilbert polynomial is P ( z ) = ( z + n n ) {\displaystyle P(z)={\binom {z+n}{n}}} ; its arithmetic genus is zero. If the homogeneous coordinate ring R is an integrally closed domain , then

486-414: Is a functor . The prototypical theorem relating X and X says that for any two coherent sheaves F {\displaystyle {\mathcal {F}}} and G {\displaystyle {\mathcal {G}}} on X , the natural homomorphism: is an isomorphism. Here O X {\displaystyle {\mathcal {O}}_{X}} is the structure sheaf of

540-432: Is a canonical surjective map π : V ∖ { 0 } → P ( V ) {\displaystyle \pi :V\setminus \{0\}\to \mathbb {P} (V)} , which is defined using the chart described above. One important use of the construction is this (cf., § Duality and linear system ). A divisor D on a projective variety X corresponds to a line bundle L . One then set it

594-507: Is a scheme which it is a union of ( n + 1) copies of the affine n -space k . More generally, projective space over a ring A is the union of the affine schemes in such a way the variables match up as expected. The set of closed points of P k n {\displaystyle \mathbb {P} _{k}^{n}} , for algebraically closed fields k , is then the projective space P n ( k ) {\displaystyle \mathbb {P} ^{n}(k)} in

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648-454: Is an affine curve given by, say, y 2 = x 3 + a x + b {\displaystyle y^{2}=x^{3}+ax+b} in the affine plane, then its projective completion in the projective plane is given by y 2 z = x 3 + a x z 2 + b z 3 . {\displaystyle y^{2}z=x^{3}+axz^{2}+bz^{3}.} For various applications, it

702-575: Is an algebraic variety covered by ( n +1) open affine charts X ∩ U i {\displaystyle X\cap U_{i}} . Note that X is the closure of the affine variety X ∩ U 0 {\displaystyle X\cap U_{0}} in P n {\displaystyle \mathbb {P} ^{n}} . Conversely, starting from some closed (affine) variety V ⊂ U 0 ≃ A n {\displaystyle V\subset U_{0}\simeq \mathbb {A} ^{n}} ,

756-555: Is called projective over S if it factors as a closed immersion followed by the projection to S . A line bundle (or invertible sheaf) L {\displaystyle {\mathcal {L}}} on a scheme X over S is said to be very ample relative to S if there is an immersion (i.e., an open immersion followed by a closed immersion) for some n so that O ( 1 ) {\displaystyle {\mathcal {O}}(1)} pullbacks to L {\displaystyle {\mathcal {L}}} . Then

810-475: Is called the Hilbert polynomial of X . It is a numerical invariant encoding some extrinsic geometry of X . The degree of P is the dimension r of X and its leading coefficient times r! is the degree of the variety X . The arithmetic genus of X is (−1) ( P (0) − 1) when X is smooth. For example, the homogeneous coordinate ring of P n {\displaystyle \mathbb {P} ^{n}}

864-466: Is called the complete linear system of D . Projective space over any scheme S can be defined as a fiber product of schemes If O ( 1 ) {\displaystyle {\mathcal {O}}(1)} is the twisting sheaf of Serre on P Z n {\displaystyle \mathbb {P} _{\mathbb {Z} }^{n}} , we let O ( 1 ) {\displaystyle {\mathcal {O}}(1)} denote

918-470: Is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased as "any analytic subspace of complex projective space that 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. Foundations for the many relations between the two theories were put in place during

972-474: Is coherent, a result known as the Oka coherence theorem , and also, it was proved in “Faisceaux Algebriques Coherents” that the structure sheaf of the algebraic variety O X {\displaystyle {\mathcal {O}}_{X}} is coherent. Another important statement is as follows: For any coherent sheaf F {\displaystyle {\mathcal {F}}} on an algebraic variety X

1026-468: Is complete. The converse is not true in general. However: Some properties of a projective variety follow from completeness. For example, for any projective variety X over k . This fact is an algebraic analogue of Liouville's theorem (any holomorphic function on a connected compact complex manifold is constant). In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as

1080-489: Is discussed further below. The product of two projective spaces is projective. In fact, there is the explicit immersion (called Segre embedding ) As a consequence, the product of projective varieties over k is again projective. The Plücker embedding exhibits a Grassmannian as a projective variety. Flag varieties such as the quotient of the general linear group G L n ( k ) {\displaystyle \mathrm {GL} _{n}(k)} modulo

1134-501: 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

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1188-457: Is explained below. Quasi-projective varieties are, by definition, those which are open subvarieties of projective varieties. This class of varieties includes affine varieties . Affine varieties are almost never complete (or projective). In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globally regular functions on a projective variety. By definition, any homogeneous ideal in

1242-392: Is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e., P n ( k ) {\displaystyle \mathbb {P} ^{n}(k)}

1296-581: Is particularly rich, including a classification by the genus of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties. Hilbert schemes parametrize closed subschemes of P n {\displaystyle \mathbb {P} ^{n}} with prescribed Hilbert polynomial. Hilbert schemes, of which Grassmannians are special cases, are also projective schemes in their own right. Geometric invariant theory offers another approach. The classical approaches include

1350-521: 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 . There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in

1404-628: 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 . This principle permits

1458-503: The Teichmüller space and Chow varieties . A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials defining X have complex coefficients. Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example,

1512-403: 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 . In the twentieth century,

1566-619: The pullback of O ( 1 ) {\displaystyle {\mathcal {O}}(1)} to P S n {\displaystyle \mathbb {P} _{S}^{n}} ; that is, O ( 1 ) = g ∗ ( O ( 1 ) ) {\displaystyle {\mathcal {O}}(1)=g^{*}({\mathcal {O}}(1))} for the canonical map g : P S n → P Z n . {\displaystyle g:\mathbb {P} _{S}^{n}\to \mathbb {P} _{\mathbb {Z} }^{n}.} A scheme X → S

1620-409: The quotient ring is called the homogeneous coordinate ring of X . Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring . Projective varieties arise in many ways. They are complete , which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but Chow's lemma describes

1674-451: The Zariski topology are defined to be the common zero-locus of a finite collection of homogeneous polynomial functions. Given a polynomial f ∈ k [ x 0 , … , x n ] {\displaystyle f\in k[x_{0},\dots ,x_{n}]} , the condition does not make sense for arbitrary polynomials, but only if f is homogeneous , i.e.,

Algebraic geometry and analytic geometry - Misplaced Pages Continue

1728-712: The algebraic variety X and O X an {\displaystyle {\mathcal {O}}_{X}^{\text{an}}} is the structure sheaf of the analytic variety X . More precisely, the category of coherent sheaves on the algebraic variety X is equivalent to the category of analytic coherent sheaves on the analytic variety X , and the equivalence is given on objects by mapping F {\displaystyle {\mathcal {F}}} to F an {\displaystyle {\mathcal {F}}^{\text{an}}} . (Note in particular that O X an {\displaystyle {\mathcal {O}}_{X}^{\text{an}}} itself

1782-574: The canonical surjection induces the closed immersion Compared to projective varieties, the condition that the ideal I be a prime ideal was dropped. This leads to a much more flexible notion: on the one hand the topological space X = Proj ⁡ R {\displaystyle X=\operatorname {Proj} R} may have multiple irreducible components . Moreover, there may be nilpotent functions on X . Closed subschemes of P k n {\displaystyle \mathbb {P} _{k}^{n}} correspond bijectively to

1836-410: 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. (e.g. Kodaira type vanishing 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

1890-527: The close relation of these two notions. Showing that a variety is projective is done by studying line bundles or divisors on X . A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties, Serre duality can be viewed as an analog of Poincaré duality . It also leads to the Riemann–Roch theorem for projective curves, i.e., projective varieties of dimension 1. The theory of projective curves

1944-610: The closure of V in P n {\displaystyle \mathbb {P} ^{n}} is the projective variety called the projective completion of V . If I ⊂ k [ y 1 , … , y n ] {\displaystyle I\subset k[y_{1},\dots ,y_{n}]} defines V , then the defining ideal of this closure is the homogeneous ideal of k [ x 0 , … , x n ] {\displaystyle k[x_{0},\dots ,x_{n}]} generated by for all f in I . For example, if V

1998-430: 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

2052-455: 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. In slightly lesser generality, the GAGA theorem asserts that the category of coherent algebraic sheaves on

2106-535: The coordinate ring Say i = 0 for the notational simplicity and drop the superscript (0). Then X ∩ U 0 {\displaystyle X\cap U_{0}} is a closed subvariety of U 0 ≃ A n {\displaystyle U_{0}\simeq \mathbb {A} ^{n}} defined by the ideal of k [ y 1 , … , y n ] {\displaystyle k[y_{1},\dots ,y_{n}]} generated by for all f in I . Thus, X

2160-417: The definition of projective varieties is projective space P n {\displaystyle \mathbb {P} ^{n}} , which can be defined in different, but equivalent ways: A projective variety is, by definition, a closed subvariety of P n {\displaystyle \mathbb {P} ^{n}} , where closed refers to the Zariski topology . In general, closed subsets of

2214-469: The degree of X is the multiplicity of the vertex of the affine cone over X . Oka coherence theorem In mathematics, the Oka coherence theorem , proved by Kiyoshi Oka  ( 1950 ), states that the sheaf O C n {\displaystyle {\mathcal {O}}_{\mathbb {C} ^{n}}} of holomorphic functions on C n {\displaystyle \mathbb {C} ^{n}} (and subsequently

Algebraic geometry and analytic geometry - Misplaced Pages Continue

2268-457: The degrees of all the monomials (whose sum is f ) are the same. In this case, the vanishing of is independent of the choice of λ ≠ 0 {\displaystyle \lambda \neq 0} . Therefore, projective varieties arise from homogeneous prime ideals I of k [ x 0 , … , x n ] {\displaystyle k[x_{0},\dots ,x_{n}]} , and setting Moreover,

2322-507: 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 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

2376-449: The homogeneous ideals I of k [ x 0 , … , x n ] {\displaystyle k[x_{0},\ldots ,x_{n}]} that are saturated ; i.e., I : ( x 0 , … , x n ) = I . {\displaystyle I:(x_{0},\dots ,x_{n})=I.} This fact may be considered as a refined version of projective Nullstellensatz . We can give

2430-454: The homogeneous polynomial defining X . The "general positions" can be made precise, for example, by intersection theory ; one requires that the intersection is proper and that the multiplicities of irreducible components are all one. The other definition, which is mentioned in the previous section, is that the degree of X is the leading coefficient of the Hilbert polynomial of X times (dim X )!. Geometrically, this definition means that

2484-418: The homomorphisms are isomorphisms for all q' s. This means that the q -th cohomology group on X is isomorphic to the cohomology group on X . 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 . Algebraic varieties are locally defined as

2538-481: The nineteenth century. Some of the more important advances are listed here in chronological order. Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an (smooth projective) algebraic curve . Under the name Riemann's existence theorem 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

2592-458: The projective variety X is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study of X (e.g., singularity) reduces to that of an affine variety. The explicit structure is as follows. The projective space P n {\displaystyle \mathbb {P} ^{n}} is covered by the standard open affine charts which themselves are affine n -spaces with

2646-481: The projective variety X is said to be projectively normal . Note, unlike normality , projective normality depends on R , the embedding of X into a projective space. The normalization of a projective variety is projective; in fact, it's the Proj of the integral closure of some homogeneous coordinate ring of X . Let X ⊂ P N {\displaystyle X\subset \mathbb {P} ^{N}} be

2700-525: The subgroup of upper triangular matrices , are also projective, which is an important fact in the theory of algebraic groups . As the prime ideal P defining a projective variety X is homogeneous, the homogeneous coordinate ring is a graded ring , i.e., can be expressed as the direct sum of its graded components: There exists a polynomial P such that dim ⁡ R n = P ( n ) {\displaystyle \dim R_{n}=P(n)} for all sufficiently large n ; it

2754-495: The theory of holomorphic vector bundles (more generally coherent analytic sheaves ) on X coincide with that of algebraic vector bundles. Chow's theorem says that a subset of projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as Hodge theory . Let k be an algebraically closed field. The basis of

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2808-406: The time of GAGA's publication. Projective variety In algebraic geometry , a projective variety is an algebraic variety that is a closed subvariety of a projective space . That is, it is the zero-locus in P n {\displaystyle \mathbb {P} ^{n}} of some finite family of homogeneous polynomials that generate a prime ideal , the defining ideal of

2862-474: The usual sense. An equivalent but streamlined construction is given by the Proj construction , which is an analog of the spectrum of a ring , denoted "Spec", which defines an affine scheme. For example, if A is a ring, then If R is a quotient of k [ x 0 , … , x n ] {\displaystyle k[x_{0},\ldots ,x_{n}]} by a homogeneous ideal I , then

2916-410: The variety. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial . If X is a projective variety defined by a homogeneous prime ideal I , then

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