Chow’s Theorem Yohannes D. A Advisor: Fernando Q. Gouvêa May 25, 2017 Contents 1 Introduction 2 2 Sheaves and Ringed Spaces 2 2.1 Sheaves . 2 2.2 Ringed Spaces . 6 2.3 Sheaves of Modules . 7 3 Schemes 10 3.1 A˘ne Schemes . 11 3.2 General Schemes . 13 3.3 Projective Schemes . 14 4 Analytic Spaces 18 4.1 Definitions . 18 4.2 Analytic Sheaves . 19 4.3 Coherent Analytic Sheaves . 19 5 Analytification 19 5.1 Associating an analytic sheaf to a sheaf on a variety . 20 5.2 Analytic space associated to a scheme . 21 6 GAGA & Chow’s Theorem 21 6.1 GAGA . 21 6.2 Chow’s Theorem . 21 6.3 Consequences of Chow’s Theorem . 22 7 Acknowledgments 22 8 References 23 1 1 Introduction In 1949, W.-L. Chow published the paper On Compact Complex Analytic Varieties in which he proved the theorem that now bears his name Pn Theorem 1.0.1 (Chow). Every closed analytic subspace of C is an algebraic set. In that paper, he gave a completely analytic proof of this result. His proof, as opposed to the one J.-P. Serre gave as a consequence his famous correspondence theorem GAGA, did not involve the scheme-theoretic or homological algebraic methods. In 1956, a year after he published the paper Faisceaux algebriques coherents, or Coherent algebraic sheaves, Serre proved the GAGA correspondence theorem; the theorem takes its name from the paper—Géometrie Algébrique et Géométrie Ana- lytique. The GAGA correspondence states that an equivalence of categories that preserves cohomology exists between coherent sheaves on a complex projective space X and on its associated analytic space X ℎ . Among other consequences of GAGA that bridge complex algebraic geome- try and complex analytic geometry is Chow’s theorem. The subject of this thesis is the proof of Chow’s theorem using GAGA. We will introduce the necessary sheaf theory, scheme theory and complex analysis background before stating GAGA and proving Chow’s theorem. Our rings are commutative with a unity. 2 Sheaves and Ringed Spaces 2.1 Sheaves Sheaves are tools that allow us to track local algebraic structures of topological spaces. We will begin by defining presheaves. 2.1.1. Presheaves. Let X be a topological space. Definition 2.1.1. A presheaf F of abelian groups on X consists of the following data: 1. For every open set V X, F V is an abelian group and ⇢ ( ) 2 2. If V ,U are open subsets of X such that V U, then we have a restriction ⇢ morphism ⇢ : F U F V , UV ( )! ( ) such that (a) F = 0 (zero group). (ú) (b) ⇢UU = 1 (c) If V U W are open in X, then ⇢ = ⇢ ⇢ . ⇢ ⇢ WV UV ◦ WU In other words, Definition 2.1.2. A presheaf F of abelian groups over X is a contravariant functor from the category of Top X to the category of abelian groups Ab, where ( ) the open subsets of X are the objects and inclusion maps are the morphisms of Top X ; moreover, Hom V ,U is empty if V * U and is a singleton (an ( ) ( ) inclusion homomorphism) if V U . ⇢ Definition 2.1.3. If F is a presheaf on X and x X, the stalk F of F at x 2 x is the direct limit via the restriction maps ⇢ of the groups F U , where U are ( ) open neighborhoods of x. Elements of the abelian group F U for U X are known as sections of ( ) ⇢ the presheaf F over U . So, F U , or as it is sometimes denoted Γ U, F is a ( ) ( ) group of sections of F over U . Finally, a common shorthand for ⇢ s is s , UV ( ) |V as s F U . 2 ( ) 2.1.2. Morphisms of presheaves. Let F, G be two presheaves on X.Amor- phism of presheaves : F G consists of, for every open subset U of X,a ! group homomorphism U : F U G U which is compatible with the ( ) ( )! ( ) restrictions ⇢UV . The map is injective if for every open subset U of X, U is injective. ( ) On the other hand, is surjective if for any open subset U of X, s F U , 2 ( ) and x U , the canonically induced group homomorphism : F G that 2 x x ! x maps U s to s is surjective. But is an isomorphism if U is an ( ( )( ))x x ( x ) ( ) isomorphism for every open subset U X . ⇢ 3 2.1.3. Sheaves. Putting certain conditions on the sections of a presheaf gives rise to sheaves; precisely, sheaves are presheaves with extra data on their sections. As we will see in this section, sheaves are more useful than presheaves in telling us about local data. Definition 2.1.4. A sheaf F of abelian groups on X is a presheaf with the following properties. 1. (Locality) Let V be an open cover of U . Suppose that s F U and { i } 2 ( ) that we have s = 0 for all i. Then s = 0. |Vi 2. (Gluing) If V is an open cover of U , and suppose s F V . Also { i } i 2 ( i) suppose that si V V = s j V V for all i, j. Then, there exists s F U | i \ j | i \ j 2 ( ) such that s = s , for all i. |Vi i Example 2.1.5. (1) The constant sheaf Z of sets on any topological space X where every stalk is Z, and every homomorphism is the identity. (2) The zero sheaf is the constant sheaf 0, where 0 denotes the trivial additive group. We will give an example of a presheaf that is not a sheaf. Examples 2.1.6. Consider the space X = x, y with the discrete topology. Let { } the presheaf F be defined as follows: F = , F x = F y = R, and (ú) {ú} ({ }) ({ }) F x, y = R R R. Define the restriction in the obvious way, then we have ({ }) ⇥ ⇥ a presheaf. But although we can glue any two sections over x and y , we can’t { } { } glue them uniquely. 2.1.4. Morphisms of sheaves. A morphism of sheaves is just a morphism of presheaves. However, in the case of sheaves, Proposition 2.1.7. If : F G is a morphism of sheaves on X. Then is an ! isomorphism of sheaves if and only if is an isomorphism for every x X . x 2 Proof. See prop 2.12 in [Liu06]. ⇤ This is in general false for presheaves. That is, sheaves are more descriptive of what is going on locally than presheaves. 4 Definition 2.1.8. Let f : X Y be a continuous mapping of topological ! spaces. Then the direct image functor f sends a sheaf F on X to its direct image 1 ⇤ presheaf f F U = F f − U which is in fact a sheaf on Y . ⇤ ( ) ( ( )) Definition 2.1.9. A subsheaf of a sheaf F is a sheaf F 0 such that for every open set U X, F 0 U is a subgroup of F U , and the restriction maps of the sheaf ⇢ ( ) ( ) F 0 are induced by those of F . If G is another sheaf on X, the quotient sheaf F G is the associated1 sheaf to the presheaf U F U G U for all open / 7! ( )/ ( ) U X . ⇢ Proposition 2.1.10. Let F, G be sheaves, and let be a morphism from F to G . For all x X, let N be the kernel and I the image of . Then N = N is a 2 x x x x subsheaf of F, I = I is a subsheaf of G and defines an isomorphism of F N x – / and I . – Proof. See prop. 7 in §8 [Ser55]. ⇤ We call the sheaf N above the kernel of , I the image of , and the quotient G I the cokernel of . / Definition 2.1.11. Given a sheaf morphism : F G , we say is injective ! if ker = 0, surjective if coker = 0, and bijective if it is both injective and surjective. Definition 2.1.12. If U is an open subset of X every sheaf F on X induces a subsheaf F on U in an obvious way, where F V = F V for every open |U |U ( ) ( ) V U . We call F the restriction of F to U. ⇢ |U Another way to characterize the sheaf conditions stated earlier is as follows. Proposition 2.1.13. The presheaf F over the topological space X is a sheaf if and only if, for every open subset V X and any open cover V = i I Ui, the following ⇢ 2 sequence – ↵ β 0 F V F U F U U ! ( ) −− − − − −! ( i) −− − − − ! ( i \ j ) i I i,j I ÷2 ÷2 1that is, identifying the sheaf of sections of an appropriate topological space (see sheafification §II.1 prop-defn 1.2[Har97]). 5 where ↵ takes s F V to the product of its restrictions, and 2 ( ) β : si i I ⇢U ,U U si ⇢U ,U U s j i,j I { } 2 7! { i i \ j ( )− j i \ j ( )} 2 is exact. Proof. Omitted. ⇤ 2.2 Ringed Spaces Definition 2.2.1. A ringed space is a pair X, O consisting of a topological ( X ) space X and a sheaf of rings OX on X. We call the sheaf the structure sheaf of the space. Definition 2.2.2. A locally ringed space is a ringed space X, O such that for ( X ) each point x X, the stalk O is a local ring. 2 x Many types of spaces can be identified as certain types of ringed spaces.