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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Ringed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Sheaves of Modules . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Schemes 10

3.1 A˘ne Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 General Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Projective Schemes . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Analytic Spaces 18

4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Analytic Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Coherent Analytic Sheaves . . . . . . . . . . . . . . . . . . . . 19

5 Analytification 19

5.1 Associating an analytic sheaf to a sheaf on a variety . . . . . . . 205.2 Analytic space associated to a scheme . . . . . . . . . . . . . . 21

6 GAGA & Chow’s Theorem 21

6.1 GAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Chow’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 216.3 Consequences of Chow’s Theorem . . . . . . . . . . . . . . . . 22

7 Acknowledgments 22

8 References 23

1

1 Introduction

In 1949, W.-L. Chow published the paperOnCompact Complex Analytic Varietiesin which he proved the theorem that now bears his name

Theorem 1.0.1 (Chow). Every closed analytic subspace of PnC is an algebraic set.

In that paper, he gave a completely analytic proof of this result. His proof, asopposed to the one J.-P. Serre gave as a consequence his famous correspondencetheorem GAGA, did not involve the scheme-theoretic or homological algebraicmethods.

In 1956, a year after he published the paper Faisceaux algebriques coherents, orCoherent algebraic sheaves, Serre proved the GAGA correspondence theorem; thetheorem takes its name from the paper—Géometrie Algébrique et Géométrie Ana-lytique. The GAGA correspondence states that an equivalence of categories thatpreserves cohomology exists between coherent sheaves on a complex projectivespace X and on its associated analytic space X � .

Among other consequences of GAGA that bridge complex algebraic geome-try and complex analytic geometry is Chow’s theorem. The subject of this thesisis the proof of Chow’s theorem using GAGA. We will introduce the necessarysheaf theory, scheme theory and complex analysis background before statingGAGA 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 topologicalspaces. 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 followingdata:

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 restrictionmorphism ⇢UV : F (U ) ! F (V ),

such that

(a) F (ú) = 0 (zero group).

(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 contravariantfunctor from the category of Top(X ) to the category of abelian groups Ab, wherethe open subsets of X are the objects and inclusion maps are the morphismsof Top(X ); moreover, Hom(V ,U ) is empty if V * U and is a singleton (aninclusion homomorphism) if V ⇢ U .

Definition 2.1.3. If F is a presheaf on X and x 2 X , the stalk Fx of F at xis the direct limit via the restriction maps ⇢ of the groups F (U ), whereU areopen neighborhoods of x .

Elements of the abelian group F (U ) for U ⇢ X are known as sections ofthe presheaf F over U . So, F (U ), or as it is sometimes denoted �(U ,F ) is agroup of sections of F overU . Finally, a common shorthand for ⇢UV (s) is s |V ,as s 2 F (U ).

2.1.2. Morphisms of presheaves. Let F ,G be two presheaves on X . A mor-phism of presheaves : F ! G consists of, for every open subset U of X , agroup homomorphism (U ) : F (U ) ! G (U ) which is compatible with therestrictions ⇢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 2 F (U ),and x 2 U , the canonically induced group homomorphism x : Fx ! Gx thatmaps ( (U )(s))x to x (sx ) is surjective. But is an isomorphism if (U ) is anisomorphism for every open subsetU ⇢ X .

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2.1.3. Sheaves. Putting certain conditions on the sections of a presheaf givesrise 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 tellingus about local data.

Definition 2.1.4. A sheaf F of abelian groups on X is a presheaf with thefollowing properties.

1. (Locality) Let {Vi} be an open cover of U . Suppose that s 2 F (U ) andthat we have s |Vi = 0 for all i. Then s = 0.

2. (Gluing) If {Vi} is an open cover of U , and suppose si 2 F (Vi). Alsosuppose that si |Vi\Vj = s j |Vi\Vj for all i, j . Then, there exists s 2 F (U )such that s |Vi = si , for all i .

Example 2.1.5. (1) The constant sheaf Z of sets on any topological space Xwhere every stalk is Z, and every homomorphism is the identity.

(2) The zero sheaf is the constant sheaf 0, where 0 denotes the trivial additivegroup.

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. Letthe presheaf F be defined as follows: F (ú) = {ú},F ({x}) = F ({y}) = R, andF ({x, y}) = R ⇥R ⇥R. Define the restriction in the obvious way, then we havea presheaf. But although we can glue any two sections over {x} and {y}, we can’tglue them uniquely.

2.1.4. Morphisms of sheaves. A morphism of sheaves is just a morphism ofpresheaves. However, in the case of sheaves,

Proposition 2.1.7. If : F ! G is a morphism of sheaves on X. Then is anisomorphism of sheaves if and only if x is an isomorphism for every x 2 X .

Proof. See prop 2.12 in [Liu06]. ⇤

This is in general false for presheaves. That is, sheaves are more descriptiveof what is going on locally than presheaves.

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Definition 2.1.8. Let f : X ! Y be a continuous mapping of topologicalspaces. Then the direct image functor f⇤ sends a sheaf F on X to its direct imagepresheaf f⇤F (U ) = F ( f �1(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 opensetU ⇢ X , F 0(U ) is a subgroup of F (U ), and the restriction maps of the sheafF 0 are induced by those of F . If G is another sheaf on X , the quotient sheafF/G is the associated1 sheaf to the presheaf U 7! F (U )/G (U ) for all openU ⇢ X .

Proposition 2.1.10. Let F ,G be sheaves, and let be a morphism from F to G .For all x 2 X , let Nx be the kernel and Ix the image of x . Then N =

–Nx is a

subsheaf of F ,I =–

Ix is a subsheaf of G and defines an isomorphism of F/Nand I .

Proof. See prop. 7 in §8 [Ser55]. ⇤

We call the sheaf N above the kernel of , I the image of , and thequotient G /I the cokernel of .

Definition 2.1.11. Given a sheaf morphism : F �! G , we say is injectiveif ker = 0, surjective if coker = 0, and bijective if it is both injective andsurjective.

Definition 2.1.12. If U is an open subset of X every sheaf F on X induces asubsheaf F |U onU in an obvious way, where F |U (V ) = F (V ) for every openV ⇢ U .We call F |U the restriction of F to 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 andonly if, for every open subset V ⇢ X and any open cover V =

–i2I Ui, the following

sequence

0 �! F (V ) ↵������!÷i2I

F (Ui)�������!

÷i, j2I

F (Ui \Uj )

1that is, identifying the sheaf of sections of an appropriate topological space (see sheafification§II.1 prop-defn 1.2[Har97]).

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where ↵ takes s 2 F (V ) to the product of its restrictions, and

� : {si}i2I 7! {⇢Ui,Ui\Uj (si) � ⇢Uj ,Ui\Uj (s j )}i, j2Iis exact.

Proof. Omitted. ⇤

2.2 Ringed Spaces

Definition 2.2.1. A ringed space is a pair (X ,OX ) consisting of a topologicalspace 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 ,OX ) such that foreach point x 2 X , the stalk Ox is a local ring.

Many types of spaces can be identified as certain types of ringed spaces.

Examples 2.2.3. (1) Let X be a topological space. Then there is a correspondingringed space (X ,CX ) where CX is the sheaf of germs of continuous functions onX .

(2) If X is an algebraic variety carrying the Zariski topology (§3.1.1), we candefine a locally ringed space by taking OX (U ) to be the ring of rational mappingsdefined on the Zariski-open setU that don’t become infinite withinU .

(3) Schemes are locally ringed spaces that are locally isomorphic to thespectrum of a ring; see §3.

(4) A complex analytic space (definition.4.1.1) is a locally ringed space whosestructure is a C-algebra, where C is the constant sheaf on a topological spacewith value C.

2.2.1. Morphisms of ringed spaces.

Definition 2.2.4. A morphism of ringed spaces

( , #) : (X ,OX ) ! (Y ,OY )

consists of a continuous map : X ! Y and a morphism of sheaves of rings # : OY ! ⇤OX . A morphism ( , #) is an isomorphism if and only if is ahomeomorphism of the underlying topological spaces, and # is an isomorphismof the sheaves.

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Definition 2.2.5. A morphism of locally ringed spaces (X ,OX ) �! (Y ,OY ) is amorphism of ringed spaces (�, �#) such that for all x 2 X the induced homomor-phism on stalks

�#x : (��1OY )x = OY ,�(x) �! OX ,x

is a local ring homomorphism.

That it is a local homomorphism means that �#�1x (mx ) = m�(x), where mx isthe maximal ideal of OX ,x .

The composition of two morphisms of locally ringed spaces is again a mor-phism of locally ringed spaces. Therefore locally ringed spaces form a category.

Example 2.2.6. Let X be a topological space and consider the sheaf CX of R-valued continuous functions on X (i.e., forU ⇢ X open, CX (U ) is the R-algebraof continuous functions s : U ! R). For x 2 X ,CX ,x is the ring of germs [s]of continuous functions s in a neighborhood of x . Let mx ⇢ CX ,x be the set ofgerms [s] such that s(x) = 0. This is a unique maximal ideal and (X ,CX ) is alocally ringed space [UG10].

Lastly,

Definition 2.2.7. A morphism ( , #) : (X ,OX ) ! (Y ,OY ) is an open immer-sion (resp. closed immersion) if f is a topological open immersion (resp. closedimmersion) and if f #x is an isomorphism (resp. if f #x is surjective) for everyx 2 X .

Moreover, if V is open in Y , the restriction (V ,OY |V ) is a ringed space. So,when we consider an open subset, it will inherit this structure induced by OY .Hence, ( , #) is an open immersion if and only if there exists an open subset Vof Y such that ( , #) induces an isomorphism from (X ,OX ) onto (V ,OY |V ).

2.3 Sheaves of Modules

These are special types of sheaves that we almost always work with in this paper.

2.3.1. Definitions.

Definition 2.3.1. Let (X ,OX ) be a ringed space. A sheaf of OX -modules is a sheafF on X such that F (U ) is an OX (U )-module for every open subsetU , and thatif V ⇢ U , then (s t )|V = s |V t |V for every s 2 OX (U ) and every t 2 F (U ).

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Examples 2.3.2. (1) If O is the constant sheaf Z, then the sheaf of O -moduleson it is the same as the sheaf of the abelian groups Z.

(2) If X is a smooth variety, then the cotangent sheaf is a sheaf of modules.

Definition 2.3.3. An OX -submodule G of F is an OX -module F such thatF (U ) is a subset of G (U ) for all open setsU ⇢ X and such that the inclusions◆U : F (U ) ,! G (U ) form a homomorphism ◆ : F �! G of OX -modules.

The OX -submodules of OX are called ideals ofOX .

2.3.2. Operations on sheaves of modules. Let O be a sheaf of rings on X andN ,M be two sheaves of O -modules.

Definition 2.3.4. The direct sum N � M is the sheaf of O -modules defined by(N � M )(U ) = N (U ) � M (U ).

By recurrence, the definition of direct sum extends to a finite number ofO -modules; if we have n sheaves isomorphic to one sheaf F , we denote theirdirect sum by F n .

Definition 2.3.5. An OX -module F on a ringed space X is called locally freeof finite rank if every point in X has an open neighborhood U such that therestriction F |U is isomorphic to a finite direct sum of copies of OX |U .

Definition 2.3.6. The tensor product N ⌦M is the sheaf ofO -modules associatedto the presheafU 7! N (U ) ⌦O(U ) M (U ).

2.3.3. Support of an OX -module.

Definition 2.3.7. Let F be an OX -module. Then

Supp(F ) = {x 2 X : Fx , 0}

is called the support of F .

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2.3.4. Coherent Sheaves. Coherent sheaves are generalizations of vector bun-dles. But unlike vector bundles, the category of coherent sheaves on a topologicalspace X forms an abelian category. So we can take kernels, images, and cokernels.

Let (X ,OX ) be a ringed space, and let F be an OX -module. We say that Fis finitely generated if for every x 2 X , there exists an open neighborhoodU ofx , an integer n � 1, and a surjective homomorphism On

X |U �! F |U .On the other hand, we say F is finitely presented if in addition there exists

an integer m � 1, and a morphism of sheaves OmX |U ! On

X |U such that thesequence

OmX |U �! On

X |U �! F

is exact.

Definition 2.3.8. The sheaf F is coherent if it is finitely generated, and if forevery open subsetU of X , and for every homomorphism : On

X |U �! F |U ,the kernel ker is finitely generated.

Examples 2.3.9. (1) If X is a complex analytic variety, the sheaf of germs ofholomorphic functions on X is a coherent sheaf of rings, by Oka’s CoherenceTheorem (Theorem 6.4.1[Hor90]).

(2) If X is an algebraic variety, the sheaf of local rings of X is a coherentsheaf of rings.

Coherent sheaves are actually a specific type of a more general class of sheavesof modules called quasi-coherent sheaves.

Definition 2.3.10. A sheaf G of OX -modules is called quasi-coherent if it is,locally, isomorphic to the cokernel of a map between free OX -modules.

It is apparent that coherence is a local property as the sheaf F is coherentif and only if F |U� is coherent for every � 2 ⇤, where {U� |� 2 ⇤} is an opencover of X .We observe a few more properties

Proposition 2.3.11. —

1. Any subsheaf of finite type of a coherent sheaf is coherent.

2. Let 0 ! F↵�! G

��! K ! 0 be an exact sequence of homomorphisms. Iftwo of the sheaves F ,G ,K are coherent, so is the third.

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3. Let � be a homomorphism from a coherent sheaf F to a coherent sheaf G . Thekernel, the cokernel and the image of � are also coherent sheaves.

Proof. —

1. no. 12, prop. 3[Ser55]

2. no. 13, theorem 1 [Ser55]

3. Because F is coherent, the image is of finite type and by (1) coherent; andapplying (2) to the following exact sequences

0 ! ker � ! F ! Im � ! 0,

0 ! Im � ! F ! coker � ! 0,

we get that so are the cokernel and the kernel of �. ⇤

The key property of modules of finite type is highlighted by the followingproposition.

Proposition 2.3.12. Let (X ,OX ) be a ringed space and let F be an OX -module offinite type. Let x 2 X be a point and let si 2 F (U ) for i = 1, . . . , n be sections oversome open neighborhood of x such that the germs (si)x generate the stalk Fx . Thenthere exists an open neighborhood V ⇢ U such that the si |V generate F |V .

Proof. prop. 7.29, [UG10] ⇤

The following result applied to coherent sheaves will be crucial in the proofof Chow’s theorem.

Corollary 2.3.13. Let (X ,OX ) be a ringed space and let F be an OX -module offinite type. Then Supp(F ) is closed in X.

3 Schemes

Schemes were introduced by Alexander Grothendieck. They generalize thenotion of variety. They are constructed locally from objects called a�ne schemes.An a˘ne scheme is a ringed space; so, we will start by specifying the topologicalspace before we construct a structure sheaf on it.

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3.1 A�ne Schemes

3.1.1. Spec. The spectrum Spec R of a ring R is, as a space, the set of all primeideals of R; as a topological space, it carries the Zariski topology. Declare theclosed sets to be sets of the form V(I ) = {p � I | p 2 Spec (R)}, where I is anideal in R.

To see that this actually defines a topology, first note that ú = V(Spec (R))and Spec (R) = V(ú), while for any arbitrary collection of closed sets indexed byI , Ÿ

◆2IV (a◆) = V (b),

where b is the ideal generated by the union of a◆. Furthermore, for any closedV (a), V (b),

V (a) [V (b) = V (ab).Hence, we have a topology on Spec (R) as required.

If we let X = Spec (R) and R f be the ideal generated by f 2 R, then the set

X f = X �V (R f )

for any f is open in X . These X f in fact make an open base for the Zariskitopology and are referred to as the distinguished open sets of this topology.

An interesting property of this topology is that it is not Hausdor¯. The setp 2 Spec (R) is Z-closed if and only if p is maximal. For instance, if R = k[X ],where k is a field, we get Spec (R) = A1

k ; but the point 0 is not Z-closed because{0} is not a maximal ideal.

3.1.2. The sheaves eM . Let R be a ring and X = Spec (R) . Also suppose thatM is an R-module. We will now define a canonical sheaf of modules on X . Thecase of M = R reduces to eM = OSpec(R).

Recall that the distinguished open sets X f for f 2 R form a basis for theZariski topology on X . To define a sheaf on X , then it su˘ces to work withthe X f ’s. To see this, let B be a base of open subsets on X and let U = [Ui ,where Ui 2 B. Then A (U ) contains elements (si)i 2 Œ

i A0(Ui) such thatsi |Ui\Uj = s j |Ui \Uj, where A is the extension of a B�sheaf A0. Furthermore,this extension is unique up to isomorphism.

Let Mf = M ⌦R R f where R f is the localization of R at f . That is, Mf is theset of symbolsm/ f n,m 2 M , n 2 Zmodulo the relation thatm1/ f n1 = m2/ f n2

11

if and only if f n2+k ·m1 = f n1+k ·m2 for some k 2 Z.We claim that this definesa presheaf on X .

It only remains to show the existence of restriction functions. SupposeX � ⇢ X f . This implies that there exists m � 1 and b 2 R such that �m = f b .As f is then invertible, we have a restriction homomorphism

R f ! R� : a f �n 7! (abn)��mn .

In fact, eM is also a sheaf. We will reproduce the proof from [Liu06]:p.42. LetX fi = Xi . Then

V ’

ifiR

!=

ŸiV ( fiR) = ú,

and R =Õ

i fiR.Hence, there exists a finite subset F ⇢ I such that 1 2 Õi2F fiR.

To check for Definition 1.1.4 (1), suppose s 2 R and s |Vi = 0 for alli . Then there exists an m � 1 such that f mi s = 0 for every i 2 F . ThenX =

–i2F D( fi) =

–i2F D( f mi ). It follows that 1 2 Õ

i2F f mi R and so,

s 2’i2F

s f mi R = {0}

as required.To check for Definition 1.1.4 (2), suppose si 2 eM (Ui) be sections that

agree on the intersections Ui \ Uj . Then, there exists an m � 1 such thatsi = bi f �mi 2 R fi for every i 2 F . But since si |Ui\Uj = s j |Ui\Uj , there existsan r � 1 such that (bi f mj � bj f mi )( fi f j )r = 0 for every i, j 2 F . As above,

1 =Õ

j2F a j f m+rj with a j 2 R. Let us set s =Õ

j2F a jb j f rj 2 R. For every i 2 F ,we have

f m+ri s =’j2F

a jb j f rj f m+ri =’j2F

a jbi f ri f m+rj = bi f ri .

Hence, s |Ui = si for every i 2 F . That is, if i 2 I , then (s |Ui )|Ui\Uj = si |Ui\Uj forevery j 2 F . Since Ui =

–j2F Ui \Uj, we have s |Ui = si . This concludes the

proof that eM is a sheaf.

Definition 3.1.1. An a�ne scheme is a locally ringed space isomorphic as alocally ringed space to (Spec (R) , eR) for some ring R.

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3.2 General Schemes

Definition 3.2.1. A scheme is a ringed topological space (X ,OX ) admitting anopen covering {Ui}i such that (Ui,OX |Ui ) is an a˘ne scheme for every i .

Remark 3.2.2. In other words, A scheme is a ringed space that is locally a�ne.If (X ,OX ) is a ringed space and X is covered by open sets Ui , then (X ,OX ) islocally a˘ne if there exist rings Ri and homeomorphisms Spec (Ri) � Ui withOX |Ui � OSpec(Ri ) As a consequence, we have that (1) OSpec(Ri )(X f ) = R[ f �1]; (2)The stalks Ox are local rings, and (3) X and sp(Spec (O(X ))) are homeomorphic.

Proposition 3.2.3. Let X be a scheme2. Then for any open subset U of X , the ringedspace (U ,OX |U ) is also a scheme.

Proof. prop 3.9 - [DE07] ⇤

3.2.1. Maps between schemes.

Definition 3.2.4. A morphism or map between schemes X and Y is a pair( , #), where : X ! Y is a continuous map on the underlying topologicalspaces and

# : OY ! ⇤OX

is a map of sheaves on Y satisfying the condition that for any point p 2 X andany neighborhoodU of q = (p) in Y , a section f 2 OY (U ) vanishes at q if andonly if the section # f of

⇤OX (U ) = OX ( �1U )

vanishes at p .

3.2.2. Subschemes. We are particularly interested in the closed ones; so we willfirst define

Definition 3.2.5. A morphism of schemes � : Y ! X is a closed embeddingif every point x 2 X has an a˘ne neighborhood U such that ��1(U ) ⇢ Y isan a˘ne subscheme and the homomorphism U : OX (U ) �! OY (��1(U )) issurjective.

In this case, Y is said to be a closed subscheme of X .2the ringed space (X ,OX ) by abuse of notation

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3.2.3. S-Schemes.

Definition 3.2.6. Let S be a fixed scheme. A scheme over S is a scheme X ,together with a morphism X ! S . If X and Y are schemes over S , a morphismof X to Y as schemes over S , called an S-morphism is a morphism � : X ! Ywhich is compatible with the given morphisms to S .

3.3 Projective Schemes

Both GAGA and Chow’s Theorem are about projective spaces. Unlike thesection on a˘ne schemes, where we built up them from the ground up, this time,to enrich our discussion of projective spaces, we will define projective schemes assubschemes in a projective space. Another approach is to construct projectiveschemes from graded algebras. This is done by using the Proj functor. We willalso demonstrate this in the end.

3.3.1. Projective n-Space. We will be working in the field C. For this construc-tion we will closely follow [Har97].

Definition 3.3.1. The set of equivalence classes of (n + 1)-tuples (a0, . . . , an) ofelements of C, not all zero, under the equivalence relation given by

(a0, . . . , an) ⇠ (�a0, . . . , �an)

for all � 2 C, � , 0 is called the complex projective n-space and it is denoted byPn. Unless otherwise stated, Pn refers to PnC.

Equivalently, Pn is the set of all one-dimensional subspaces of An+1. Or inother words, it is the quotient of the set An+1� {(0, . . . , 0)} under the equivalencerelation which identifies points lying on the same line through the origin.

We call these equivalence classes points of Pn and the elements of an equiva-lence class is referred to as a set of homogeneous coordinates for the class.

Now that we have defined the main subject, we remind the reader that GAGAtells us that analytic (or holomorphic) objects on the complex projective space Pn

are indeed algebraic. This hints that projective spaces are unique and importantas they seem bridge the gap between algebraic and complex geometry.

Definition 3.3.2. A subset Y of Pn is an algebraic set if there exists a set T ofhomogeneous elements of C[x0, . . . , xn] such that Y is the vanishing set of T .

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3.3.2. Structure of Pn. Just like the a˘ne space, we can make it into a topo-logical space by endowing it the Zariski topology; take the open sets to be thecomplements of the algebraic sets.

Definition 3.3.3. A projective variety is a zero locus of a finite set of homoge-neous polynomials, with the Zariski topology, the closed sets being projectivealgebraic sets.

Proposition 3.3.4. Pn is a compact Hausdor� space with the quotient topology(analytic topology).

Proof. [RCG65]:p.150. ⇤

3.3.3. Projective Schemes.

Definition 3.3.5. A scheme isomorphic to a closed subscheme of Pn is a projec-tive scheme over C.

One can obtain a closed subscheme X ⇢ Pn by glueing n + 1 a˘ne schemesVi = Ui \ X for i = 0, . . . , n, where Vi = X \ An

i ; the structure sheaf ofVi is the restriction OX |V i . That is, Vi = Spec (C )i where Ci = Ri/ai withRi = R[X0/Xi, . . . ,Xn/Xi] and ai is an ideal of Ri . More generally,

Definition 3.3.6. If R is a ring, a projective scheme over R is an R-scheme that isisomorphic to a closed subscheme of PnR for some n � 0.

The main di¯erence between algebraic and projective varieties is that thelatter are compact over C in the usual topology. That is, if f : X ! Y is amorphism of projective varieties, then f (X ) is a closed subspace of Y . Moregenerally,

Proposition 3.3.7. Every complex projective scheme is compact in the complextopology.

Proof. theorem 13.40, [UG10]. ⇤

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3.3.4. Proj. The functor Proj gives an alternative way of producing a projectivescheme (from a graded ring.) The construction is analogous to the Spec con-struction, where we produced an a˘ne scheme from a commutative ring. Incontrast, the points of the space from Proj are homogeneous prime ideals of Rthat do not contain the irrelevant ideal.

For convenience, we will first recall a few definitions.

Definition 3.3.8. A graded ring is a ring S , together with a decompositionS =

…d�0 Sd of S into the direct sum of abelian groups Sd , such that for any

d, e � 0, Sd · Se ⇢ Sd+e .

An element of Sd is called a homogeneous element of degree d.

Definition 3.3.9. An ideal a ⇢ S is a homogeneous ideal if

a = d�0

(a \ Sd ) .

We callS+ =

 d>0

Sd

the irrelevant ideal.

Also recall that a homogeneous ideal p is prime if and only if for any twohomogeneous elements a, b 2 S, ab 2 p implies a 2 p or b 2 p.

We are now ready to begin. Let S be a graded ring. We will first define theset Proj S and specify a topology before constructing the structure sheaf thatmakes it a projective scheme.

Define Proj S to be the set of all homogeneous prime ideals that do notcontain the irrelevant ideal S+. Next, we specify a topology on Proj S by settingthe closed sets to be the sets of the form

V (a) = {p 2 Proj S | a ⇢ p}

where a is a homogeneous ideal of S . To see that this indeed defines a topology,let {ai}i2I be a family of homogeneous ideals of S , then we clearly have

ŸV (ai) = V

⇣’ai

⌘.

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On the other hand if we have a and b two homogeneous ideals in S , we haveV (b) [V (a) = V (ab).

We will now attach to the space Proj S a structure of a sheaf P . This sheafwill in the end give us a structure of a scheme on Proj S .We will follow [DE07].As we did (§3.1.2) for the sheaves eM , we will just specify the structure on eachof a basis of open sets in the topology we just assigned.

Denote by X f the open set sp(Proj S) � V�h f i

�, where f is any homoge-

neous element of S of degree 1. So, X f does not contain f and S+. We canidentify the points of X f with the homogeneous primes of S[ f �1]. In fact, wecan identify X f with Spec

�S[ f �1]0

�, where S[ f �1]0 denotes the primes of the

ring of elements of degree 0 in S[ f �1]. That is, we have an open a˘ne subschemeof Proj S with the structure of the a˘ne scheme we just identified. Denote thissubscheme by (Proj S) f .

Now Proj S is a scheme via the a˘ne open covering of Proj S given by opensets (Proj S)xi = Proj S �V (xi), where xi are elements of degree 1 generating anideal whose radical is S+. To verify this we will check for the gluing condition;let � 2 S be another element of degree 1. Then, with the spectrum

S[ f �1]0[(�/ f )�1] = S[ f �1, ��1]0,

the open set (Proj S) f \ (Proj S)� is an open a˘ne subset of (Proj S) f . By sym-metry of the expressions, we have verified that

((Proj S) f )(�/ f ) = ((Proj S)� )( f /� ).

Hence, (Proj S,P) is a scheme.In fact, if S = C[x0, . . . , xn] the polynomial ring graded by the degree of

homogeneous polynomials, then Pn = ProjC [x0, . . . , xn] with

Xxi = Spec (R[x0/xi, . . . , xn/xi])

for all i = 0, . . . , n. More generally, we can redefine the projective n-space over aring R.

Definition 3.3.10. If R is a ring, we define projective n-space over R to be thescheme PnA = Proj R[X0, . . . ,Xn].

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4 Analytic Spaces

4.1 Definitions

If the coherent algebraic sheaf is defined on an algebraic variety X , then acoherent analytic sheaf is defined on the associated analytic space X � of X .Recall that the algebraic variety X intrinsically comes with the Zariski topology;the same set with the usual topology is what we call the associated analytic spaceX � . First we will define analytic varieties.

Definition 4.1.1. IfU ⇢ Cn is a domain, Z is an analytic variety if it is the zeroset of functions f1, . . . , fq regular onU .

Lastly, we will introduce the structure sheaf OZ on the analytic variety Z . IfU is open in Z , let OZ = HU /IZ , where HU is the restriction of the sheaf ofgerms of holomorphic functions ontoU , and IZ = ( f1, . . . , fq ) for some q 2 Z+.Then

Definition 4.1.2. An analytic space (X ,OX ) is a ringed space such that aroundevery point x 2 X , there exists an open neighborhoodU ⇢ X such that (U ,OU )is isomorphic to an analytic variety (Z,OZ ).

If U ⇢ Cn, then we say that U is analytic if, for each x 2 U , there arefunctions f1, . . . , fq , holomorphic in a neighborhood V of x, such that V \Uis identical to the set of points x 2 V satisfying the equations fi(z) = 0 fori = 1, . . . , q .

An interesting revelation of Chow’s theorem is that while it is not surprisingthat algebraic subspaces could be analytic, on the other hand, analytic subspacesof a projective space are indeed algebraic. To illustrate our point consider thefollowing example.

Certainly, the set where sin x vanishes in A1 is analytic. However, oneobserves that if V is a set of zeroes of a finite number of polynomials (that arenot all zero), V is finite. Hence, algebraic sets are either finite or the wholeparent space. Going back to sin x, its set of roots is infinite and as a result, thisanalytic set is not algebraic! Granted, A is not compact or projective and Chow’stheorem still holds.

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4.2 Analytic Sheaves

We call a sheaf that is a sheaf of H -modules an analytic sheaf.Let X ⇢ C n. If C (X )x denotes the sheaf of germs of continuous functions de-

fined on X defined at x , then the sheaf Hx,X of germs of holomorphic functionson X at x is a subsheaf of C (X )x . Now, if

✏ x : C (Cn)x �! C (X )x

is the restriction of the germs of functions in Cn to X , then clearly ✏ x [Hx ] =Hx,X , where H is the holomorphic functions defined on Cn. Moreover, ifAx (X ) denotes ker ✏ x, we can identify Hx,X as the quotient Hx/Ax (X ). Lastly,as HX is the union of Hx,X over x 2 X , we would like HX to be identical tothe restriction of H /A (X ) to X .

Now, suppose Y is a closed analytic subset of X , then under a similarconstruction as above, we identify Hx,Y as Hx,X/Ax,Y z , which is a stalk ofHX/Ax (Y ). But since Y closed, HX/Ax (Y ) is zero outside of Y , i.e., it isconcentrated on Y ; hence, its restriction to Y is HY .

4.3 Coherent Analytic Sheaves

An analytic sheaf is a sheaf of modules over the sheaf of rings HX ; that is,it is asheaf of HX -modules.

Proposition 4.3.1. —

1. The sheaf HX is a coherent sheaf of rings.

2. If Y is a closed analytic subspace of X , the sheaf A (Y ) is a coherent analyticsheaf.

Proof. no.3, prop. 1 [Ser56] ⇤

5 Analytification

In the rest of this section, we will construct a coherent analytic sheaf from acoherent algebraic sheaf on any scheme X . In the next chapter we will focus onjust Z-closed subvarieties of the projective space PnC

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By analytification, we mean two things. One, the process of assigning ananalytic space structure to a scheme of finite type over C; on the other hand, wealso mean associating an analytic sheaf to any (algebraic) sheaf on an a variety.Both processes are functorial.

5.1 Associating an analytic sheaf to a sheaf on a variety

In this section, we will be working with the equivalent definition of sheaves dueto Serre [Ser55] Let X be an algebraic variety, and F is any sheaf on X .

Now X � denotes the analytic space associated with X . Let F be any sheafon X .We want to make it into a sheaf on X � . Further, we remind the reader⇡ : F ! X is a projection.

To make F into a sheaf on X � , we will assign to it an appropriate topology3.To do so, consider the injection of F into X � ⇥ F by sending f 2 F to(⇡( f ), f ). That is, F inherits the product topology. We denote this new sheafon sp(F ) by F 0. This in fact makes F into a structure sheaf on X � as for anyx 2 X , the stalk F 0

x sits inside the product as (x,Fx ) which we can identify asFx .

Remark 5.1.1. The sheaf F 0 is the inverse image of F under the continuousmap X � ! X .

Definition 5.1.2. Let F be an algebraic sheaf on X . Then the analytic sheafassociated to F is defined by

F � = F 0 ⌦OX H ,

where the tensor is taken over the sheaf of rings and H is the sheaf of germs ofholomorphic functions defined on X .

As the tensor results a change in the ring of operators of F 0 to H , the newsheaf F � is a sheaf of H -modules.

Proposition 5.1.3. The functor F � is exact and for every algebraic sheaf F , thehomomorphism ↵ : F 0 ! F � is injective.

Proof. no.9, prop. 10 [Ser56]. ⇤3For now, by F we mean sp(F ), that is F as a set, unless otherwise stated.

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5.2 Analytic space associated to a scheme

Given a scheme X , we will attach to it an analytic structure. We will denotethe resulting analytic space by X � . Cover X with a˘ne subsets Spec (Ai) whereeach Ai is an algebra of finite type over C. That is, each

Ai � C[x1, . . . , xn]/( f1, . . . , fq )

for some polynomials f1, . . . , fq .Since each polynomial f1, . . . , fq is holomorphic their set of zeros is a com-

plex analytic subspace (Yi)� ⇢ Cn . As we glue Spec (Ai) to get the scheme X ,with the same gluing data, gluing the Yi gives an analytic space X � .

6 GAGA & Chow’s Theorem

6.1 GAGA

Theorem 6.1.1. In full generality, if X is a projective variety, then the followinghold:

1. For every coherent algebraic sheaf F on X, and for every integer q � 0, thenatural maps

✏ : H q (X ,F ) ! H q (X �,F �)are isomorphisms.

2. If F and G are two coherent algebraic sheaves on X , every analytic homomor-phism of F � into G � comes from a unique algebraic homomorphism of Finto G .

3. For every coherent analytic sheaf M on X � , there exists a coherent algebraicsheaf F on X such that F � is isomorphic to M . Moreover, this propertydetermines F in a unique fashion, up to isomorphism.

6.2 Chow’s Theorem

Corollary 6.2.1 (Chow). Let P be a scheme of finite type over C. Assume P iscompact. Then any closed analytic subspace of P is algebraic.

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Proof. Let X be a projective space, and let Y be a closed analytic subset of X � .Suppose i : Y ! X � is the inclusion map. By Cartan’s theorem [Ser56, no.3, prop 1], HY = HX/A (Y ) is a coherent analytic sheaf on X � . By GAGA,there exists a coherent algebraic sheaf F on X such that HY = F � . Since thehomomorphism ↵ : F 0 �! F � is injective, the support of F � equals thesupport of F . Since F is coherent, the support is Z-closed by Corollary 2.3.13.But F � =HY , so Y is Z-closed as required. ⇤

6.3 Consequences of Chow’s Theorem

Chow’s theorem a˘rms the existence of nonconstant meromorphic functionson a compact Riemann surface. Embed the surface into a complex n-dimensionalprojective space Pn as a complex submanifold for a suitable n. Then the surfaceacquires the structure of a smooth projective algebraic curve [Bal10]. Hence,embedded Riemann surfaces as well us meromorphic functions on them can alsobe described algebraically.

Corollary 6.3.1. Every compact Riemann surface is an algebraic curve.

That is, as any analytic subset of Pn is an algebraic subset, any complexsubmanifold of Pn is an algebraic submanifold [Kod12]. On the other hand,we have that any algebraic curve is a compact Riemann surface—an algebraicmanifold of dimension 1 (since we are working over C).

Another consequence is the following:

Corollary 6.3.2 ( [Ara65]:p.264). A holomorphic map between nonsingular projec-tive algebraic varieties is a morphism of varieties.

7 Acknowledgments

I would like to first thank Prof. Fernando Gouvêa for his patience and guidanceduring this year long project. I have learned a lot of Algebraic geometry and itwould not have been possible without the extremely helpful discussions we havehad. I would also like to thank Prof. David Krumm for agreeing to be a facultyreader of this thesis and providing helpful suggestions.

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8 References

[Ara65] Donu Arapura. Algebraic Geometry over the Complex Numbers. Uni-versitext. Springer, 1965.

[Bal10] T. E. Venkata Balaji. An Introduction to Families, Deformations andModuli. Universitätsverlag Göttingen, 2010.

[DE07] Joe Harris David Eisenbud. The Geometry of Schemes. Graduate Textsin Mathematics. Springer, 2007.

[Har97] Robert Hartshorne. Algebraic Geometry. Graduate Texts in Mathe-matics. Springer, 1997.

[Hor90] Lars Hormander. An Introduction to Complex Analysis in SeveralVariables. North-Holland Mathematical Library. North Holland,1990.

[Kod12] Kunihiko Kodaira. Complex Manifolds and Deformation of Com-plex Structures. A Series of Comprehensive Studies in Mathematics.Springer, 2012.

[Liu06] Qing Liu. Algebraic Geometry and Arithmetic Curves. Oxford Gradu-ate Texts in Mathematics. Oxford University Press, 2006.

[RCG65] Hugo Rossi Robert C. Gunning. Analytic Functions of Several ComplexVariables. Prentice-Hall series in Modern Analysis. Prentice-Hall,1965.

[Ser55] J.-P. Serre. Faisceaux algebriques coherents. The Annals of Mathematics,61:197–278, 1955.

[Ser56] J.-P. Serre. Géometrie algébrique et géométrie analytique. Annales del’Institut Fourier, 6:1–42, 1956.

[UG10] Torsten Wedhorn Ulrich Görtz. Algebraic Geometry: Part I: Schemes.Advanced Lectures in Mathematics. Vieweg+Teubner Verlag, 2010.

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