Introduction
This chapter develops the complex-analytic foundations needed to state and prove the comparison theorems usually called GAGA. The initial sections use the reduced notion of complex analytic space in [GAGA, §1, nos. 1–4, pp. 3–7]. This scope is recorded explicitly because a structure sheaf contained in a sheaf of functions has no nilpotents. The abstract language of ringed spaces and coherent modules is that of Sheaves, Definition 0091 and Modules, Section 01BU.
Analytic subsets of complex affine space
Definition
Let \(n \geq 0\) and endow \(\mathbf C^n\) with its usual topology. A subset \(U \subset \mathbf C^n\) is an analytic subset in the sense of Serre if, for every \(x \in U\), there are an open neighbourhood \(W\) of \(x\) and holomorphic functions \(f_1, \ldots, f_k\) on \(W\) such that \[U \cap W = \{z \in W \mid f_1(z)=\ldots=f_k(z)=0\}.\]
Let \(\mathcal C_U\) be the sheaf of germs of all complex-valued functions on \(U\). Let \(\mathcal O_{\mathbf C^n}\) be the sheaf of holomorphic functions on \(\mathbf C^n\). The sheaf of holomorphic functions on \(U\) is the image sheaf \[\mathcal O_U = \mathop{\rm Im}\left( \mathcal O_{\mathbf C^n}|_U \longrightarrow \mathcal C_U \right).\] Thus, if \(x \in U\) and \(\mathcal I_{U,x}\) is the ideal of germs in \(\mathcal O_{\mathbf C^n,x}\) which vanish on \(U\) near \(x\), then \[\mathcal O_{U,x}= \mathcal O_{\mathbf C^n,x}/\mathcal I_{U,x}.\]
Definition
Let \(U \subset \mathbf C^r\) and \(V \subset \mathbf C^s\) be analytic subsets. A map \(\varphi:U\to V\) is holomorphic if it is continuous and composition with \(\varphi\) sends every germ in \(\mathcal O_{V,\varphi(x)}\) to a germ in \(\mathcal O_{U,x}\) for every \(x\in U\). Equivalently, the \(s\) coordinate functions of \(\varphi\) are sections of \(\mathcal O_U\) locally on \(U\).
A bijective holomorphic map is an analytic isomorphism if its inverse is holomorphic.
Lemma
Let \(U \subset \mathbf C^n\) and \(U' \subset \mathbf C^{n'}\) be analytic subsets in the sense of Definition gaga-definition-analytic-subset.
The subset \(U\) is locally closed and locally compact.
The stalk \(\mathcal O_{U,x}\) is a reduced local \(\mathbf C\)-algebra whose residue homomorphism is evaluation at \(x\).
The subset \(U\times U'\) is analytic in \(\mathbf C^{n+n'}\), its topology is the product topology, and its structure sheaf is the one obtained from the product analytic charts.
Products of holomorphic maps are holomorphic.
Proof
After shrinking around \(x\in U\), the set \(U\) is the common zero locus of finitely many continuous functions and hence is closed in an open subset of \(\mathbf C^n\). This proves (1). The displayed quotient in Definition gaga-definition-analytic-subset embeds into \(\mathcal C_{U,x}\), and a germ is invertible exactly when its value at \(x\) is nonzero. This proves (2).
Near \((x,x')\), concatenate local defining equations for \(U\) and \(U'\). This proves analyticity of the product. The remaining assertions in (3) and (4) follow directly from the induced topology, the coordinate criterion in Definition gaga-definition-holomorphic-map-analytic-subsets, and composition of holomorphic functions.
Serre complex analytic spaces
Definition
A complex analytic space in the sense of Serre is a Hausdorff topological space \(X\) together with a subsheaf of \(\mathbf C\)-algebras \[\mathcal O_X \subset \mathcal C_X\] such that \(X\) has an open covering \(X=\bigcup V_i\) for which every ringed space \((V_i,\mathcal O_X|_{V_i})\) is analytically isomorphic to an analytic subset of some \(\mathbf C^{n_i}\) endowed with the sheaf of Definition gaga-definition-analytic-subset.
A map of such spaces is holomorphic if it is continuous and pullback by composition sends germs of holomorphic functions to germs of holomorphic functions. An isomorphism is a holomorphic map with holomorphic inverse.
Remark
Because \(\mathcal O_X\) is a subsheaf of a sheaf of functions, every \(\mathcal O_{X,x}\) is reduced. Thus Definition gaga-definition-serre-analytic-space is the reduced 1956 notion used by Serre. It does not include nilpotent analytic thickenings. Any later extension to nonreduced complex analytic spaces must be introduced separately and must not be attributed to GAGA, Definition 1.
Definition
Let \(X\) be a Serre complex analytic space. A subset \(Y\subset X\) is an analytic subspace if, in every analytic chart \(V\to U\), the image of \(Y\cap V\) is an analytic subset of \(U\). It has the induced analytic-space structure. When \(Y\) is closed in \(X\), we call it a closed analytic subspace.
Lemma
Let \(X\) and \(X'\) be Serre complex analytic spaces.
Every analytic subspace of \(X\) is locally closed and the induced structure is independent of the chosen charts.
There is a unique Serre complex analytic-space structure on \(X\times X'\) for which products of charts are charts. Its topology is the product topology.
Serre complex analytic spaces and holomorphic maps form a category.
Proof
All three assertions are local on the source spaces. They therefore follow from Lemma gaga-lemma-analytic-subset-basic-properties, compatibility of the restriction maps defining the structure sheaves, and the coordinate criterion for holomorphic maps.
Analytic modules and Oka–Cartan coherence
Definition
Let \(X\) be a Serre complex analytic space. An analytic module on \(X\) is an \(\mathcal O_X\)-module. It is coherent if it is coherent in the sense of Modules, Definition 01BV.
Definition
Let \(i:Y\to X\) be a closed analytic subspace. The vanishing ideal \(\mathcal J_Y\subset\mathcal O_X\) has stalk \[(\mathcal J_Y)_x= \{f\in\mathcal O_{X,x}\mid f|_Y=0\text{ near }x\}.\] There is a canonical isomorphism \[\mathcal O_X/\mathcal J_Y \longrightarrow i_*\mathcal O_Y.\]
Theorem
Let \(X\) be a Serre complex analytic space.
The \(\mathcal O_X\)-module \(\mathcal O_X\) is coherent.
If \(Y\subset X\) is a closed analytic subspace, then its vanishing ideal \(\mathcal J_Y\) is a coherent \(\mathcal O_X\)-module.
Proof
The analytic input is the Oka–Cartan theorem that, for an open subset \(\Omega\subset\mathbf C^n\), the sheaf \(\mathcal O_\Omega\) and the vanishing ideal of every closed analytic subset of \(\Omega\) are coherent.
The assertions are local on \(X\). Choose a chart and shrink it so that \(X=Z(\mathcal I_X)\) is a closed analytic subset of an open \(\Omega\subset\mathbf C^n\). If \(Y\subset X\) is closed analytic, shrink again and write \(Y=Z(\mathcal I_Y)\) with \(\mathcal I_X\subset\mathcal I_Y\subset\mathcal O_\Omega\). Then \[\mathcal O_X=\mathcal O_\Omega/\mathcal I_X, \qquad \mathcal J_Y=\mathcal I_Y/\mathcal I_X.\] The Oka–Cartan input makes \(\mathcal O_\Omega\), \(\mathcal I_X\), and \(\mathcal I_Y\) coherent. Modules, Lemmas 01BY and modules-lemma-coherent-change-rings now give (1) and (2).
Remark
The sheaf of holomorphic sections of a holomorphic vector bundle is finite locally free and hence coherent by Theorem gaga-theorem-oka-cartan-coherence; compare [Tohoku, Chapter V, Section 5.6, p. 219]. Serre also records the sheaves of germs of automorphic functions from the Cartan seminar as coherent examples; this cited analytic construction is not needed for the foundation statements above.
Local analytic algebras
Lemma
Let \(X\) be a Serre complex analytic space and \(x\in X\). There are an integer \(n\geq 0\) and a radical ideal \(\mathfrak a\subset\mathbf C\{z_1,\ldots,z_n\}\) such that \[\mathcal O_{X,x}\cong \mathbf C\{z_1,\ldots,z_n\}/\mathfrak a\] as local \(\mathbf C\)-algebras. Its maximal ideal consists of germs vanishing at \(x\), its residue field is \(\mathbf C\), and it is reduced and Noetherian.
Proof
Choose an analytic chart at \(x\) and translate its image point to the origin. Definition gaga-definition-analytic-subset gives the displayed quotient. The ideal is radical because it is the ideal of germs vanishing on a set. The residue and reducedness statements also follow from Lemma gaga-lemma-analytic-subset-basic-properties. The remaining analytic input is the Weierstrass theorem that \(\mathbf C\{z_1,\ldots,z_n\}\) is Noetherian; the quotient is then Noetherian by Algebra, Lemma 00FN.
Lemma
The local analytic \(\mathbf C\)-algebra \(\mathcal O_{X,x}\) determines the germ of a Serre complex analytic space \(X\) at \(x\). More precisely, an isomorphism of local analytic \(\mathbf C\)-algebras \(\mathcal O_{X,x}\cong\mathcal O_{Y,y}\) is induced contravariantly by an isomorphism of analytic germs \((X,x)\cong(Y,y)\).
Proof
Present both local algebras as quotients of convergent power-series rings. Images of the coordinate germs under a local homomorphism are convergent germs and hence define a holomorphic map germ. The relations force its image to lie in the required analytic germ. Applying the same construction to the inverse algebra homomorphism gives inverse holomorphic germs.
Lemma
Let \(X\) be a Serre complex analytic space and \(x\in X\). The following are equivalent for \(n\geq 0\):
the germ \((X,x)\) is isomorphic to \((\mathbf C^n,0)\);
\(\mathcal O_{X,x}\cong\mathbf C\{z_1,\ldots,z_n\}\);
\(\mathcal O_{X,x}\) is a regular local ring of dimension \(n\) in the sense of Algebra, Definition 00KU.
Such an \(x\) is a simple point of dimension \(n\). If every point of \(X\) is simple, then \(X\) is an analytic variety.
Proof
The equivalence of (1) and (2) follows from Lemma gaga-lemma-analytic-germ-local-algebra. The implication (2) \(\Rightarrow\) (3) is the standard dimension computation for convergent power-series rings. The converse is the regular analytic local-algebra theorem, proved from the analytic inverse-function theorem (equivalently, by choosing a basis of \(\mathfrak m_x/\mathfrak m_x^2\) and applying the analytic implicit function theorem). This analytic theorem is an explicit external dependency of the present foundation block.
Lemma
Let \(X\) be a Serre complex analytic space and \(x\in X\).
The irreducible components \(X_i\) of the germ \((X,x)\) correspond to the finitely many minimal primes \(\mathfrak p_i\) of \(\mathcal O_{X,x}\), and \[0=\bigcap_i\mathfrak p_i, \qquad \mathcal O_{X_i,x}=\mathcal O_{X,x}/\mathfrak p_i.\]
The analytic dimension of \(X\) at \(x\) (one half of its local topological dimension) is the supremum of the dimensions of the \(X_i\) and equals the Krull dimension of \(\mathcal O_{X,x}\).
Proof
Reducedness and Noetherianity are given by Lemma gaga-lemma-local-analytic-algebra. The algebraic assertion about the finiteness and intersection of minimal primes follows from Algebra, Lemmas 00FR and 00E0. The correspondence with analytic components uses the local analytic Nullstellensatz. For (2), the analytic normalization/parameter theorem says that an irreducible analytic local algebra of analytic dimension \(r\) is finite over \(\mathbf C\{z_1,\ldots,z_r\}\). Dimension invariance under finite integral extensions then gives Krull dimension \(r\). The analytic Nullstellensatz and local parameter theorem are explicit external dependencies; no algebraic Stacks result alone supplies them.
Analytification of algebraic varieties
Throughout this section an algebraic variety over \(\mathbf C\) means a reduced separated scheme of finite type over \(\mathbf C\), not necessarily irreducible. Its classical points are its closed points, equivalently its \(\mathbf C\)-points. This convention is the scheme-theoretic translation of the convention in [GAGA, §2, no. 5] and is compatible with the reduced analytic spaces of Section gaga-section-serre-analytic-spaces.
Lemma
Let \(U \subset \mathbf C^n\) and \(U' \subset \mathbf C^{n'}\) be Zariski locally closed reduced subsets.
The Zariski topology on \(\mathbf C^n\) is coarser than its usual topology.
The set \(U\), with its reduced induced structure, is a Serre complex analytic space.
A regular map \(f : U \to U'\) is holomorphic.
A biregular isomorphism \(U \to U'\) is an analytic isomorphism.
Proof
A Zariski closed subset is the common zero set of polynomials, and polynomials are continuous and holomorphic. This proves (1) and proves (2) for a closed subset; a locally closed subset is open in its closure, and restriction gives the general case. A regular function on a Zariski open subset is locally a quotient \(g/h\) of polynomials with \(h\) nonvanishing. Hence it is holomorphic. Applying this coordinate by coordinate proves (3). Applying (3) to \(f\) and to \(f^{-1}\) proves (4).
Proposition
Let \(X\) be an algebraic variety over \(\mathbf C\). There is a unique Serre complex analytic space \(X^{an}\) with the following property: for every algebraic chart \[\varphi : V \longrightarrow U \subset \mathbf C^n\] from a Zariski open \(V \subset X\) onto a Zariski locally closed reduced subset, \(V\) is open in \(X^{an}\) and \(\varphi\) is an analytic isomorphism.
The construction is functorial. Every morphism \(f : X \to Y\) induces a holomorphic map \(f^{an} : X^{an} \to Y^{an}\), and there is a canonical morphism of locally ringed spaces \[a_X : (X^{an}, \mathcal O_{X^{an}}) \longrightarrow (X, \mathcal O_X)\] whose map on points sends a classical point to the corresponding closed scheme point and whose map on structure sheaves sends a regular function to the same function viewed as holomorphic.
Proof
Cover \(X\) by finitely many algebraic charts \(V_i \to U_i\). Transport the analytic structure of \(U_i\) to \(V_i\). On \(V_i \cap V_j\) the transition maps and their inverses are holomorphic by Lemma gaga-lemma-algebraic-charts-analytic; the overlap is open for the usual topology by the same lemma. The cocycle condition is inherited from the algebraic charts. The sheaves and the locally ringed spaces therefore glue by Sheaves, Lemma 00AM.
It remains to check the Hausdorff axiom, rather than hide it in the word “glue”. The inverse image of the diagonal of the glued point set in every \(V_i \times V_j\) is the graph of the algebraic identification on the overlap. Since \(X\) is separated, this graph is Zariski closed; it is closed in the usual topology by Lemma gaga-lemma-algebraic-charts-analytic. These products form an open cover, so the diagonal is closed and the glued space is Hausdorff. Uniqueness follows because the charts cover \(X\) and determine both its topology and its structure sheaf.
For a morphism \(f : X \to Y\), its restrictions in algebraic charts are holomorphic by Lemma gaga-lemma-algebraic-charts-analytic; they glue to \(f^{an}\), compatibly with identities and composition. The usual topology is finer than the Zariski topology, so the point map underlying \(a_X\) is continuous. Regular functions are holomorphic in charts, and the induced maps on stalks are local. This constructs the asserted morphism of locally ringed spaces.
Lemma
Let \(X\) and \(Y\) be algebraic varieties over \(\mathbf C\).
The space \(X^{an}\) is locally compact, second countable, and countable at infinity.
There is a canonical analytic isomorphism \((X \times_{\mathbf C} Y)^{an} \cong X^{an} \times Y^{an}\).
If \(Z \subset X\) is a Zariski locally closed reduced subvariety, then \(Z^{an}\) is the reduced analytic subspace induced on \(Z(\mathbf C)\) by \(X^{an}\).
Proof
A Zariski locally closed subset of \(\mathbf C^n\) is locally compact, second countable, and a countable union of compact subsets. The same properties pass to a finite union of open chart domains, proving (1). Products of algebraic charts are algebraic charts and the usual topology on a product of subsets of complex affine spaces is the product topology; uniqueness in Proposition gaga-proposition-analytification proves (2). Restricting charts of \(X\) to \(Z\) and using Lemma gaga-lemma-algebraic-charts-analytic proves (3).
Local algebra of analytification
Let \(x \in X(\mathbf C)\). We write \[\theta_x : \mathcal O_{X,x} \longrightarrow \mathcal O_{X^{an},x}\] for the local homomorphism induced by \(a_X\). Both rings are Noetherian local rings with residue field \(\mathbf C\): this is standard for the algebraic ring, and for the analytic ring it is Lemma gaga-lemma-local-analytic-algebra.
Proposition
Let \(Z \subset X\) be a Zariski locally closed reduced subvariety and let \(x \in Z(\mathbf C)\). After replacing \(X\) by a Zariski neighbourhood of \(x\), view \(Z\) as closed. If \(\mathcal J_{Z,x} \subset \mathcal O_{X,x}\) is its algebraic ideal and \(\mathcal I_{Z^{an},x} \subset \mathcal O_{X^{an},x}\) is the ideal of holomorphic germs vanishing on \(Z^{an}\), then \[\mathcal I_{Z^{an},x} = \mathcal J_{Z,x}\mathcal O_{X^{an},x}.\] Consequently \[\mathcal O_{Z^{an},x} \cong \mathcal O_{X^{an},x}/ \mathcal J_{Z,x}\mathcal O_{X^{an},x}.\]
Proof
First take \(X=\mathbf A^n_{\mathbf C}\) and translate \(x\) to the origin. Put \[R=\mathbf C[z_1,\ldots,z_n]_{(z_1,\ldots,z_n)}, \qquad H=\mathbf C\{z_1,\ldots,z_n\}.\] Let \(J \subset R\) be the radical ideal of \(Z\). The local analytic Nullstellensatz says that the ideal of holomorphic germs vanishing on \(Z\) is \(\sqrt{JH}\). We show that \(JH\) is already radical.
The Taylor homomorphisms identify the maximal-adic completions of both \(R\) and \(H\) with \(\mathbf C[[z_1,\ldots,z_n]]\). Completion is exact on finite modules by Algebra, Lemma 00MA; hence the completion of \(H/JH\) is \[\mathbf C[[z_1,\ldots,z_n]]/ J\mathbf C[[z_1,\ldots,z_n]],\] which is also the completion of \(R/J\). The field \(\mathbf C\) is excellent by More on Algebra, Proposition 07QW, and \(R/J\), being a localization of a finite type \(\mathbf C\)-algebra, is excellent by More on Algebra, Lemma 07QU. It is therefore Nagata by More on Algebra, Lemma 07QV. Its completion is reduced by More on Algebra, Lemma 07NZ. Thus the completion of \(H/JH\) is reduced. The ring \(H/JH\) injects into its completion by Krull’s intersection theorem, Algebra, Lemma 00IP; hence \(H/JH\) is reduced and \(JH=\sqrt{JH}\). The analytic Nullstellensatz now gives the desired equality in affine space.
For general \(X\), work in an algebraic chart which realizes \(X\) and \(Z\) as reduced locally closed subsets of one affine space. Apply the affine result to their ambient ideals and pass to the quotient. This gives both displayed formulas.
Theorem
For every algebraic variety \(X\) over \(\mathbf C\) and every \(x \in X(\mathbf C)\), the local homomorphism \(\theta_x\) induces an isomorphism of completed local rings \[\widehat{\theta}_x : \widehat{\mathcal O}_{X,x} \xrightarrow{\ \sim\ } \widehat{\mathcal O}_{X^{an},x}.\] In particular, \(\theta_x\) is injective.
Proof
For affine space at the origin the map is \[\mathbf C[z_1,\ldots,z_n]_{(z_1,\ldots,z_n)} \longrightarrow \mathbf C\{z_1,\ldots,z_n\},\] and both maximal-adic completions are canonically \(\mathbf C[[z_1,\ldots,z_n]]\) by Taylor expansion and truncation. For a locally closed reduced \(X\) in affine space, Proposition gaga-proposition-analytification-closed-ideal identifies its analytic local ring with the quotient by the extended algebraic ideal. Exactness of completion on finite modules, Algebra, Lemma 00MA, identifies both completions with the same quotient of the formal power-series ring. The assertion is local, so charts prove the general case. Finally, \(\mathcal O_{X,x}\) injects into its completion by Algebra, Lemma 00IP, which proves the last sentence.
Theorem
The local homomorphism \[\theta_x : \mathcal O_{X,x} \longrightarrow \mathcal O_{X^{an},x}\] is faithfully flat. In Serre’s terminology, \((\mathcal O_{X,x},\mathcal O_{X^{an},x})\) is a flat pair: the quotient \(\mathcal O_{X^{an},x}/\mathcal O_{X,x}\) is flat over \(\mathcal O_{X,x}\), and for every ideal \(I \subset \mathcal O_{X,x}\) one has \[I\mathcal O_{X^{an},x}\cap\mathcal O_{X,x}=I.\]
Proof
The map on completions is an isomorphism by Theorem gaga-theorem-analytification-completed-local-rings, hence is flat. More on Algebra, Lemma 0C4G implies that \(\theta_x\) is flat, and Algebra, Lemma 00HR makes this local flat map faithfully flat. The contraction formula is Algebra, Lemma 05CK. Universal injectivity after tensoring the exact sequence \(0\to\mathcal O_{X,x}\to\mathcal O_{X^{an},x}\) shows that the displayed quotient is flat, which is precisely the flat-pair assertion.
Lemma
For every \(x \in X(\mathbf C)\), \[\dim(\mathcal O_{X,x})=\dim(\mathcal O_{X^{an},x}).\]
Proof
Each ring has the same dimension as its completion by More on Algebra, Lemma 07NV, and the two completions are isomorphic by Theorem gaga-theorem-analytification-completed-local-rings.
Lemma
If \(X\) is irreducible of dimension \(r\), then \(X^{an}\) has analytic dimension \(r\) at every point.
Proof
For a closed point \(x\), Varieties, Lemma 0A21, parts 0B18 and 0B19, gives \[\dim(\mathcal O_{X,x})=r.\] Apply Lemma gaga-lemma-analytification-local-dimension, followed by Lemma gaga-lemma-components-dimension-analytic-space, which identifies analytic dimension with the Krull dimension of the analytic local ring.
The classical and Zariski topologies
Lemma
Let \(U \subset X\) be Zariski open and Zariski dense. Then \(U^{an}\) is dense in \(X^{an}\).
Proof
Put \(Z=X\setminus U\) with its reduced induced structure. If a point \(x \in X^{an}\) were not in the closure of \(U^{an}\), then \(Z^{an}\) would contain an analytic neighbourhood of \(x\). Its local vanishing ideal at \(x\) would therefore be zero. Proposition gaga-proposition-analytification-closed-ideal and the injectivity in Theorem gaga-theorem-analytification-completed-local-rings would give \(\mathcal J_{Z,x}=0\). Thus \(Z\) would agree with \(X\) on a nonempty Zariski neighbourhood of \(x\), contradicting the Zariski density of \(U\).
Proposition
An algebraic variety \(X\) over \(\mathbf C\) is proper over \(\mathbf C\) if and only if the topological space \(X^{an}\) is compact.
Proof
Apply Chow’s lemma in the precise form of Limits, Lemma 0202. After replacing the source by its reduction as allowed by the following remark there, we obtain a proper surjective morphism \(\pi : U \to X\) and an immersion \(U \to \mathbf P^n_{\mathbf C}\). Let \(Y\) be the reduced closure of \(U\) in \(\mathbf P^n_{\mathbf C}\). Then \(Y\) is projective and \(U\) is Zariski open and dense in \(Y\). Moreover the graph of \(\pi\) is closed in \(X\times Y\): over \(X\) it is the graph of a morphism from the proper \(X\)-scheme \(U\) to the separated \(X\)-scheme \(X\times Y\).
If \(X\) is proper, then \(U\) is proper over \(\mathbf C\). Its immersion into projective space is proper and hence closed, so \(U=Y\). The space \(Y^{an}\) is a closed subset of the compact Hausdorff space \(\mathbf P^n(\mathbf C)\), hence compact. The continuous surjection \(\pi^{an}:Y^{an}\to X^{an}\) shows that \(X^{an}\) is compact.
Conversely, assume \(X^{an}\) compact. Then \(X^{an}\times Y^{an}\) is compact, and the analytification of the closed graph is closed in this product by Lemma gaga-lemma-analytification-properties. Its projection to \(Y^{an}\) is \(U^{an}\), hence \(U^{an}\) is compact and therefore closed in the Hausdorff space \(Y^{an}\). It is dense by Lemma gaga-lemma-zariski-dense-open-analytically-dense, so \(U^{an}=Y^{an}\). Thus \(U=Y\) because a nonempty closed subset of a finite type \(\mathbf C\)-scheme has a closed \(\mathbf C\)-point. Now the projective variety \(Y\) is proper and maps surjectively to \(X\); Morphisms, Lemma 03GN implies that \(X\) is proper.
Lemma
Let \(f : X \to Y\) be a morphism of algebraic varieties over \(\mathbf C\). If \(f(X)\) is Zariski dense in \(Y\), then \(f(X)\) contains a Zariski open dense subset of \(Y\).
Proof
The image is constructible by Morphisms, Lemma 054J. A dense constructible subset contains a dense open subset by Topology, Lemma topology-lemma-dense-in-constructible.
Proposition
Let \(f : X \to Y\) be a morphism of algebraic varieties. The closure of \(f(X(\mathbf C))\) in \(Y^{an}\) is the set of classical points of the Zariski closure of \(f(X)\) in \(Y\).
Proof
Let \(T\) be the reduced Zariski closure of \(f(X)\). Applying Lemma gaga-lemma-dense-image-contains-open to \(X\to T\) gives a Zariski open dense \(V\subset T\) contained in \(f(X)\). By Lemma gaga-lemma-zariski-dense-open-analytically-dense, \(V^{an}\) is dense in \(T^{an}\), so \(T^{an}\) is contained in the analytic closure of \(f(X)\). The opposite inclusion holds because \(T^{an}\) is closed in \(Y^{an}\) by Lemma gaga-lemma-analytification-properties.
An analytic criterion for regularity
Proposition
Let \(X\) and \(Y\) be algebraic varieties and let \(f : X^{an}\to Y^{an}\) be holomorphic. If its graph is the classical point set of a Zariski locally closed reduced subvariety \(T\subset X\times_{\mathbf C}Y\), then \(f\) is induced by a morphism \(X\to Y\).
Proof
The first projection \(p:T\to X\) is a morphism and is bijective on classical points. Its inverse \(x\mapsto(x,f(x))\) is holomorphic. By Lemma gaga-lemma-analytification-properties, \(p^{an}\) is therefore an analytic isomorphism. Proposition gaga-proposition-analytic-isomorphism-algebraic below shows that \(p\) is an isomorphism. Hence \(f=\operatorname{pr}_Y\circ p^{-1}\) is regular.
Proposition
Let \(p:T\to X\) be a morphism of algebraic varieties which is bijective on classical points. If \(p^{an}:T^{an}\to X^{an}\) is an analytic isomorphism, then \(p\) is an isomorphism.
Proof
First \(p\) is a homeomorphism for the Zariski topology on classical points. Indeed, if \(F\subset T\) is Zariski closed, then \(F^{an}\) is closed in \(T^{an}\). Its image under the analytic homeomorphism is closed in \(X^{an}\). By Proposition gaga-proposition-image-closure, applied to \(F\to X\), this image is Zariski closed.
Both schemes are Jacobson by Varieties, Lemma 0478. Hence their closed subsets are determined by their closed points. Passing from irreducible closed subsets to their unique generic points (Schemes, Lemma 01IS) extends the preceding homeomorphism uniquely to a homeomorphism of the full underlying scheme spaces.
Fix \(t\in T(\mathbf C)\) and put \(x=p(t)\). The Zariski homeomorphism makes \[A=\mathcal O_{X,x}\longrightarrow A'=\mathcal O_{T,t}\] injective. The analytic isomorphism identifies \(B=\mathcal O_{X^{an},x}\) with \(\mathcal O_{T^{an},t}\). By Theorem gaga-theorem-analytification-local-faithfully-flat, both \(A\) and \(A'\) embed in this common ring \(B\).
The irreducible components through \(x\) correspond to those through \(t\). Write their minimal primes as \(\mathfrak p_i\subset A\) and \(\mathfrak p'_i\subset A'\), with \(\mathfrak p'_i\cap A=\mathfrak p_i\). Put \[K_i=\operatorname{Frac}(A/\mathfrak p_i), \qquad K'_i=\operatorname{Frac}(A'/\mathfrak p'_i).\] The corresponding irreducible varieties are homeomorphic and have the same dimension, so \(K'_i/K_i\) is a finite extension. To see that its degree is one, use Morphisms, Lemma 02NX and generic flatness, Morphisms, Proposition 052A, to shrink the target to a nonempty open over which the component map is finite locally free. Since the ground field has characteristic zero, \(K'_i/K_i\) is separable; after shrinking once more the map is finite etale of degree \([K'_i:K_i]\). Every closed fibre then consists of that many reduced \(\mathbf C\)-points. Bijectivity of \(p\) forces \([K'_i:K_i]=1\).
Let \(S\) be the complement in \(A\) of the union of its minimal primes and define \(S'\) similarly for \(A'\). Lemma gaga-lemma-total-fractions-reduced below gives \[A_S=\prod_iK_i=\prod_iK'_i=A'_{S'}.\] If \(h\in A'\), write \(h=g/s\) in this common total ring of fractions with \(g\in A\) and \(s\in S\). Thus \(g=sh\in sB\). Faithful flatness and Algebra, Lemma 05CK, give \(sB\cap A=sA\), so \(g=sa\) for some \(a\in A\). Hence \(s(h-a)=0\). The element \(s\) lies in no minimal prime of the reduced ring \(A'\), and is therefore a nonzerodivisor by Algebra, Lemma 00EW. Thus \(h=a\) and \(A'=A\).
We have proved that the local maps are isomorphisms at every closed point. Thus \(p\) is etale at every closed point by Etale, Lemma 039L. The etale locus is open; its closed complement, if nonempty, would contain a closed point because \(T\) is Jacobson. Hence \(p\) is etale everywhere.
Finally, repeat the finite-degree argument above for the restriction of \(p\) to the reduced closure of an arbitrary point of \(T\). The Zariski homeomorphism identifies this closure with the closure of its image, and the same argument shows that the induced extension of their function fields, namely the residue field extension at the two generic points, has degree one. Thus \(p\) is injective on points with trivial residue extensions, hence radicial by Morphisms, Lemma 01S4. An etale radicial morphism is an open immersion by Etale, Theorem 025G. Since \(p\) is also surjective, it is an isomorphism.
Lemma
Let \(A\) be a reduced ring with finitely many minimal primes \(\mathfrak p_1,\ldots,\mathfrak p_r\). Let \(S\) be the set of elements lying in none of the \(\mathfrak p_i\) and put \(K_i=\operatorname{Frac}(A/\mathfrak p_i)\). Then \[A_S=\prod_{i=1}^rK_i.\]
Proof
Algebra, Lemma 00EW says that \(S\) is exactly the set of nonzerodivisors and that \(A_{\mathfrak p_i}=K_i\). Algebra, Lemma 02LX then identifies the total ring of fractions \(A_S\) with the displayed product.
Analytification of modules
Definition
Let \(X\) be an algebraic variety over \(\mathbf C\) and let \(a_X:X^{\mathrm{an}}\to X\) be the canonical morphism of locally ringed spaces of Proposition gaga-proposition-analytification. For an \(\mathcal O_X\)-module \(\mathcal F\), its it analytification is \[\mathcal F^{\mathrm{an}}=a_X^*\mathcal F =\mathcal O_{X^{\mathrm{an}}} \otimes_{a_X^{-1}\mathcal O_X}a_X^{-1}\mathcal F.\] There is a canonical map of \(a_X^{-1}\mathcal O_X\)-modules \[\eta_{\mathcal F}:a_X^{-1}\mathcal F\longrightarrow \mathcal F^{\mathrm{an}},\qquad s\longmapsto1\otimes s.\] Every \(\mathcal O_X\)-linear map \(\mathcal F\to\mathcal G\) induces an \(\mathcal O_{X^{\mathrm{an}}}\)-linear map \(\mathcal F^{\mathrm{an}}\to\mathcal G^{\mathrm{an}}\), functorially, and \(\mathcal O_X^{\mathrm{an}}=\mathcal O_{X^{\mathrm{an}}}\) canonically.
Lemma
Let \(x\in X(\mathbf C)=X^{\mathrm{an}}\). There are canonical identifications \[(\mathcal F^{\mathrm{an}})_x =\mathcal F_x\otimes_{\mathcal O_{X,x}} \mathcal O_{X^{\mathrm{an}},x}\] under which \((\eta_{\mathcal F})_x\) is the map \(m\mapsto1\otimes m\).
Proof
Stalks commute with inverse image and tensor product. Since \(x\) is a closed point, \((a_X^{-1}\mathcal F)_x=\mathcal F_x\). The displayed identification and the description of \(\eta_{\mathcal F}\) follow directly from Definition gaga-definition-analytification-module.
Lemma
Let \(X\) be an algebraic variety over \(\mathbf C\).
The functor \(\mathcal F\mapsto\mathcal F^{\mathrm{an}}\) on \(\mathcal O_X\)-modules is exact.
The map \(\eta_{\mathcal F}:a_X^{-1}\mathcal F\to \mathcal F^{\mathrm{an}}\) is injective for every \(\mathcal F\).
If \(\mathcal F\) is coherent, then \(\mathcal F^{\mathrm{an}}\) is a coherent analytic module.
Proof
The stalk formula of Lemma gaga-lemma-analytification-module-stalks and faithful flatness in Theorem gaga-theorem-analytification-local-faithfully-flat prove (1); compare Modules, Lemma 02N4. The same stalk formula and Algebra, Lemma 05CK prove (2).
For (3), a coherent module on the locally Noetherian scheme \(X\) is locally finitely presented. Pullback preserves finite presentations. The analytic structure sheaf is coherent by Theorem gaga-theorem-oka-cartan-coherence; hence a finitely presented analytic module is coherent by Modules, Lemma 01BZ.
Remark
If \(\mathcal J\subset\mathcal O_X\) is a quasi-coherent ideal, exactness identifies \(\mathcal J^{\mathrm{an}}\) with the ideal of \(\mathcal O_{X^{\mathrm{an}}}\) generated by the image of \(a_X^{-1}\mathcal J\). For the ideal of a reduced closed subvariety this is also Proposition gaga-proposition-analytification-closed-ideal.
Closed immersions and analytification
Lemma
Let \(i:Z\to X\) be a closed immersion of algebraic varieties over \(\mathbf C\), and let \(\mathcal F\) be a coherent \(\mathcal O_Z\)-module. There is a canonical isomorphism \[(i_*\mathcal F)^{\mathrm{an}} \longrightarrow i^{\mathrm{an}}_*(\mathcal F^{\mathrm{an}}).\]
Proof
The commutative analytification square and adjunction give the canonical map. Both sides vanish away from \(Z^{\mathrm{an}}\). At \(z\in Z(\mathbf C)\) the source stalk is \[\mathcal F_z\otimes_{\mathcal O_{X,z}} \mathcal O_{X^{\mathrm{an}},z},\] whereas the target stalk is \[\mathcal F_z\otimes_{\mathcal O_{Z,z}} \mathcal O_{Z^{\mathrm{an}},z}.\] Proposition gaga-proposition-analytification-closed-ideal gives \[\mathcal O_{Z^{\mathrm{an}},z} =\mathcal O_{Z,z}\otimes_{\mathcal O_{X,z}} \mathcal O_{X^{\mathrm{an}},z}.\] Associativity of tensor product identifies the two stalks, and the identification is the stalk of the canonical map.
The cohomology comparison map
Definition
Let \(\mathcal F\) be an \(\mathcal O_X\)-module. Exactness of \(a_X^*\) and the derived adjunction unit give \[\mathcal F\longrightarrow Ra_{X,*}a_X^*\mathcal F.\] After applying derived global sections and using \[R\Gamma(X,Ra_{X,*}\mathcal G) =R\Gamma(X^{\mathrm{an}},\mathcal G),\] we obtain canonical maps \[\epsilon^q_{\mathcal F}: H^q(X,\mathcal F)\longrightarrow H^q(X^{\mathrm{an}},\mathcal F^{\mathrm{an}}), \qquad q\geq0.\] For \(q=0\), this sends a section \(s\) to \(1\otimes s\). Thus it agrees with Serre’s map obtained from Zariski covers and their analytifications.
Lemma
The maps \(\epsilon^q_{\mathcal F}\) are natural in \(\mathcal F\). For every short exact sequence of \(\mathcal O_X\)-modules, they form a morphism from the algebraic long exact cohomology sequence to the analytic long exact cohomology sequence. In particular, they commute with connecting maps.
Proof
The derived adjunction unit is a natural transformation. Because analytification is exact by Lemma gaga-lemma-analytification-exact-faithful-coherent, it carries short exact sequences to short exact sequences. Naturality of the associated distinguished triangles gives the asserted morphism of long exact sequences.
The GAGA comparison theorems
Theorem
Let \(X\) be a projective algebraic variety over \(\mathbf C\) and let \(\mathcal F\) be a coherent \(\mathcal O_X\)-module. For every \(q\geq0\), the map \[\epsilon^q_{\mathcal F}:H^q(X,\mathcal F) \longrightarrow H^q(X^{\mathrm{an}},\mathcal F^{\mathrm{an}})\] is an isomorphism. In particular, \[\Gamma(X,\mathcal F)\longrightarrow \Gamma(X^{\mathrm{an}},\mathcal F^{\mathrm{an}})\] is an isomorphism.
Theorem
Let \(X\) be a projective algebraic variety over \(\mathbf C\) and let \(\mathcal F,\mathcal G\) be coherent \(\mathcal O_X\)-modules. Analytification induces a bijection \[\mathop{\rm Hom}\nolimits_{\mathcal O_X}(\mathcal F,\mathcal G) \longrightarrow \mathop{\rm Hom}\nolimits_{\mathcal O_{X^{\mathrm{an}}}} (\mathcal F^{\mathrm{an}},\mathcal G^{\mathrm{an}}).\]
Theorem
Let \(X\) be a projective algebraic variety over \(\mathbf C\). Every coherent \(\mathcal O_{X^{\mathrm{an}}}\)-module \(\mathcal M\) is isomorphic to \(\mathcal F^{\mathrm{an}}\) for a coherent \(\mathcal O_X\)-module \(\mathcal F\). The module \(\mathcal F\) is unique up to isomorphism; moreover, every analytic isomorphism between two such analytifications is induced by a unique algebraic isomorphism.
The proofs of these three theorems occupy the following sections.
Remark
For \(X=\mathbf A^1_{\mathbf C}\) and \(\mathcal F=\mathcal O_X\), the degree-zero comparison map is \[\mathbf C[z]\longrightarrow \Gamma(\mathbf C,\mathcal O_{\mathbf C}),\] whose target is the ring of entire functions. It is not surjective. Thus the projectivity hypothesis in the comparison theorems cannot simply be discarded.
Remark
The comparison map factors canonically as \[H^q(X,\mathcal F) \longrightarrow H^q(X^{\mathrm{an}},a_X^{-1}\mathcal F) \longrightarrow H^q(X^{\mathrm{an}},\mathcal F^{\mathrm{an}}).\] The first arrow need not be an isomorphism. For example, let \(X\) be an irreducible projective variety whose analytification has a nonzero Betti number \(b_q\) with \(q>0\), and put \(K=\mathbf C(X)\). The sheaf \(K\) is flasque on the irreducible Zariski space \(X\), so \(H^q(X,K)=0\). Its inverse image is the constant sheaf with value \(K\) on \(X^{\mathrm{an}}\), whose degree-\(q\) cohomology is a \(K\)-vector space of dimension \(b_q\). This is the obstruction recorded after the three theorems in [GAGA, §3, no. 12].
Proof of the cohomology comparison theorem
We first record the two analytic inputs used to start and bound the proof. They are deliberately visible: neither is a formal consequence of the algebraic cohomology calculation.
Lemma
Let \(r \geq 0\) and let \(\mathcal A\) be a sheaf of abelian groups on \(\mathbf P^r(\mathbf C)\) with its analytic topology. Then \[H^q(\mathbf P^r(\mathbf C),\mathcal A)=0 \quad\text{for }q>2r.\]
Proof
The underlying space of \(\mathbf P^r(\mathbf C)\) is a compact triangulable real manifold of dimension \(2r\). We use the topological cohomological dimension theorem saying that a paracompact Hausdorff space of covering dimension \(d\) has vanishing sheaf cohomology in degrees greater than \(d\). This covering-dimension theorem is an external topological input; the current Stacks chapter on Cohomology defines cohomological dimension but does not prove this comparison with covering dimension.
Lemma
For every \(r \geq 0\) and every \(q \geq 0\), the comparison map \[H^q(\mathbf P^r_{\mathbf C},\mathcal O) \longrightarrow H^q(\mathbf P^r(\mathbf C),\mathcal O^{\mathrm{an}})\] is an isomorphism.
Proof
By Cohomology of Schemes, Lemma 01XT, the group on the left is \(\mathbf C\) for \(q=0\) and is zero for \(q>0\). On the analytic side we use the Dolbeault theorem together with the standard computation \[H^{0,q}_{\overline\partial}(\mathbf P^r(\mathbf C))= \begin{cases} \mathbf C & q=0,\\ 0 & q>0. \end{cases}\] This Dolbeault–projective-space calculation is the external analytic input in this lemma; see [GAGA, §2, no. 13]. In degree zero the comparison map sends a constant algebraic function to the same constant holomorphic function, and hence is the identity on \(\mathbf C\). In positive degree both sides vanish.
Lemma
Let \(r \geq 0\) and \(n \in \mathbf Z\). For every \(q \geq 0\), the comparison map \[H^q(\mathbf P^r_{\mathbf C},\mathcal O(n)) \longrightarrow H^q(\mathbf P^r(\mathbf C),\mathcal O(n)^{\mathrm{an}})\] is an isomorphism.
Proof
We argue by induction on \(r\). The assertion for \(r=0\) is immediate. Suppose \(r>0\), choose a hyperplane \(i:E\longrightarrow\mathbf P^r_{\mathbf C}\), and identify \(E\) with \(\mathbf P^{r-1}_{\mathbf C}\) as in Varieties, Lemma 089Z. Multiplication by an equation of \(E\) gives, for every \(n\), a short exact sequence \[0\longrightarrow\mathcal O(n-1)\longrightarrow\mathcal O(n) \longrightarrow i_*\mathcal O_E(n)\longrightarrow0.\] Analytification is exact by Lemma gaga-lemma-analytification-exact-faithful-coherent and commutes with the closed-immersion pushforward by Lemma gaga-lemma-analytification-pushforward-closed-immersion. Thus this sequence and its analytification give two long exact cohomology sequences. The comparison maps form a morphism between them by Lemma gaga-lemma-analytification-cohomology-map-functorial.
The induction hypothesis gives the comparison isomorphisms for \(\mathcal O_E(n)\) in all degrees. The four and five lemmas (Homology, Lemmas 05QA and 05QB) now show that comparison for \(\mathcal O(n)\) in every degree is equivalent to comparison for \(\mathcal O(n-1)\) in every degree. The case \(n=0\) is Lemma gaga-lemma-projective-structure-sheaf-comparison; induction upward and downward in \(n\) proves the result for every integer.
Proof
Let \(X\) be projective over \(\mathbf C\), let \(\mathcal F\) be coherent, and choose a closed immersion \(i:X\to\mathbf P^r_{\mathbf C}\). The direct image \(i_*\mathcal F\) is coherent, and Cohomology, Lemma 02UV identifies its algebraic and analytic cohomology with the corresponding cohomology on \(X\) and \(X^{\mathrm{an}}\). Lemma gaga-lemma-analytification-pushforward-closed-immersion identifies \((i_*\mathcal F)^{\mathrm{an}}\) with \(i_*\mathcal F^{\mathrm{an}}\), compatibly with the comparison map. We may therefore assume \(X=\mathbf P^r_{\mathbf C}\).
We use descending induction on \(q\), simultaneously for all coherent \(\mathcal F\). For \(q>2r\), the algebraic group vanishes by Cohomology of Schemes, Lemma 01YS, and the analytic group vanishes by Lemma gaga-lemma-projective-analytic-cohomological-dimension.
Assume comparison is known in degree \(q+1\) for every coherent sheaf. By Cohomology of Schemes, Lemma 01YS, there is a short exact sequence \[0\longrightarrow\mathcal R\longrightarrow \mathcal L\longrightarrow\mathcal F\longrightarrow0\] with \(\mathcal R\) coherent and \(\mathcal L\) a finite direct sum of twists \(\mathcal O(n)\). Comparison is an isomorphism for \(\mathcal L\) in every degree by Lemma gaga-lemma-projective-twist-comparison. In the morphism of long exact sequences, the four lemma applied to \[H^q(\mathcal L)\longrightarrow H^q(\mathcal F) \longrightarrow H^{q+1}(\mathcal R)\longrightarrow H^{q+1}(\mathcal L)\] shows first that the comparison map for \(\mathcal F\) in degree \(q\) is surjective. Since this holds for every coherent sheaf, it holds for \(\mathcal R\). Apply the injective half of the four lemma to \[H^q(\mathcal R)\longrightarrow H^q(\mathcal L) \longrightarrow H^q(\mathcal F)\longrightarrow H^{q+1}(\mathcal R).\] The comparison map for \(\mathcal F\) is injective as well. This completes the descending induction and the proof.
Proof of full faithfulness
Lemma
Let \(X\) be a variety over \(\mathbf C\) and let \(\mathcal F,\mathcal G\) be coherent \(\mathcal O_X\)-modules. There is a canonical isomorphism \[\SheafHom_{\mathcal O_X}(\mathcal F,\mathcal G)^{\mathrm{an}} \longrightarrow \SheafHom_{\mathcal O_{X^{\mathrm{an}}}} (\mathcal F^{\mathrm{an}},\mathcal G^{\mathrm{an}}).\]
Proof
Write \(a_X:X^{\mathrm{an}}\to X\) for the canonical morphism of locally ringed spaces. A coherent module on the locally Noetherian scheme \(X\) is finitely presented, and \(a_X\) is flat by Theorem gaga-theorem-analytification-local-faithfully-flat. The displayed map is therefore the flat-pullback internal-Hom isomorphism of Modules, Lemma 0C6I. Equivalently, on a stalk it is the flat base-change isomorphism of More on Algebra, Lemma 087R. Coherence of the source follows from Cohomology of Schemes, Lemma 01Y2; the displayed isomorphism therefore gives coherence of the target as well.
Proof
Put \(\mathcal A=\SheafHom_{\mathcal O_X}(\mathcal F,\mathcal G)\). Then \[\Hom_{\mathcal O_X}(\mathcal F,\mathcal G) =H^0(X,\mathcal A).\] Theorem gaga-theorem-gaga-cohomology in degree zero and Lemma gaga-lemma-analytification-internal-hom give canonical isomorphisms \[H^0(X,\mathcal A) \longrightarrow H^0(X^{\mathrm{an}},\mathcal A^{\mathrm{an}}) \longrightarrow \Hom_{\mathcal O_{X^{\mathrm{an}}}} (\mathcal F^{\mathrm{an}},\mathcal G^{\mathrm{an}}).\] Their composite sends \(f\) to \(f^{\mathrm{an}}\) by construction of the cohomology comparison map. Hence analytification is fully faithful.
Essential surjectivity: preliminaries and reduction
Lemma
Let \(i:Y\to X\) be a closed immersion of complex analytic spaces whose structure sheaves and defining ideal are coherent. Then \(i_*\) is exact and fully faithful on modules, carries coherent \(\mathcal O_Y\)-modules to coherent \(\mathcal O_X\)-modules, and its essential image consists of the coherent modules annihilated by the defining ideal. Moreover, \[H^q(Y,\mathcal M)=H^q(X,i_*\mathcal M)\] for every \(q\geq0\).
Proof
Exactness, full faithfulness, and the description of the essential image are Modules, Lemma 08KS. To check coherence, first regard \(i_*\mathcal M\) as a module over \(i_*\mathcal O_Y=\mathcal O_X/\mathcal I\). Modules, Lemma 01C4, applied with this quotient structure sheaf, identifies coherence on \(Y\) with coherence over \(i_*\mathcal O_Y\). Modules, Lemma modules-lemma-coherent-change-rings then identifies this with coherence over \(\mathcal O_X\). The cohomology equality is Cohomology, Lemma 02UV.
Lemma
Uniqueness in Theorem gaga-theorem-gaga-essential-surjectivity follows from Theorem gaga-theorem-gaga-fully-faithful. For existence, it is enough to prove the theorem when \(X=\mathbf P^r_{\mathbf C}\).
Proof
If \(\mathcal F^{\mathrm{an}}\cong\mathcal G^{\mathrm{an}}\), full faithfulness lifts the isomorphism and its inverse uniquely to algebraic maps. Faithfulness shows that their composites are the identity, proving uniqueness.
For the reduction, let \(i:Y\to\mathbf P^r_{\mathbf C}\) be a closed immersion and let \(\mathcal M\) be coherent on \(Y^{\mathrm{an}}\). By Lemma gaga-lemma-coherent-analytic-closed-immersion, \(i_*\mathcal M\) is coherent on \(\mathbf P^r(\mathbf C)\). Assuming essential surjectivity on projective space, choose a coherent algebraic module \(\mathcal G\) with \(\mathcal G^{\mathrm{an}}\cong i_*\mathcal M\).
Let \(\mathcal I\) be the ideal of \(Y\). Compatibility of analytification with ideals, tensor products, and images identifies \((\mathcal I\mathcal G)^{\mathrm{an}}\) with \(\mathcal I^{\mathrm{an}}\mathcal G^{\mathrm{an}}\), which is zero because \(i_*\mathcal M\) is annihilated by \(\mathcal I^{\mathrm{an}}\). Exactness and faithfulness in Lemma gaga-lemma-analytification-exact-faithful-coherent therefore give \(\mathcal I\mathcal G=0\). Morphisms, Lemma 01QY and Cohomology of Schemes, Lemma 087T give a coherent algebraic \(\mathcal F\) on \(Y\) with \(i_*\mathcal F=\mathcal G\). Finally, Lemma gaga-lemma-analytification-pushforward-closed-immersion gives \[i_*\mathcal F^{\mathrm{an}} \cong(i_*\mathcal F)^{\mathrm{an}} \cong\mathcal G^{\mathrm{an}} \cong i_*\mathcal M.\] Full faithfulness of analytic closed-immersion pushforward yields \(\mathcal F^{\mathrm{an}}\cong\mathcal M\).
Analytic twists and global generation
We now fix \(X=\mathbf P^r_{\mathbf C}\) and argue by induction on \(r\) for essential surjectivity. The case \(r=0\) is the equivalence between finite dimensional complex vector spaces and coherent modules on a point.
Definition
For a coherent analytic module \(\mathcal M\) on \(X^{\mathrm{an}}\) and \(n\in\mathbf Z\), set \[\mathcal M(n)= \mathcal M\otimes_{\mathcal O_{X^{\mathrm{an}}}} \mathcal O_X(n)^{\mathrm{an}}.\] This is coherent, and the tensor compatibility of pullback gives the canonical identity \[\mathcal F^{\mathrm{an}}(n)\cong\mathcal F(n)^{\mathrm{an}}\] for every coherent algebraic \(\mathcal F\).
Lemma
Let \(E\cong\mathbf P^{r-1}_{\mathbf C}\) be a hyperplane and let \(\mathcal A\) be a coherent analytic module on \(E^{\mathrm{an}}\). Then \[H^q(E^{\mathrm{an}},\mathcal A(n))=0\] for every \(q>0\) and all sufficiently large \(n\).
Proof
By the induction hypothesis for essential surjectivity there is a coherent algebraic \(\mathcal F\) on \(E\) with \(\mathcal A\cong\mathcal F^{\mathrm{an}}\). Definition gaga-definition-coherent-analytic-twist and Theorem gaga-theorem-gaga-cohomology identify the displayed group with \(H^q(E,\mathcal F(n))\). The latter vanishes for \(q>0\) and \(n\) sufficiently large by Cohomology of Schemes, Lemma 01YS. There are only finitely many possibly nonzero degrees, so one lower bound on \(n\) works for all \(q>0\).
Lemma
Let \(\mathcal M\) be a coherent analytic module on \(X^{\mathrm{an}}=\mathbf P^r(\mathbf C)\). There is an integer \(n_0\) such that \(\mathcal M(n)\) is generated by global sections for every \(n\geq n_0\).
Proof
We first make two elementary observations. If global sections generate \(\mathcal M(n)\) at \(x\), choose a homogeneous coordinate \(T_k\) nonzero at \(x\). Multiplication by \(T_k^{m-n}\) sends global sections of \(\mathcal M(n)\) to global sections of \(\mathcal M(m)\) and is an isomorphism on the stalk at \(x\); generation at \(x\) therefore persists for every \(m\geq n\). Generation at a point is open: choose finitely many global sections generating the stalk, take their evaluation map from a finite free module, and use coherence of its cokernel to see that it vanishes near the point. Since \(X^{\mathrm{an}}\) is compact, it is consequently enough to find, for every \(x\), one twist generated at \(x\).
Fix \(x\) and choose a hyperplane \(i:E\to X\) through it, cut out by a linear form \(t\). Tensoring \(0\to\mathcal O_X(-1)^{\mathrm{an}}\xrightarrow{t} \mathcal O_{X^{\mathrm{an}}}\to i_*\mathcal O_{E^{\mathrm{an}}}\to0\) with \(\mathcal M\) gives an exact sequence \[0\longrightarrow\mathcal C\longrightarrow\mathcal M(-1) \xrightarrow{t}\mathcal M\longrightarrow\mathcal B\longrightarrow0,\] where \(\mathcal C\) and \(\mathcal B\) are coherent. Both are annihilated by \(t\), so Lemma gaga-lemma-coherent-analytic-closed-immersion regards them as coherent modules on \(E^{\mathrm{an}}\).
After twisting by \(n\), let \(\mathcal L_n\) be the image of \(\mathcal M(n-1)\to\mathcal M(n)\). We have short exact sequences \[0\to\mathcal C(n)\to\mathcal M(n-1)\to\mathcal L_n\to0, \qquad 0\to\mathcal L_n\to\mathcal M(n)\to\mathcal B(n)\to0.\] Lemma gaga-lemma-hyperplane-analytic-serre-vanishing gives, for all large \(n\), \[H^2(X^{\mathrm{an}},\mathcal C(n))=0, \qquad H^1(X^{\mathrm{an}},\mathcal B(n))=0.\] The two long exact sequences then give \[\dim H^1(\mathcal M(n-1)) \geq\dim H^1(\mathcal L_n) \geq\dim H^1(\mathcal M(n)).\] All these dimensions are finite by Grauert’s finiteness theorem for coherent analytic cohomology on compact complex spaces [Grauert-direct-image, Hauptsatz I, p. 57]. Thus the nonnegative integers \(\dim H^1(\mathcal M(n))\) eventually stabilize. For \(n\) in the stable range, the surjection \(H^1(\mathcal L_n)\to H^1(\mathcal M(n))\) is an isomorphism, and the long exact sequence yields a surjection \[H^0(X^{\mathrm{an}},\mathcal M(n)) \longrightarrow H^0(X^{\mathrm{an}},\mathcal B(n)).\]
By induction on \(r\), the module on \(E^{\mathrm{an}}\) corresponding to \(\mathcal B\) is the analytification of a coherent algebraic module \(\mathcal G\) on \(E\). For all sufficiently large \(n\), the module \(\mathcal G(n)\) is generated by global sections by Properties, Proposition 01Q3. Theorem gaga-theorem-gaga-cohomology in degree zero says the same for \(\mathcal B(n)\), and the preceding surjection lifts those generators to global sections of \(\mathcal M(n)\).
At \(x\), put \(A=\mathcal O_{X^{\mathrm{an}},x}\), \(M=\mathcal M(n)_x\), and let \(N\subset M\) be the submodule generated by global sections. If \(\mathfrak p=(t)\subset A\), then \(\mathcal B(n)_x=M/\mathfrak pM\). The lifted sections generate this quotient, so \(M=N+\mathfrak pM\). Since \(\mathfrak p\) lies in the maximal ideal of the local ring \(A\), Nakayama’s lemma (Algebra, Lemma 00DV) gives \(M=N\). This supplies the required twist at \(x\); the two opening observations and compactness supply one \(n_0\) valid at every point and every \(n\geq n_0\).
Completion of essential surjectivity
Proof
It remains, by Lemma gaga-lemma-gaga-essential-surjectivity-projective-reduction, to treat \(X=\mathbf P^r_{\mathbf C}\). Let \(\mathcal M\) be coherent on \(X^{\mathrm{an}}\). Lemma gaga-lemma-projective-analytic-global-generation and compactness give a finite set of global generators after a twist. Untwisting produces a surjection \[\mathcal L_0^{\mathrm{an}}\longrightarrow\mathcal M, \qquad \mathcal L_0=\mathcal O_X(-n)^{\oplus p}.\] Its kernel \(\mathcal R\) is coherent by Modules, Lemma 01BY. Apply the same global-generation argument to \(\mathcal R\) to obtain a finite direct sum of twists \(\mathcal L_1\) and a surjection \(\mathcal L_1^{\mathrm{an}}\to\mathcal R\). Composing with the inclusion of \(\mathcal R\) gives an exact sequence \[\mathcal L_1^{\mathrm{an}}\xrightarrow{g} \mathcal L_0^{\mathrm{an}}\longrightarrow\mathcal M\longrightarrow0.\]
Theorem gaga-theorem-gaga-fully-faithful supplies a unique algebraic map \(f:\mathcal L_1\to\mathcal L_0\) with \(f^{\mathrm{an}}=g\). Put \(\mathcal F=\mathop{\rm Coker}(f)\). This is coherent, and exactness of analytification gives \[\mathcal F^{\mathrm{an}} =\mathop{\rm Coker}(f^{\mathrm{an}}) =\mathop{\rm Coker}(g) \cong\mathcal M.\] Together with the reduction and uniqueness already proved, this completes Theorem gaga-theorem-gaga-essential-surjectivity.
Applications of the comparison theorems
Conjugation and Betti numbers
Let \(\sigma\in\mathop{\rm Aut}(\mathbf C)\). For a scheme \(X\) over \(\mathbf C\), write \[X^\sigma=X\times_{\mathop{\rm Spec}(\mathbf C),\sigma} \mathop{\rm Spec}(\mathbf C).\]
Proposition
Let \(X\) be a smooth projective algebraic variety over \(\mathbf C\). Then \(X^{\rm an}\) and \((X^\sigma)^{\rm an}\) have the same Betti numbers.
Proof
For \(p,q\geq 0\), flat base change gives a \(\sigma\)-semilinear isomorphism \[H^q(X,\Omega^p_{X/\mathbf C}) \longrightarrow H^q(X^\sigma,\Omega^p_{X^\sigma/\mathbf C});\] see Cohomology of Schemes, Lemma 02KH. In particular, the two vector spaces have the same dimension. For a smooth variety, local coordinates give the canonical comparison \[(\Omega^p_{X/\mathbf C})^{\rm an} \cong \Omega^p_{X^{\rm an}},\] where the module on the right is the module of holomorphic \(p\)-forms. Thus the cohomology comparison theorem gaga-theorem-gaga-cohomology identifies these dimensions with \[h^{p,q}(X^{\rm an}) \quad\hbox{and}\quad h^{p,q}((X^\sigma)^{\rm an}),\] respectively, where \[h^{p,q}(M)=\dim_{\mathbf C}H^q(M,\Omega^p_M).\] Finally, a smooth projective analytification is compact Kähler. Hodge decomposition (using the Dolbeault resolution and the Kähler identities), which is an external analytic input here, gives \[b_n(X^{\rm an})=\sum_{p+q=n}h^{p,q}(X^{\rm an})\] and the analogous formula for \(X^\sigma\). The assertion follows.
Proposition
Let \(V\) be a smooth projective variety over a field \(K\) algebraic over \(\mathbf Q\). The Betti numbers of \(V\times_{K,\iota}\mathbf C\) are independent of the embedding \(\iota:K\longrightarrow\mathbf C\).
Proof
The standard extension theorem for field embeddings shows that two embeddings of \(K\) in \(\mathbf C\) differ by an automorphism of \(\mathbf C\). Apply Proposition gaga-proposition-conjugate-smooth-projective-betti-numbers.
Remark
Serre observed in [GAGA, §4, no. 18] that a conjugate pair need not be analytically isomorphic (elliptic curves already show this), and asked there whether such a pair must be homeomorphic. This historical question is not an input to Proposition gaga-proposition-conjugate-smooth-projective-betti-numbers.
Chow’s theorem and algebraicity
Theorem
Let \(X\) be a projective algebraic variety over \(\mathbf C\). Every closed reduced analytic subspace \(Y\subset X^{\rm an}\) is the analytification of a unique closed reduced subscheme of \(X\).
Proof
The vanishing ideal \(\mathcal J_Y\subset\mathcal O_{X^{\rm an}}\) is coherent by Theorem gaga-theorem-oka-cartan-coherence. By Theorem gaga-theorem-gaga-essential-surjectivity, there is a coherent \(\mathcal O_X\)-module \(\mathcal F\) with \(\mathcal F^{\rm an}\cong\mathcal J_Y\). Compose this isomorphism with the inclusion into \(\mathcal O_{X^{\rm an}}\). The full-faithfulness theorem gaga-theorem-gaga-fully-faithful algebraizes the composite to an \(\mathcal O_X\)-linear map \[u:\mathcal F\longrightarrow\mathcal O_X.\] Its image \(\mathcal I\) is a coherent ideal. Exactness and faithfulness of analytification identify \(\mathcal I^{\rm an}\) with \(\mathcal J_Y\). Hence the ideal \(\mathcal I\) is radical. Indeed, at every closed point \(x\in X(\mathbf C)\), if \(a^m\in\mathcal I_x\), then the image of \(a\) in \(\mathcal O_{X^{\rm an},x}\) belongs to the radical ideal \(\mathcal J_{Y,x}=\mathcal I_x\mathcal O_{X^{\rm an},x}\). Faithful flatness in Theorem gaga-theorem-analytification-local-faithfully-flat contracts this ideal back to \(\mathcal I_x\). Since \(X\) is Noetherian and Jacobson, radicality at all closed stalks implies that \(\mathcal I\) is radical. The closed reduced subscheme defined by \(\mathcal I\) therefore has analytic ideal \(\mathcal J_Y\), and hence has analytification \(Y\).
If two reduced closed subschemes have analytification \(Y\), their ideal sheaves have equal extensions at every analytic local ring. Faithful flatness contracts these extensions to equal stalks at every closed point; coherence and the Jacobson property then identify the ideal sheaves. This proves uniqueness.
Proposition
Let \(X\) be an algebraic variety over \(\mathbf C\). Every compact closed reduced analytic subspace \(Y\subset X^{\rm an}\) is algebraic: there is a unique closed reduced subscheme \(Z\subset X\) with \(Z^{\rm an}=Y\).
Proof
Nagata compactification and Chow’s lemma (More on Flatness, Theorem 0F41 and Limits of Schemes, Lemma 0202) give an open immersion \(X\subset\overline X\) with \(\overline X\) proper and a proper surjection \(p:X'\to\overline X\) together with an immersion \(X'\to\mathbf P^n_{\mathbf C}\). Replacing \(X'\) by its reduction preserves these properties. The resulting variety \(X'\) is proper over \(\mathbf C\), so its immersion is closed and \(X'\) is projective. Since \(Y\) is compact, it is closed in the Hausdorff space \(\overline X^{\rm an}\). Its analytic structure on \(X^{\rm an}\) glues with the empty closed analytic subspace on \(\overline X^{\rm an}\setminus Y\), and therefore makes \(Y\) a closed reduced analytic subspace of \(\overline X^{\rm an}\).
The reduced inverse image \[Y'=\left((X')^{\rm an}\times_{\overline X^{\rm an}}Y\right)_{\rm red}\] is a closed reduced analytic subspace of \((X')^{\rm an}\). Theorem gaga-theorem-chow-closed-analytic-projective algebraizes it to a closed reduced subscheme \(Z'\subset X'\). Let \(Z\subset\overline X\) be the reduced Zariski closure of \(p(Z')\). The analytic map sends \((Z')^{\rm an}=Y'\) onto \(Y\). Since \(Y\) is closed, Proposition gaga-proposition-image-closure shows that the classical point set of \(Z^{\rm an}\) is exactly \(Y\). Reduced closed analytic subspaces are determined by their underlying analytic sets, so \(Z^{\rm an}=Y\). All closed points of \(Z\) lie in \(X\); the Jacobson property therefore shows that \(Z\subset X\).
For uniqueness, if \(Z_1,Z_2\subset X\) have analytification \(Y\), their ideal stalks have equal extensions to every \(\mathcal O_{X^{\rm an},x}\). Faithful flatness in Theorem gaga-theorem-analytification-local-faithfully-flat contracts them to equal stalks at every closed point. Coherence and the Jacobson property give \(Z_1=Z_2\).
Proposition
Let \(X\) be a proper algebraic variety over \(\mathbf C\) and let \(Y\) be an algebraic variety over \(\mathbf C\). Every holomorphic map \[f:X^{\rm an}\longrightarrow Y^{\rm an}\] is the analytification of a unique morphism \(X\to Y\).
Proof
The graph \(\Gamma_f\) is a compact closed analytic subspace of \((X\times_{\mathbf C}Y)^{\rm an}\). Proposition gaga-proposition-compact-analytic-subspace-algebraic makes it the analytification of a closed reduced subvariety of \(X\times_{\mathbf C}Y\). Proposition gaga-proposition-algebraic-graph-holomorphic-regular therefore defines the required algebraic map. For uniqueness, two candidate maps have closed reduced graphs with the same classical points; the graphs, and hence the maps, are equal.
Proposition
A compact reduced analytic space admits at most one algebraic-variety structure: any analytic isomorphism between two algebraizations is the analytification of a unique algebraic isomorphism.
Proof
Every algebraization is proper by Proposition gaga-proposition-proper-if-and-only-if-compact-analytification. Apply Proposition gaga-proposition-holomorphic-map-from-proper-is-regular to the analytic isomorphism and to its inverse.
Algebraic and analytic principal bundles
Definition
Let \(G\) be an algebraic group over \(\mathbf C\) and let \(X\) be an algebraic variety over \(\mathbf C\). Write \[H^1_{\rm Zar}(X,G)\] for the pointed set of isomorphism classes of Zariski locally trivial principal \(G\)-bundles and \[H^1_{\rm an}(X^{\rm an},G^{\rm an})\] for the pointed set of isomorphism classes of holomorphically locally trivial principal \(G^{\rm an}\)-bundles. Analytification defines a map \[\varepsilon_G:H^1_{\rm Zar}(X,G) \longrightarrow H^1_{\rm an}(X^{\rm an},G^{\rm an}).\] This is the torsor interpretation of first cohomology from Cohomology on Sites, Section 03AG.
Proposition
If \(X\) is a proper algebraic variety over \(\mathbf C\), then \(\varepsilon_G\) is injective. More precisely, every analytic isomorphism between the analytifications of two algebraic principal \(G\)-bundles is algebraic.
Proof
For two algebraic principal \(G\)-bundles \(P\) and \(P'\) on \(X\), the associated bundle \[\mathop{\rm Isom}\nolimits_G(P,P')\] is an algebraic fibre bundle over \(X\), and its sections are precisely the \(G\)-equivariant isomorphisms \(P\to P'\). An analytic isomorphism gives a holomorphic section of its analytification. Proposition gaga-proposition-holomorphic-map-from-proper-is-regular makes the section algebraic.
Proposition
If \(X\) is a projective algebraic variety over \(\mathbf C\) and \(G=\mathbf G_a\), then \(\varepsilon_G\) is bijective.
Proof
Principal \(\mathbf G_a\)-bundles are torsors under the additive sheaf \(\mathcal O_X\), and hence are classified by \(H^1(X,\mathcal O_X)\); see Cohomology, Lemma 02FQ. The analytic statement uses \(H^1(X^{\rm an},\mathcal O_{X^{\rm an}})\). The comparison isomorphism of Theorem gaga-theorem-gaga-cohomology identifies these groups.
Proposition
If \(X\) is a projective algebraic variety over \(\mathbf C\) and \(G=\mathop{\rm GL}_n\), then \(\varepsilon_G\) is bijective.
Proof
Principal \(\mathop{\rm GL}_n\)-bundles are equivalent to rank \(n\) vector bundles. Let \(\mathcal E\) be the locally free coherent analytic module associated with an analytic vector bundle. By Theorem gaga-theorem-gaga-essential-surjectivity, it is isomorphic to \(\mathcal F^{\rm an}\) for a coherent \(\mathcal O_X\)-module \(\mathcal F\). For every closed point \(x\in X(\mathbf C)\), the faithfully flat local map of Theorem gaga-theorem-analytification-local-faithfully-flat makes \[\mathcal F_x\otimes_{\mathcal O_{X,x}} \mathcal O_{X^{\rm an},x}\] free of rank \(n\). Faithfully flat descent for finite projectivity (Algebra, Proposition 058S) shows that \(\mathcal F_x\) is finite projective. Algebra, Theorem 0593 makes it free, and faithful base change fixes its rank as \(n\). Since \(X\) is Jacobson and \(\mathcal F\) is coherent, local freeness at every closed point makes \(\mathcal F\) locally free of rank \(n\) everywhere. Its frame bundle algebraizes the given analytic principal bundle. Injectivity is Proposition gaga-proposition-principal-bundle-analytification-injective.
Remark
For \(n=1\), Proposition gaga-proposition-general-linear-principal-bundles-gaga is the isomorphism \[\mathop{\rm Pic}(X)\longrightarrow\mathop{\rm Pic}(X^{\rm an}).\] Serre’s remarks after Proposition 18 in [GAGA, §4, no. 20] express this in terms of divisor classes on a normal variety and record the earlier smooth result of Kodaira–Spencer. Those historical attributions are not additional hypotheses here. Serre also notes that Proposition 18 extends Kodaira’s Theorems 7 and 8 to arbitrary projective varieties, possibly with singularities, without pursuing those applications.
Proposition
Let \(H\subset G\) be algebraic groups over \(\mathbf C\). Assume that the quotient \(G/H\) exists as an algebraic variety and that \(G\to G/H\) has a rational section. Let \(X\) be a proper algebraic variety over \(\mathbf C\) and let \(P\) be an analytic principal \(H^{\rm an}\)-bundle on \(X^{\rm an}\). Then \(P\) is algebraizable if and only if the extended bundle \[P\times^H G\] is algebraizable.
Proof
A rational section over a nonempty open of \(G/H\), and the translates of its domain by \(G\), show that \(G\to G/H\) is Zariski locally trivial as an \(H\)-bundle. Necessity is therefore immediate by extension of structure group.
Conversely, choose an algebraic principal \(G\)-bundle \(Q\) and an analytic isomorphism \(Q^{\rm an}\cong P\times^H G\). Form the algebraic associated bundle \[E=Q\times^G(G/H).\] The analytic reduction \(P\) determines a holomorphic section \(s:X^{\rm an}\to E^{\rm an}\). By Proposition gaga-proposition-holomorphic-map-from-proper-is-regular, this section is algebraic. Pulling the principal \(H\)-bundle \(Q\to E\) back along \(s\) gives an algebraic principal \(H\)-bundle whose analytification is \(P\).
Definition
A closed algebraic subgroup \(G\subset\mathop{\rm GL}_n\) satisfies (R) if the quotient exists and the projection \[\mathop{\rm GL}_n/G\longleftarrow\mathop{\rm GL}_n\] has a rational section.
Proposition
Let \(G\subset\mathop{\rm GL}_n\) satisfy (R). For every projective algebraic variety \(X\) over \(\mathbf C\), the map \[\varepsilon_G:H^1_{\rm Zar}(X,G) \longrightarrow H^1_{\rm an}(X^{\rm an},G^{\rm an})\] is bijective.
Proof
Given an analytic principal \(G\)-bundle \(P\), Proposition gaga-proposition-general-linear-principal-bundles-gaga algebraizes \(P\times^G\mathop{\rm GL}_n\). Proposition gaga-proposition-algebraize-reduction-structure-group, with \(H=G\) and the ambient group \(\mathop{\rm GL}_n\), then algebraizes \(P\). Injectivity is Proposition gaga-proposition-principal-bundle-analytification-injective.
Remark
Condition (R) holds for solvable subgroups by Rosenlicht’s theorem, for \(\mathop{\rm SL}_n\) via the determinant section, and for \(\mathop{\rm Sp}_{2m}\) by rational symplectic Gram–Schmidt. Serre also records in [GAGA, §4, no. 20] the then-open simply connected semisimple conjecture and the failure of (R) for the unimodular orthogonal group in dimensions at least \(3\); in the latter case he leaves bijectivity of \(\varepsilon_G\) open. These historical observations are not used in Proposition gaga-proposition-rational-section-principal-bundles-gaga.