Xiaolong Liu
Xiaolong Liu's Blog
I'm interested in higher dimensional algebraic geometry, birational geometry, moduli space of varieties and bundles and derived categories of sheaves.

A Quick Tour of Géométrie algébrique et géométrie analytique

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In this blog, we will introduce some basic fact about GAGA-principle. Actually I only vaguely knew that this is a correspondence between analytic geometry and algebraic geometry over $\mathbb{C}$ before. So as we may use GAGA frequently, we will summarize in this blog to facilitate learning and use.

The original paper is J.P. Serre’s famous paper [Serre56]1 which consider the projective integral case(there is a book [AA07]2 for this). Then Grothendieck extended the theory to proper $\mathbb{C}$-schemes locally of finite types with analytic spaces in [SGA-I]3. Here we mainly follows the surveys [GAGA13]4, [Wiki]5.

There is much more development of GAGA in arithmatic analytic geometry (Conrad-Temkin) and even in stacks and moduli spaces (see GAGA in nlab).

1. Basic facts about analytic spaces

1.1. Basic definitions

Definition 1.1.1. Pick $U\subset\mathbb{C}^n$ and let $f_1, . . . , f_k$ be holomorphic functions on $U$ and $\mathscr{I}$ be the coherent sheaf of ideals generated by these functions. An affine analytic space $(X, \mathscr{O}^{Hol}_{X})$ is a locally ringed space whose underlying space is \[X=\{y\in U:f_1(y)=\cdots=f_k(y)=0\}\subset U\] and whose sheaf of rings is defined by $i^{−1}\mathscr{O}^{Hol}_{U}/\mathscr{I}$ where $i:X\to U$.

Definition 1.1.2. An analytic space $(X, \mathscr{O}^{Hol}_{X})$ is a locally ringed space such that $X$ is a Hausdorff (separated) topological space and there is an open cover $\{V_i\}$ of $X$ such that each $(V_i, \mathscr{O}^{Hol}_X|_{V_i})$ is an affine analytic space.

Remark. If $X$ is smooth, then it is actually a complex manifold. Moreover, we have $\mathscr{O}^{Hol}_{X,x}\cong\mathbb{C}\{x_1,…,x_k\}/I$.

Proposition 1.1.3. Let $\mathscr{O}^{Reg}_{X}\subset \mathscr{O}^{Hol}_{X}$ be the subsheaf of regular functions on $X$. Then for any $x\in X$ we have \[(\mathscr{O}^{Reg}_{X,x})^{\wedge}\cong(\mathscr{O}^{Hol}_{X,x})^{\wedge}.\]

Proof. Omitted, see paper [Serre56]1. $\blacksquare$

Remark. One can also show that $\mathscr{O}^{Reg}_{X,x}\to\mathscr{O}^{Hol}_{X,x}$ is faithfully flat (see4 Proposition 1.8)!

1.2. Analytifications of $\mathbb{C}$-schemes

Definition 1.2.1. (Analytification). Fix $X$ be a $\mathbb{C}$-scheme locally of finite type and its space $(X(\mathbb{C}),\mathscr{O}_X|_{X(\mathbb{C})})$ of $\mathbb{C}$-rational points. In the affine case, we have $X\cong\mathrm{Spec}\mathbb{C}[x_1,…,x_n]/I$. Then we endow $X(\mathbb{C})$ with a finer topology induced as a subspace topology of $\mathbb{C}^n$ as zeros of $I$, we call it $X^{\mathrm{an}}$. Let \[\mathscr{O}^{Hol}_{X^{\mathrm{an}},x}:=\mathscr{O}^{Hol}_{\mathbb{C}^n,x}/I\cdot \mathscr{O}^{Hol}_{\mathbb{C}^n,x}\] and we get the analytification of $X$ be an analytic space $(X^{\mathrm{an}},\mathscr{O}^{Hol}_{X^{\mathrm{an}}})$.

Remark. There is a natural morphism of locally ringed spaces $\phi:(X^{\mathrm{an}},\mathscr{O}^{Hol}_{X^{\mathrm{an}}})\to(X,\mathscr{O}_X)$ as $X^{\mathrm{an}}\to X(\mathbb{C})\subset X$ and $\phi^{-1} \mathscr{O}_X\to\mathscr{O}^{Hol}_{X^{\mathrm{an}}}$ as $f\mapsto f\circ\phi$. Actually by the Proposition 1.1.3 we get the map $\phi_x: \mathscr{O}_{X,x}\to\mathscr{O}^{Hol}_{X^{\mathrm{an}},x}$ induce $\mathscr{O}_{X,x}^{\wedge}\cong(\mathscr{O}^{Hol}_{X^{\mathrm{an}},x})^{\wedge}$. Moreover, we have $\mathscr{O}_{X,x}\to\mathscr{O}^{Hol}_{X^{\mathrm{an}},x}$ is faithfully flat by previous Remark.

Theorem 1.2.2. Fix $X$, a $\mathbb{C}$-scheme locally of finite type. Consider the functor \[\Phi_X:\mathbf{AnSp}\to\mathbf{Sets},\mathcal{X}\mapsto\mathrm{Hom}_{\mathbb{C}}(\mathcal{X},X)\] where $\mathbf{AnSp}$ be the category of analytic spaces and $\mathrm{Hom}$ be the set of morphisms of locally ringed spaces. Then $\Phi_X$ representable by the analytic space $X^{\mathrm{an}}$ and the equivalence $\Phi_X\cong\mathrm{Hom}(-,X^{\mathrm{an}})$ induced by $\phi$.

Sketch of proof. WLOG we let $X$ is affine. We find that:

(i) If $X\subset Y$ be an open immersion and $Y$ satisfies this property, then so is $X$. This is because $X^{\mathrm{an}}\cong Y^{\mathrm{an}}\times_YX$ and locally $X^{\mathrm{an}}$ and $Y^{\mathrm{an}}$ are isomorphism;

(ii) If $X\subset Y$ be an closed immersion and $Y$ satisfies this property, then so is $X$. This is because locally if $\mathscr{O}_{X,x}\cong\mathscr{O}_{Y,x}/I$, then $\mathscr{O}^{Hol}_{X^{\mathrm{an}},x}\cong\mathscr{O}^{Hol}_{Y^{\mathrm{an}},x}/I\cdot\mathscr{O}^{Hol}_{Y^{\mathrm{an}},x}$;

(iii) We have $(X\times Y)^{\mathrm{an}}\cong X^{\mathrm{an}}\times Y^{\mathrm{an}}$.

Hence we just show the case when $X=\mathbb{A}^1_{\mathbb{C}}$. One can show that

\begin{align*} &\mathrm{Hom}_{\mathbb{C}}(\mathcal{X},\mathbb{A}^1_{\mathbb{C}})\cong\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}[x],\Gamma(\mathcal{X},\mathscr{O}^{Hol}_{\mathcal{X}}))\\ &\cong\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}\{x\},\Gamma(\mathcal{X},\mathscr{O}^{Hol}_{\mathcal{X}}))\cong \mathrm{Hom}_{\mathbb{C}}(\mathcal{X},(\mathbb{A}^1_{\mathbb{C}})^{\mathrm{an}}) \end{align*}

and well done. $\blacksquare$

Remark. Using this theorem, we can get that for any morphism of these schemes $f:X\to Y$ lifts to a unique morphism of locally ringed spaces $f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}$ such that $f\circ\phi_X=\phi_Y\circ f^{\mathrm{an}}$. We also have $X^{\mathrm{an}}\times_{Z^{\mathrm{an}}}X^{\mathrm{an}}\cong(X\times_ZY)^{\mathrm{an}}$.

2. Collection of properties and morphisms between $\mathbb{C}$-schemes and analytic spaces

Proposition 2.1. Let$X$ be a $\mathbb{C}$-scheme locally of finite type and $T\subset X$ a constructible subset. Then $\phi^{-1}(\overline{T})=\overline{\phi^{-1}(T)}$. In particular, $T$ is closed (open, dense, respectively) if and only if $\phi^{-1}(T)$ is closed (open, dense, respectively).

Proof. See [GAGA13]4 Proposition 2.3. $\blacksquare$

Theorem 2.2. Let $X$ be a $\mathbb{C}$-scheme locally of finite type. Let $\mathbf{P}$ be one of the following properties: non-empty; Cohen-Macaulay; $S_n$; regular; $R_n$; normal; reduced; dimension $n$; discrete; connected; irreducible; integral. Then $X$ has property $\mathbf{P}$ if and only if $X^{\mathrm{an}}$ has property $\mathbf{P}$.

Proof. See [GAGA13]4 Proposition 2.2, 2.5. $\blacksquare$

Theorem 2.3. Let $f:X\to Y$ be a morphism between $\mathbb{C}$-schemes $X,Y$ locally of finite type and $f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}$ induced by $f$. Let $\mathbf{Q}$ be one of the following properties: flat; unramified; étale; smooth; injective; open immersion; isomorphism; a monomorphism. Let $\mathbf{R}$ be one of the following properties: surjective; dominant; closed immersion; immersion; projective; proper; finite. Then $f$ has property $\mathbf{Q}$ if and only if $f^{\mathrm{an}}$ has property $\mathbf{Q}$. Moreover, if $f$ is of finite type, then $f$ has property $\mathbf{R}$ if and only if $f^{\mathrm{an}}$ has property $\mathbf{R}$.

Proof. See [GAGA13]4 Proposition 3.2. $\blacksquare$

3. The GAGA theorems

3.1. Analytifications of coherent sheaves

Proposition-Definition 3.1.1. Let $\phi:X^{\mathrm{an}}\to X$ be the canonical map. Pick $\mathscr{F}\in\mathbf{Coh}(X)$, we define \[\mathscr{F}^{\mathrm{an}}:=\phi^* \mathscr{F}=\phi^{-1}\mathscr{F}\otimes_{\phi^{-1}\mathscr{O}_X}\mathscr{O}^{Hol}_{X^{\mathrm{an}}}.\] This defines a functor \[\phi^* :\mathbf{Coh}(X)\to \mathbf{Coh}(X^{\mathrm{an}}),\mathscr{F}\mapsto\mathscr{F}^{\mathrm{an}}.\] Then $\phi^* $ is exact, faithful and conservative.

Proof. Exactness follows from the fact that $\phi^{-1}$ is exact and $\mathscr{O}_{X,x}\to\mathscr{O}^{Hol}_{X^{\mathrm{an}},x}$ is flat. It is faithful by the faithfully flatness of $\phi^{-1}\mathscr{O}_X\to\mathscr{O}^{Hol}_{X^{\mathrm{an}}}$ and the fact that $X(\mathbb{C})\subset X$ is dense. Conservativeness follows from faithfulness and flatness. $\blacksquare$

Remark. We have adjoint pair $(\phi^* ,\phi_* )$, hence $\mathscr{F}\to\phi_* \phi^* \mathscr{F}=\phi_* \mathscr{F}^{\mathrm{an}}$.

3.2. Two main results of GAGA

Construction 3.2.1. Consider $f:X\to Y$, $\phi_X:X^{\mathrm{an}}\to X$, $\phi_Y:Y^{\mathrm{an}}\to Y$ and $f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}$. Pick $\mathscr{F}\in\mathbf{Coh}(X)$ and as $\phi_X^* $ exact, we get

\begin{align*} &\phi_Y^*Rf_*\mathscr{F}\to \phi_Y^*Rf_*(\phi_{X,*}\mathscr{F}^{\mathrm{an}})\\ &\to \phi_Y^*R(f\circ\phi_X)_*\mathscr{F}^{\mathrm{an}}=\phi_Y^*R(\phi_Y\circ f^{\mathrm{an}})_*\mathscr{F}^{\mathrm{an}}\\ &=R(\phi_Y^*\circ\phi_{Y,*}\circ f^{\mathrm{an}}_*)\mathscr{F}^{\mathrm{an}}\to Rf^{\mathrm{an}}_*\mathscr{F}^{\mathrm{an}}. \end{align*}

Hence we get \[\theta^i:(R^if_* \mathscr{F})^{\mathrm{an}}\to R^if^{\mathrm{an}}_* \mathscr{F}^{\mathrm{an}}.\]

Remark. One can describe this by relative Čech cohomology when $f$ is proper. In both category $\mathbf{AnSp}$ (for any sheaves) and $\mathbf{Sch}/\mathbb{C}$ (for quasi-coherent sheaves), we have: for open covering $\mathfrak{U}:=\{U_j\}_{j\in I}$, let \[\mathscr{C}^i(\mathfrak{U},\mathscr{F},f_* )=\bigoplus_{|J|=i,J\subset I}(f|_{U_J})_* \mathscr{F}|_{U_J},\] then we get \[R^if_* \mathscr{F}=\varinjlim_{\mathfrak{U}}\mathscr{H}^i(\mathscr{C}^i(\mathfrak{U},\mathscr{F},f_* )).\] When taking some affine open sets (good cover), we can get \[R^if_* \mathscr{F}=\mathscr{H}^i(\mathscr{C}^i(\mathfrak{U},\mathscr{F},f_* )).\] Then the $\eta^i$ is trivial.

Theorem 3.2.2. (GAGA-I). Let $f:X\to Y$ be a proper morphism of $\mathbb{C}$-schemes locally of finite type and $\mathscr{F}\in\mathbf{Coh}(X)$. Then, for any $i\geq 0$, the morphism \[\theta^i:(R^if_* \mathscr{F})^{\mathrm{an}}\to R^if^{\mathrm{an}}_* \mathscr{F}^{\mathrm{an}}\] is an isomorphism.

Sketch of Proof. When $f$ is projective, this is not hard and very standard (if you have no idea about this, please go to review the basic AG). First consider the case when $X=\mathbb{P}^n_Y$ and prove this when $\mathscr{F}=\mathscr{O}_X$ by calculation and then prove the case when $\mathscr{F}=\mathscr{O}_X(n)$ by induction. As for any $\mathscr{F}\in\mathbf{Coh}(X)$, we get the exact sequence \[0\to\mathscr{G}\to\bigoplus_i\mathscr{O}_X(n_i)\to\mathscr{F}\to0.\] By inducting on the rank of $\mathscr{F}$ and using the long exact sequence, we may get the result. Next consider the general case. Using the fact that if $i:X\to\mathbb{P}^n_Y$ be a closed immersion, then $(i_* \mathscr{F})^{\mathrm{an}}=i_* ^{\mathrm{an}}\mathscr{F}^{\mathrm{an}}$ by checking stalks.

For general proper $f$, we need use lemme de dévissage and Chow’s Lemma. It’s not so hard but we will omit it. We refer Théorème XII.4.2 in [SGA-I]3. $\blacksquare$

Theorem 3.2.3. (GAGA-II). Let $X$ be a proper $\mathbb{C}$-scheme and canonical map $\phi:X^{\mathrm{an}}\to X$, then the functor \[\phi^* :\mathbf{Coh}(X)\to \mathbf{Coh}(X^{\mathrm{an}}),\mathscr{F}\mapsto\mathscr{F}^{\mathrm{an}}\] is an equivalence of categories.

Sketch of Proof. We only show that the functor is fully faithful. The proof of essential surjectivity is complicated and we refer Théorème XII.4.4 in [SGA-I]3. By GAGA-I, let $\mathscr{F},\mathscr{G}\in\mathbf{Coh}(X)$ we get

\begin{align*} &\mathrm{Hom}_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})\cong H^0(X,\mathscr{H}om_{\mathscr{O}_X}(\mathscr{F},\mathscr{G}))\\ &\cong H^0(X^{\mathrm{an}},\mathscr{H}om_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})^{\mathrm{an}})\\ &\cong H^0(X^{\mathrm{an}},\mathscr{H}om_{\mathscr{O}^{Hol}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}},\mathscr{G}^{\mathrm{an}}))\\ &\cong \mathrm{Hom}_{\mathscr{O}^{Hol}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}},\mathscr{G}^{\mathrm{an}}) \end{align*}

where the last isomorphism is because $\mathscr{F}$ is coherent and $\mathscr{O}^{Hol}_{X^{\mathrm{an}}}$ is flat over $\phi^{-1}\mathscr{O}_X$. $\blacksquare$

4. Applications: generalized Chow’s theorem and a glimpse of Riemann’s existence theorem

Theorem 4.1. (Generalized Chow’s Theorem). Let $X$ be a proper $\mathbb{C}$-scheme. Then for any closed analytic subspace $\mathcal{Y}\subset X^{\mathrm{an}}$, there exists a closed subscheme $Y\subset X$ such that $\mathcal{Y}=Y^{\mathrm{an}}$.

Proof. Use ideal sheaves and GAGA-I, well done. $\blacksquare$

Theorem 4.2. Let $X$ be a separated $\mathbb{C}$-scheme locally of finite type. Then, all proper analytic subspaces of $X^{\mathrm{an}}$ are analytifications of some closed subschemes of $X$.

Proof. Checking locally we may let $X$ is of finite type. Then by Nagata compactification theorem and Generalized Chow’s Theorem, well done. $\blacksquare$

Theorem 4.3. Let $S$ be a $\mathbb{C}$-scheme locally of finite type. Let $X$ be a proper $S$-scheme and $Y$ be a $S$-scheme locally of finite type. Then we have the canonical isomorphism \[\mathrm{Hom}_S(X,Y)\cong \mathrm{Hom}_{S^{\mathrm{an}}}(X^{\mathrm{an}},Y^{\mathrm{an}}).\]

Proof. Omitted, see Theorem 5.4 in [GAGA13]4. $\blacksquare$

Theorem 4.4. (Riemann’s existence theorem). Let $X$ be a $\mathbb{C}$-scheme locally of finite type. The functor \[\Phi:\mathbf{Fét}(X)\to \mathbf{Fét}(X^{\mathrm{an}}) (\cong\mathrm{FTopCov}(X^{\mathrm{an}})),f\mapsto f^{\mathrm{an}}\] induces an equivalence.

Proof. Omitted, see Théorème XII.5.1 in [SGA-I]3. $\blacksquare$

Actually, using Riemann’s existence theorem we can show that $\pi_1^{\mathrm{ét}}(X,x)\cong(\pi_1(X^{\mathrm{an}},x))^{\wedge}$, the profinite completion, see Corollaire XII.5.2 in [SGA-I]3.

  1. [Serre56] Jean-Pierre Serre, Géométrie algébrique et géométrie analytique Université de Grenoble. Annales de l’Institut Fourier 6: 1–42, (1956) doi:10.5802/aif.59, ISSN 0373-0956, MR 0082175. ↩ ↩2

  2. [AA07] Amnon Neeman, Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345, 2007. ↩

  3. [SGA-I] Alexander Grothendieck, SGA I, Exposé XII. ↩ ↩2 ↩3 ↩4 ↩5

  4. [GAGA13] Yan Zhao, Géométrie algébrique et géométrie analytique, 2013. ↩ ↩2 ↩3 ↩4 ↩5 ↩6

  5. [Wiki] Wikipedia, Algebraic geometry and analytic geometry. ↩

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