More examples of complex manifolds. Let $X$ be a complex manifold of dimension $n.$ Suppose that we have a group $G$ holomorphically acting on $X$ (i.e., we have a group homomorphism $G \rightarrow \mathrm{Aut}(X),$ where the automorphism group is taken in the category of complex manifolds).
Suppose that this $G$-action on $X$ is properly discontinuous: that is, we assume that
- for any $x \in X,$ we may find an open neighborhood $U \ni x$ in $X$ such that the only $g \in G$ that makes $g(U)$ intersect $U$ is the identity of $G,$ and
- for any $x, y \in X$ in different $G$-orbits, we have some open neighborhoods $U \ni x$ and $V \ni y$ such that $g(U) \cap V = \emptyset$ for all $g \in G.$
The first condition ensures that the quotient map $X \twoheadrightarrow X/G$ is a covering space (a.k.a "$G$-covering"). The second condition ensures that $X/G$ is Hausdorff.
Proof of the second remark. indeed, take $U' := \bigcup_{g \in G}g(U)$ and $V' := \bigcup_{g \in G}g(V)$ so that $U'$ and $V'$ become the $G$-invariant open subsets of $X.$ Then $U'/G \ni [x]$ and $V'/G \ni [y]$ are open neighborhoods of $X/G,$ and because of Condition 2, we see that $U'/G$ and $V'/G$ are disjoint. $\Box$
Remark. Given any $x \in X,$ take an open subset $U \in x$ in $X$ such that each $g(U)$ are disjoint for distinct $g \in G,$ as in the first condition. If we take $U$ small enough, it comes with a holomorphic chart map $U \hookrightarrow \mathbb{C}^{n},$ and $U$ is homeomorhic to $U'/G,$ under the quotient map, where we used the notation in the proof above. This gives $X/G$ a complex manifold structure.
Example (Complex tori). Let $\Lambda \subset \mathbb{C}^{n}$ be a lattice, a free $\mathbb{Z}$-module of rank $2n$ such that $$\Lambda \otimes_{\mathbb{Z}} \mathbb{R} = \mathbb{R}^{2n} = \mathbb{C}^{n}.$$ Consider the action of $\Lambda$ on $\mathbb{C}^{n}$ given by the translation, namely, given $w \in \Lambda$ and $v \in \mathbb{C}^{n},$ the action is given by $$w \cdot v := w + v.$$ This action is properly discontinuous, so $\mathbb{C}^{n}/\Lambda$ is a complex manifold. When $n = 1,$ we may realize $\mathbb{C}^{1}/\Lambda$ as $\mathbb{C}$-points of an elliptic curve, which is a torus when it is considered as a real manifold.
Personal remark. I have only computed examples in this Wikipedia page long time ago, but just $\mathbb{C}$-points, so I am not entirely certain on what I am talking about in the last sentence above, although this should be roughly correct. I should go back and re-understand this scheme-theoretically.
On the other hand, when $n \geq 2,$ we will see (in this course) that most complex tori $\mathbb{C}^{n}/\Lambda$ do not arise from complex algebraic varieties.
Example (Hopf surface). Let $\mathbb{Z}$ act on $\mathbb{C}^{2} \smallsetminus \{(0,0)\}$ by defining its action on a generator $\gamma$ to be $\gamma \cdot (z_{1}, z_{2}) = (2z_{1}, 2z_{2}).$ Because we do not have $(0,0)$ on the space $\mathbb{Z}$ (or $\langle \gamma \rangle$) acts, this action is properly discontinuous, so we get the complex manifold structure on the group quotient. Consider the diffeomorphism $$\varphi : \mathbb{C}^{2} \smallsetminus \{(0,0)\} \overset{\sim}{\longrightarrow} S^{3} \times \mathbb{R}$$ given by $$(z_{1}, z_{2}) \mapsto \left(\frac{(z_{1}, z_{2})}{\sqrt{|z_{1}|^{2} + |z_{2}|^{2}}}, \log(\sqrt{|z_{1}|^{2} + |z_{2}|^{2}})\right),$$ as its inverse can be given as $$(\boldsymbol{v}, a) \mapsto e^{a}\boldsymbol{v}.$$ Moreover, we have $$\begin{align*}\varphi(\gamma \cdot (z_{1}, z_{2})) &= \varphi(2z_{1}, 2z_{2}) \\ &= \left(\frac{(z_{1}, z_{2})}{\sqrt{|z_{1}|^{2} + |z_{2}|^{2}}}, \log(\sqrt{|z_{1}|^{2} + |z_{2}|^{2}}) + \log(2)\right) \\ &= \varphi(z_{1}, z_{2}) + (0, \log(2)).\end{align*}$$ Thus, we see that $\varphi$ induces $$(\mathbb{C}^{2} \smallsetminus \{(0,0)\})/\langle \gamma \rangle \overset{\sim}{\longrightarrow} S^{3} \times \mathbb{R}/\log(2)\mathbb{Z} \simeq S^{3} \times S^{1},$$ which is topologically familiar to us.
Analytic spaces. We now discuss analytic spaces. Given a complex variety $X,$ its analytification $X^{\mathrm{an}}$ will have an analytic space structure. For a complex manifold, the analytic space structure will be identical to the complex manifold structure.
We first discuss the local model for an analytic space. Given an open subset $U \subset \mathbb{C}^{n},$ our local model is the vanishing set of some holomorphic functions $f_{1}, \dots, f_{r} : U \rightarrow \mathbb{C},$ namely $$Z = \{x \in U : f_{1}(x) = \cdots = f_{r}(x) = 0.\}$$ Given any open subset $V \subset Z,$ we define $\mathscr{O}_{Z}(V)$ to be the set of functions $V \rightarrow \mathbb{C}$ that can locally extend to a holomorphic function on an open subset of $U.$ Denoting $j : Z \hookrightarrow U$ for the inclusion map, we have may consider an exact sequence of $\mathscr{O}_{U}$-modules on $U$: $$0 \rightarrow \mathscr{I} \rightarrow \mathscr{O}_{U} \rightarrow j_{*}\mathscr{O}_{Z} \rightarrow 0.$$ Given any open $W \subset U,$ the map $$\Gamma(W, \mathscr{O}_{U}) \rightarrow \Gamma(W, j_{*}\mathscr{O}_{Z}) = \Gamma(Z \cap W, \mathscr{O}_{Z})$$ is given by $f \mapsto f|_{Z \cap W}.$ The kernel $\mathscr{I}$ is necessarily given by $$\Gamma(W, \mathscr{I}) = \{f \in \mathscr{O}_{U}(W) : f|_{Z \cap W} = 0\}.$$ Note that $(Z, \mathscr{O}_{Z})$ forms a locally ringed space: given $p \in Z,$ the germs in $\mathscr{O}_{Z,p}$ vanishing at $p$ gives rise to the unique maximal ideal.
Remark. The above is technically a reduced model (and we shall use this terminology below). The ideal sheaf $\mathscr{I}$ given above determines $Z$ because it is the vanishing locus of sections in $\mathscr{I}(W) = \Gamma(W, \mathscr{I})$ in $U.$ In the above, stalks of the sheaf $\mathscr{O}_{U}/\mathscr{I} \simeq j_{*}\mathscr{O}_{Z}$ are necessarily reduced, as it is defined using germs of actual $\mathbb{C}$-valued functions.
Possible non-reduced business. It is possible to give a non-reduced model by using an ideal sheaf $\mathscr{J}$ of $\mathscr{O}_{U}$ whose stalks are not radical (still generated by $f_{1}, \dots, f_{r} \in \mathscr{O}_{U}(U)$) and defining the structure sheaf of $Z$ by $\mathscr{O}_{U}/\mathscr{J}$: given any open $W \subset U,$ we have $$\mathscr{O}_{Z}(Z \cap W) := (\mathscr{O}_{U}/\mathscr{J})(W).$$ More specifically, consider $n = r = 1$ with $U = \mathbb{C}$ and take $f_{1}(t) = t^{2}.$ In this case, we have $Z = \{0\},$ while the non-reduced ideal sheaf $\mathscr{J}$ given by $f_{1}$ satisfies $\Gamma(Z, \mathscr{J}) = (t^{2}) \subset \mathbb{C}\{t\} = \mathscr{O}_{\mathbb{C}}(\mathbb{C}),$ (which we will see later why) so $$\mathscr{O}_{Z}(Z) = \mathbb{C}[t]/(t^{2}),$$ which is not a reduced ring.
A reduced analytic space is a locally ringed space $(X, \mathscr{O}_{X})$ such that
Proof of the second remark. indeed, take $U' := \bigcup_{g \in G}g(U)$ and $V' := \bigcup_{g \in G}g(V)$ so that $U'$ and $V'$ become the $G$-invariant open subsets of $X.$ Then $U'/G \ni [x]$ and $V'/G \ni [y]$ are open neighborhoods of $X/G,$ and because of Condition 2, we see that $U'/G$ and $V'/G$ are disjoint. $\Box$
Remark. Given any $x \in X,$ take an open subset $U \in x$ in $X$ such that each $g(U)$ are disjoint for distinct $g \in G,$ as in the first condition. If we take $U$ small enough, it comes with a holomorphic chart map $U \hookrightarrow \mathbb{C}^{n},$ and $U$ is homeomorhic to $U'/G,$ under the quotient map, where we used the notation in the proof above. This gives $X/G$ a complex manifold structure.
Example (Complex tori). Let $\Lambda \subset \mathbb{C}^{n}$ be a lattice, a free $\mathbb{Z}$-module of rank $2n$ such that $$\Lambda \otimes_{\mathbb{Z}} \mathbb{R} = \mathbb{R}^{2n} = \mathbb{C}^{n}.$$ Consider the action of $\Lambda$ on $\mathbb{C}^{n}$ given by the translation, namely, given $w \in \Lambda$ and $v \in \mathbb{C}^{n},$ the action is given by $$w \cdot v := w + v.$$ This action is properly discontinuous, so $\mathbb{C}^{n}/\Lambda$ is a complex manifold. When $n = 1,$ we may realize $\mathbb{C}^{1}/\Lambda$ as $\mathbb{C}$-points of an elliptic curve, which is a torus when it is considered as a real manifold.
Personal remark. I have only computed examples in this Wikipedia page long time ago, but just $\mathbb{C}$-points, so I am not entirely certain on what I am talking about in the last sentence above, although this should be roughly correct. I should go back and re-understand this scheme-theoretically.
On the other hand, when $n \geq 2,$ we will see (in this course) that most complex tori $\mathbb{C}^{n}/\Lambda$ do not arise from complex algebraic varieties.
Example (Hopf surface). Let $\mathbb{Z}$ act on $\mathbb{C}^{2} \smallsetminus \{(0,0)\}$ by defining its action on a generator $\gamma$ to be $\gamma \cdot (z_{1}, z_{2}) = (2z_{1}, 2z_{2}).$ Because we do not have $(0,0)$ on the space $\mathbb{Z}$ (or $\langle \gamma \rangle$) acts, this action is properly discontinuous, so we get the complex manifold structure on the group quotient. Consider the diffeomorphism $$\varphi : \mathbb{C}^{2} \smallsetminus \{(0,0)\} \overset{\sim}{\longrightarrow} S^{3} \times \mathbb{R}$$ given by $$(z_{1}, z_{2}) \mapsto \left(\frac{(z_{1}, z_{2})}{\sqrt{|z_{1}|^{2} + |z_{2}|^{2}}}, \log(\sqrt{|z_{1}|^{2} + |z_{2}|^{2}})\right),$$ as its inverse can be given as $$(\boldsymbol{v}, a) \mapsto e^{a}\boldsymbol{v}.$$ Moreover, we have $$\begin{align*}\varphi(\gamma \cdot (z_{1}, z_{2})) &= \varphi(2z_{1}, 2z_{2}) \\ &= \left(\frac{(z_{1}, z_{2})}{\sqrt{|z_{1}|^{2} + |z_{2}|^{2}}}, \log(\sqrt{|z_{1}|^{2} + |z_{2}|^{2}}) + \log(2)\right) \\ &= \varphi(z_{1}, z_{2}) + (0, \log(2)).\end{align*}$$ Thus, we see that $\varphi$ induces $$(\mathbb{C}^{2} \smallsetminus \{(0,0)\})/\langle \gamma \rangle \overset{\sim}{\longrightarrow} S^{3} \times \mathbb{R}/\log(2)\mathbb{Z} \simeq S^{3} \times S^{1},$$ which is topologically familiar to us.
Analytic spaces. We now discuss analytic spaces. Given a complex variety $X,$ its analytification $X^{\mathrm{an}}$ will have an analytic space structure. For a complex manifold, the analytic space structure will be identical to the complex manifold structure.
We first discuss the local model for an analytic space. Given an open subset $U \subset \mathbb{C}^{n},$ our local model is the vanishing set of some holomorphic functions $f_{1}, \dots, f_{r} : U \rightarrow \mathbb{C},$ namely $$Z = \{x \in U : f_{1}(x) = \cdots = f_{r}(x) = 0.\}$$ Given any open subset $V \subset Z,$ we define $\mathscr{O}_{Z}(V)$ to be the set of functions $V \rightarrow \mathbb{C}$ that can locally extend to a holomorphic function on an open subset of $U.$ Denoting $j : Z \hookrightarrow U$ for the inclusion map, we have may consider an exact sequence of $\mathscr{O}_{U}$-modules on $U$: $$0 \rightarrow \mathscr{I} \rightarrow \mathscr{O}_{U} \rightarrow j_{*}\mathscr{O}_{Z} \rightarrow 0.$$ Given any open $W \subset U,$ the map $$\Gamma(W, \mathscr{O}_{U}) \rightarrow \Gamma(W, j_{*}\mathscr{O}_{Z}) = \Gamma(Z \cap W, \mathscr{O}_{Z})$$ is given by $f \mapsto f|_{Z \cap W}.$ The kernel $\mathscr{I}$ is necessarily given by $$\Gamma(W, \mathscr{I}) = \{f \in \mathscr{O}_{U}(W) : f|_{Z \cap W} = 0\}.$$ Note that $(Z, \mathscr{O}_{Z})$ forms a locally ringed space: given $p \in Z,$ the germs in $\mathscr{O}_{Z,p}$ vanishing at $p$ gives rise to the unique maximal ideal.
Remark. The above is technically a reduced model (and we shall use this terminology below). The ideal sheaf $\mathscr{I}$ given above determines $Z$ because it is the vanishing locus of sections in $\mathscr{I}(W) = \Gamma(W, \mathscr{I})$ in $U.$ In the above, stalks of the sheaf $\mathscr{O}_{U}/\mathscr{I} \simeq j_{*}\mathscr{O}_{Z}$ are necessarily reduced, as it is defined using germs of actual $\mathbb{C}$-valued functions.
Possible non-reduced business. It is possible to give a non-reduced model by using an ideal sheaf $\mathscr{J}$ of $\mathscr{O}_{U}$ whose stalks are not radical (still generated by $f_{1}, \dots, f_{r} \in \mathscr{O}_{U}(U)$) and defining the structure sheaf of $Z$ by $\mathscr{O}_{U}/\mathscr{J}$: given any open $W \subset U,$ we have $$\mathscr{O}_{Z}(Z \cap W) := (\mathscr{O}_{U}/\mathscr{J})(W).$$ More specifically, consider $n = r = 1$ with $U = \mathbb{C}$ and take $f_{1}(t) = t^{2}.$ In this case, we have $Z = \{0\},$ while the non-reduced ideal sheaf $\mathscr{J}$ given by $f_{1}$ satisfies $\Gamma(Z, \mathscr{J}) = (t^{2}) \subset \mathbb{C}\{t\} = \mathscr{O}_{\mathbb{C}}(\mathbb{C}),$ (which we will see later why) so $$\mathscr{O}_{Z}(Z) = \mathbb{C}[t]/(t^{2}),$$ which is not a reduced ring.
A reduced analytic space is a locally ringed space $(X, \mathscr{O}_{X})$ such that
- $X$ is Hausdorff and second countable, and
- there is an open cover $X = \bigcup_{i \in I}W_{i}$ such that each $(W_{i}, \mathscr{O}_{X}|_{W_{i}})$ is isomorphic to a local model, in the category of locally ringed spaces.
Global sections of $\mathscr{O}_{X}$ are called holomorphic functions of $X.$ A holomorphic map $(X, \mathscr{O}_{X}) \rightarrow (Y, \mathscr{O}_{Y})$ is defined to be a morphism of locally ringed spaces given by the following data:
- $\pi : X \rightarrow Y$ a continuous map, and
- $\mathscr{O}_{Y} \rightarrow \pi_{*}\mathscr{O}_{X},$ a map of sheaves of $\mathbb{C}$-algebras on $Y$ such that $\mathscr{O}_{Y}(V) \rightarrow \mathscr{O}_{X}(\pi^{-1}(V))$ is given by $g \mapsto g \circ \pi.$
Reduced analytic spaces form a category, and complex manifolds form a full subcategory of it. Given any $\mathbb{C}$-variety $X,$ we can consider its analytification $(X^{\mathrm{an}}, \mathscr{O}_{X^{\mathrm{an}}}),$ as an (reduced) analytic space. Here is a recipe to construct $\mathscr{O}_{X^{\mathrm{an}}}$: choose any affine open cover $X = \bigcup_{i \in I}U_{i}.$ Consider $U = U_{i},$ which comes with a closed embedding $U \hookrightarrow \mathbb{A}^{N}.$ This means that $U$ is cut out by finitely many polynomials in $N$ variables, so we have a structure sheaf $\mathscr{O}_{U^{\mathrm{an}}}$ on $U^{\mathrm{an}} = U(\mathbb{C})$ making it an analytic space. The gluing data among $\mathscr{O}_{U_{i}}$ induces some corresponding gluing data among $\mathscr{O}_{U_{i}^{\mathrm{an}}}.$ For this, it is important that sheaf maps for holomorphic maps are given by post compositions. A similar argument constrcuts a functor $\textbf{Var}_{\mathbb{C}} \rightarrow \textbf{Anal}_{\mathbb{C}}.$
Comparison. Let $X$ be any complex variety. Then we have a comparison map (or I would like to say, "unanalytifying map") $u : (X^{\mathrm{an}}, \mathscr{O}_{X^{\mathrm{an}}}) \rightarrow (X, \mathscr{O}_{X}),$ which is given by the following data:
Comparison. Let $X$ be any complex variety. Then we have a comparison map (or I would like to say, "unanalytifying map") $u : (X^{\mathrm{an}}, \mathscr{O}_{X^{\mathrm{an}}}) \rightarrow (X, \mathscr{O}_{X}),$ which is given by the following data:
- $u : X(\mathbb{C}) \hookrightarrow X$ is the inclusion on the level of topological spaces;
- $\mathscr{O}_{X} \rightarrow u_{*}\mathscr{O}_{X}$ is given as follows:
for any open $U \subset X,$ we have $\Gamma(U, \mathscr{O}_{X}) \rightarrow \Gamma(U(\mathbb{C}), \mathscr{O}_{X^{\mathrm{an}}})$ given by $f \mapsto f^{\mathrm{an}}.$ For any $x \in X(\mathbb{C}),$ the map on stalks $\mathscr{O}_{X,x} \rightarrow \mathscr{O}_{X^{\mathrm{an}},x}$ is a local map.
Why? The preimage of the maximal ideal of the target consists of $f \in \mathscr{O}_{X,x}$ that lifts to an element $f \in \Gamma(U, \mathscr{O}_{X})$ for some open $U \ni x$ such that the induced map $f : U \rightarrow \mathbb{A}^{1}$ vanishes at $x,$ namely $x \mapsto (t) \in \mathrm{Spec}(\mathbb{C}[t]) = \mathbb{A}^{1}.$ This map factors as $$\mathrm{Spec}(\mathscr{O}_{X,x}) \rightarrow U \overset{f}{\longrightarrow} \mathbb{A}^{1},$$ where the first map is given by the localization, which gives $\mathfrak{m}_{x} \mapsto x.$ The induced map $f$ is from the ring map $\mathbb{C}[t] \rightarrow \Gamma(U, \mathscr{O}_{X})$ given by $t \mapsto f,$ so saying that $f$ is vanishing at $x$ is the same as saying that when we localize this ring map to get $\mathbb{C}[t] \rightarrow \mathscr{O}_{X,x},$ the maximal ideal $\mathfrak{m}_{x}$ pulls back to $(t).$ Thus, the image of $t,$ namely $f,$ sits inside $\mathfrak{m}_{x}.$ This shows that $\mathscr{O}_{X,x} \rightarrow \mathscr{O}_{X^{\mathrm{an}},x}$ is a local map.
Given any $\mathscr{O}_{X}$-module $\mathscr{F},$ we define $$\mathscr{F}^{\mathrm{an}} := u^{*}\mathscr{F},$$ the pullback of $\mathscr{F}$ under $u.$
Remark. In other words, we have $\mathscr{F}^{\mathrm{an}} = u^{-1}\mathscr{F} \otimes_{u^{-1}\mathscr{O}_{X}} \mathscr{O}_{X^{\mathrm{an}}}.$ Or better yet, we have a bijection $$\mathrm{Hom}_{\mathscr{O}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}}, \mathscr{G}) \simeq \mathrm{Hom}_{\mathscr{O}_{X}}(\mathscr{F}, u_{*}\mathscr{G})$$ functorial in $\mathscr{G} \in \mathrm{Mod}_{\mathscr{O}_{X^{\mathrm{an}}}}$ and $\mathscr{F} \in \mathrm{Mod}_{\mathscr{O}_{X}}.$ (See Vakil 16.3.4 and 16.3.5.) With either definition, it is not difficult to check that $\mathscr{O}_{X}^{\mathrm{an}} = \mathscr{O}_{X^{\mathrm{an}}}.$ From the first definition (along with a possible proof of Vakil 2.6.J and Vakil 2.7.C), we note that $$(\mathscr{F}^{\mathrm{an}})_{x} \simeq \mathscr{F}_{x} \otimes_{\mathscr{O}_{X,x}} \mathscr{O}_{X^{\mathrm{an}},x}.$$ Since the analytification of $\mathscr{O}_{X}$-modules are given by the pullback under unanalytification, it is a functor $\mathrm{Mod}_{\mathscr{O}_{X}} \rightarrow \mathrm{Mod}_{\mathscr{O}_{X^{\mathrm{an}}}}.$ Hence, for any $\mathscr{O}_{X}$-modules $\mathscr{F}$ and $\mathscr{G},$ we have a map $$\mathrm{Hom}_{\mathscr{O}_{X}}(\mathscr{F}, \mathscr{G}) \rightarrow \mathrm{Hom}_{\mathscr{O}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}}, \mathscr{G}^{\mathrm{an}}).$$
Why? The preimage of the maximal ideal of the target consists of $f \in \mathscr{O}_{X,x}$ that lifts to an element $f \in \Gamma(U, \mathscr{O}_{X})$ for some open $U \ni x$ such that the induced map $f : U \rightarrow \mathbb{A}^{1}$ vanishes at $x,$ namely $x \mapsto (t) \in \mathrm{Spec}(\mathbb{C}[t]) = \mathbb{A}^{1}.$ This map factors as $$\mathrm{Spec}(\mathscr{O}_{X,x}) \rightarrow U \overset{f}{\longrightarrow} \mathbb{A}^{1},$$ where the first map is given by the localization, which gives $\mathfrak{m}_{x} \mapsto x.$ The induced map $f$ is from the ring map $\mathbb{C}[t] \rightarrow \Gamma(U, \mathscr{O}_{X})$ given by $t \mapsto f,$ so saying that $f$ is vanishing at $x$ is the same as saying that when we localize this ring map to get $\mathbb{C}[t] \rightarrow \mathscr{O}_{X,x},$ the maximal ideal $\mathfrak{m}_{x}$ pulls back to $(t).$ Thus, the image of $t,$ namely $f,$ sits inside $\mathfrak{m}_{x}.$ This shows that $\mathscr{O}_{X,x} \rightarrow \mathscr{O}_{X^{\mathrm{an}},x}$ is a local map.
Given any $\mathscr{O}_{X}$-module $\mathscr{F},$ we define $$\mathscr{F}^{\mathrm{an}} := u^{*}\mathscr{F},$$ the pullback of $\mathscr{F}$ under $u.$
Remark. In other words, we have $\mathscr{F}^{\mathrm{an}} = u^{-1}\mathscr{F} \otimes_{u^{-1}\mathscr{O}_{X}} \mathscr{O}_{X^{\mathrm{an}}}.$ Or better yet, we have a bijection $$\mathrm{Hom}_{\mathscr{O}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}}, \mathscr{G}) \simeq \mathrm{Hom}_{\mathscr{O}_{X}}(\mathscr{F}, u_{*}\mathscr{G})$$ functorial in $\mathscr{G} \in \mathrm{Mod}_{\mathscr{O}_{X^{\mathrm{an}}}}$ and $\mathscr{F} \in \mathrm{Mod}_{\mathscr{O}_{X}}.$ (See Vakil 16.3.4 and 16.3.5.) With either definition, it is not difficult to check that $\mathscr{O}_{X}^{\mathrm{an}} = \mathscr{O}_{X^{\mathrm{an}}}.$ From the first definition (along with a possible proof of Vakil 2.6.J and Vakil 2.7.C), we note that $$(\mathscr{F}^{\mathrm{an}})_{x} \simeq \mathscr{F}_{x} \otimes_{\mathscr{O}_{X,x}} \mathscr{O}_{X^{\mathrm{an}},x}.$$ Since the analytification of $\mathscr{O}_{X}$-modules are given by the pullback under unanalytification, it is a functor $\mathrm{Mod}_{\mathscr{O}_{X}} \rightarrow \mathrm{Mod}_{\mathscr{O}_{X^{\mathrm{an}}}}.$ Hence, for any $\mathscr{O}_{X}$-modules $\mathscr{F}$ and $\mathscr{G},$ we have a map $$\mathrm{Hom}_{\mathscr{O}_{X}}(\mathscr{F}, \mathscr{G}) \rightarrow \mathrm{Hom}_{\mathscr{O}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}}, \mathscr{G}^{\mathrm{an}}).$$
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