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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