Hodge theory: Lecture 6

Our main interest lies on the geometry of varieties $X$ over $\mathbb{C}$ (i.e., reduced and separated schemes finite type over $\mathbb{C}$), and we wanted to understand them by analytic geometry of $X^{\mathrm{an}} = X(\mathbb{C}).$ We realized that this gives the analytification functor $\textbf{Var}_{\mathbb{C}} \rightarrow \textbf{Top},$ but we can vividly see from its definition that $X(\mathbb{C})$ has more structure than just its topology. In general, it seems that we can give the structure of analytic spaces to $X^{\mathrm{an}},$ but for smooth varieties (which will probably be our main focus), such structures will be turn out to be the same notion as complex manifolds.

Complex manifolds. Given a nonempty open subset $U \subset \mathbb{C}^{n},$ let $$\mathscr{O}_{U}(V) := \{f : V \rightarrow \mathbb{C} : f\text{ holomorphic}\},$$ for any nonempty open $V \subset U.$ Note that this is a $\mathbb{C}$-algebra. We define $\mathscr{O}_{U}(\emptyset) = 0,$ the zero $\mathbb{C}$-algebra. Note that these form a sheaf $\mathscr{O}_{U},$ where we could use the actual restrictions to define restriction maps. (The key here is that holomorphicity can be checked open-locally.)

Remark. I think we have $\mathscr{O}_{\emptyset} = 0,$ in any reasonable definition. In any case, let's declare this.

A complex manifold is a pair $(X, \mathscr{O}_{X})$ such that

1. $X$ is a topological space that is Hausdorff and second countable, and
2. $\mathscr{O}_{X} \hookrightarrow \mathscr{C}_{X, \mathbb{C}},$ the sheaf of $\mathbb{C}$-algebras given by $\mathbb{C}$-valued continuous functions, is a subsheaf over $X$ with is an open cover $X = \bigcup_{i \in I}U_{i}$ such that for each $i \in I,$ we have some open subset $V_{i} \subset \mathbb{C}^{n}$ with an isomorphism of ringed spaces $(U_{i} \mathscr{O}_{U_{i}}) \simeq (V_{i} \mathscr{O}_{V_{i}}).$

Remark. The last condition is more concrete than the way it may look. It is saying that we have a homeomorphism $\varphi : U_{i} \overset{\sim}{\longrightarrow} V_{i} \subset \mathbb{C}^{n}$ with a sheaf isomoprhism $\varphi_{*}\mathscr{O}_{U_{i}} \overset{\sim}{\longrightarrow} \mathscr{O}_{V_{i}}$ over $V_{i}.$ Thus, for any open $W \subset V_{i},$ we have a $\mathbb{C}$-algebra isomorphism $$\mathscr{O}_{V_{i}}(W) \overset{\sim}{\longrightarrow} \mathscr{O}_{U_{i}}(\phi^{-1}(W)),$$ so each element in the right-hand side corresponds to a holomorphic function $W \rightarrow \mathbb{C},$ and we can treat it as if it is the composition $\varphi^{-1}(W) \simeq W \rightarrow \mathbb{C},$ which we declare to be holomorphic on $\varphi^{-1}(W).$ This way, we obtain some concrete description of $\mathscr{O}_{X}.$ By definition, global sections of $\mathscr{O}_{X}$ are called holomorphic functions on $X,$ but we also have a concrete meaning namely, each $f \in \Gamma(X, \mathscr{O}_{X})$ can be realized as a continuous map $f : X \rightarrow \mathbb{C}$ such that for every $p \in X,$ there is an open neighborhood $U \ni p$ in $X,$ with open embedding $U \hookrightarrow \mathbb{C}^{n}$ such that $f : U \rightarrow \mathbb{C}$ is holomormophic. (The gluing procedure in $\textbf{Top}$ will do the job, and the same holds for any open subset of $X$ in place of the whole $X.$)

A holomorphic map $(X, \mathscr{O}_{X}) \rightarrow (Y, \mathscr{O}_{Y})$ is a continuous map $\phi : X \rightarrow Y$ together with a sheaf map $\mathscr{O}_{Y} \rightarrow \phi_{*}\mathscr{O}_{X}$ given as follows: for any open $V \subset Y,$ we have a $\mathbb{C}$-algebra map $$\Gamma(V, \mathscr{O}_{Y}) \rightarrow \Gamma (\phi^{-1}(V), \mathscr{O}_{X})$$ given by $g \mapsto g \circ \phi.$

Remark. If $X \subset \mathbb{C}^{n}$ and $Y \subset \mathbb{C}^{m}$ are open subsets, the above coincides with our previous definition of holomorphic maps.

Remark. Note that a complex manifold $(X, \mathscr{O}_{X})$ is a locally ringed space. To see this fix $p \in X$ and consider the stalk $\mathscr{O}_{X, p}$ at $p.$ Since any germ not vanishing at $p$ does not vanish nearby $p,$ we see that $$\mathfrak{m}_{X,p} = \{f \in \mathscr{O}_{X,p} : f(p) = 0\}$$ is the unique maximal ideal of $\mathscr{O}_{X, p},$ so $\mathscr{O}_{X,p}$ is a local ring.

Given any holomorphic map $\phi : X \rightarrow Y,$ for any $p \in X,$ we get an induced $\mathbb{C}$-algebra map on stalks $\mathscr{O}_{Y,\phi(p)} \rightarrow \mathscr{O}_{X,p}$ given by $$g \mapsto g \circ \phi.$$ Hence, we have $\mathfrak{m}_{p} \rightarrow \mathfrak{m}_{\phi(p)}$ under this map, so the map on the stalks is a local map. Thus, a holomorphic map of complex manifolds is a morphism of locally ringed spaces.

Personal question. In the case of varieties, any morphism $\phi : X \rightarrow Y$ of locally ringed spaces gives the map $\mathscr{O}_{Y} \rightarrow \phi_{*}\mathscr{O}_{X}$ by the post composition $g \mapsto g \circ \phi.$ I have a feeling that this is a very special feature of varieties, but I cannot tell whether there is a counterexample for real/complex manifolds. Are there?

Remark. For any $n$-dimensional manifold, it is not difficult to see that any of its stalk is isomorphic to the subalgebra of the formal power series ring $\mathbb{C}[[z_{1}, \dots, z_{n}]]$ consisting of convergent power series at $(0, \dots, 0).$

Atlas definition. One can define complex manifolds using atlases. That is, an $n$-dimensional complex manifold $X$ is a Hausdorff and 2nd countable topological space with an open cover $X = \bigcup_{i \in I}U_{i}$ equipped with homemomorphisms $$\varphi_{i} : U_{i} \overset{\sim}{\longrightarrow} V_{i} \overset{\text{open}}{\hookrightarrow} \mathbb{C}^{n}$$ such that for all $i, j \in I,$ the map $$\varphi_{j} \circ \varphi_{i}^{-1} : \varphi_{i}(U_{i} \cap U_{j}) \rightarrow \varphi_{j}(U_{i} \cap U_{j})$$ is a biholomorphic map. (Just perform gluing of sheaves $\mathscr{O}_{U_{i}}$ according to $\varphi_{i}.$)

Based on the atlas definition, since holomorphic maps between the open subsets of Euclidean spaces are smooth, any complex manifold $X$ of dimension $n$ has an underlying real smooth manifold structure $X_{\mathbb{R}}$ of dimension $2n.$ We also have the inclusion of two different sheaves $\mathscr{O}_{X} \hookrightarrow \mathscr{C}^{\infty}_{X, \mathbb{C}}$ for $X,$ and we will use both in a tidy way to understand $X$ better.

Personal remark. We did discuss the Maximum Modulus Theorem, but I decided not to include it in these notes because we are not using the theorem here.