Vector bundles. Given a real smooth manifold M, a real (or respectively complex) vector bundle on M of rank r \geq 0 is a smooth map \pi : E \rightarrow M with an open cover E = \bigcup_{i \in I}U_{i} such that for each i \in I, we have a diffeomorphism \pi^{-1}(U_{i}) \simeq U_{i} \times k^{r} over U_{i} (where k is \mathbb{R} or \mathbb{C}) and at each x \in U_{i}, the diffeomorphism restricts to k-linear isomorphism \pi^{-1}(x) \simeq \{x\} \times k^{r}. Note that E gives us a sheaf \mathcal{S}_{E} defined on M given by \mathcal{S}_{E}(U) = \{s : U \rightarrow E \text{ smooth } | \ \pi \circ s = \mathrm{id}_{U}\}. This gives an equivalence between the category of k-vector bundles on M of rank r and the category of locally free sheaves of rank r of \mathscr{C}^{r}_{M,k}-modules.
Similarly, for a complex manifold X, we have an equivalence between the category of holomorphic vector bundles on X of rank r and the category of locally free sheaves of rank r of \mathscr{O}_{X}-modules.
Personal remark. I have checked this years ago, and it was really about understanding formal gluing well enough.
Remark. For a holomorphic vector bundle, we will study both of its smooth sections and holomorphic sections.
Submanifolds. Let X be a complex manifold of dimension n. A closed submanifold of X of codimension 0 \leq r \leq n is a closed subset Y \subset X with an open covering X = \bigcup_{i \in I}U_{i} and chart maps \varphi_{i} : U_{i} \overset{\sim}{\longrightarrow} V_{i} \overset{\text{open}}{\hookrightarrow} \mathbb{C}^{n} such that \varphi_{i}(U_{i} \cap Y) = \{z = (z_{1}, \dots, z_{n}) \in V_{i} : z_{1} = \cdots = z_{r} = 0\}. Note that by restricting such charts to Y, we can get a holomorphic atlas on Y, making it a complex manifold of dimension n - r.
Universal property. Let \iota : Y \hookrightarrow X be a complex closed submanifold. Given any holomorphic g : Z \rightarrow X such that g(Z) \subset Y, there is a unique holomorphic map g' : Z \rightarrow Y such that \iota \circ g' = g.
Personal remark. The above remark just seems to say that we can restrict the target of any holomorphic map to a closed submanifold containing the image. Perhaps there are better philosophical interpretations of this, but I am not able to come up with any at the moment.
Maximum rank Jacobian. We now study a criterion that tells us whether given holomorphic functions cut out closed submanifold. This should remind us the definition of smoothness of a variety over a field (Vakil 12.2.6).
Proposition. Let 0 \leq r \leq n. If U \subset \mathbb{C}^{n} is an open subset and f_{1}, \dots, f_{r} \in \mathscr{O}_{\mathbb{C}^{n}}(U), where the rank of the Jacobian matrix \left[\frac{\partial f_{i}}{\partial z_{j}} \right]_{\substack{1 \leq i \leq r \\ 1 \leq j \leq n}} has rank r everywhere in U. Then Z = \{z \in U : f_{1}(z) = \cdots = f_{r}(z) = 0\} \hookrightarrow U defines a closed submanifold of codimension r.
Proof. Given p \in Z, rearranging the orders of f_{1}, \dots, f_{r} if necessary, we may assume that \det\left[\frac{\partial f_{i}}{\partial z_{j}}(p) \right]_{1 \leq i, j \leq r} \neq 0. Consider \varphi : U \rightarrow \mathbb{C}^{n} defined by \varphi(z) = (f_{1}(z), \dots, f_{r}(z), z_{r+1}, \dots, z_{n}). Then \det\left[\frac{\partial \varphi_{i}}{\partial z_{j}}(p) \right]_{1 \leq i, j \leq n} = \det\left[\frac{\partial f_{i}}{\partial z_{j}}(p) \right]_{1 \leq i, j \leq r} \neq 0, so by the Inverse Function Theorem, we see \varphi is biholomorphic in some neighborhood of p, which gives a desired holomorphic chart map near p. \Box
Complex manifolds from smooth complex varieties. Let X be a smooth variety over \mathbb{C} of pure dimension n. Let U \subset X be an affine open subset and U \hookrightarrow \mathbb{A}^{N} a closed immersion. Denote r = N - n, and consider an affine open cover \mathbb{A}^{N} = \bigcup_{\alpha \in I}V_{\alpha}. If V_{\alpha} \cap U is nonempty, then V_{\alpha} \cap U \hookrightarrow V_{\alpha} is a closed subscheme cut out by r sections f_{1}, \dots, f_{r} \in \mathscr{O}_{V_{\alpha}}(U \cap V_{\alpha}) such that \left[\frac{\partial f_{i}}{\partial z_{j}} \right]_{\substack{1 \leq i \leq r \\ 1 \leq j \leq n}} has rank r everywhere in U, by the definition of smoothness. Applying the previous proposition, we see that the each V_{\alpha}(\mathbb{C}) \cap U(\mathbb{C}) \hookrightarrow V_{\alpha}(\mathbb{C}) defines a complex submanifold of codimension r, or equivalently, dimension n. The resulting transition maps are holomorphic, because sections are given as rational maps without singularities in their domain. This defines a complex manifold structure on X^{\mathrm{an}} = X(\mathbb{C}).
Remark. Note that if we had any number of sections in \mathscr{O}_{V_{\alpha}}(V_{\alpha} \cap U) cutting out V_{\alpha} \cap U \hookrightarrow V_{\alpha} bounds the codimension r from above by Krull's Height Theorem. The fact that we can have precisely r sections is the part of the definition of smoothness. I think that the last part of the previous paragraph can be seen by formal gluing as well. Moreover, also note that at least for any smooth variety X over \mathbb{C}, we may say write \mathscr{O}_{X}^{\mathrm{an}} := \mathscr{O}_{X^{\mathrm{an}}}, and think of it as if the structure sheaf \mathscr{O}_{X} is "analytified".
Analytification functor on smooth varieties. Now we note that any morphism for f : X \rightarrow Y of varieties smooth over \mathbb{C}, the induced map f^{\mathrm{an}} : X^{\mathrm{an}} \rightarrow Y^{\mathrm{an}} on complex manifolds is holomorphic. If I am not mistaken, checking this is surprisingly easy: for any affine open V \subset Y, a section \mathscr{O}_{Y^{\mathrm{an}}}(V) is given by analytifying a \mathbb{C}-scheme map V \rightarrow \mathbb{A}^{1}. The pullback of such a section in \mathscr{O}_{X^{\mathrm{an}}}(f^{-1}(V)) is given by analytifying f^{-1}(V) \rightarrow V \rightarrow \mathbb{A}^{1}, which is holomorphic because (locally) rational maps are holomorphic. This phenomenon can be seen as the analytification functor \textbf{Var}_{\mathbb{C}} \rightarrow \textbf{Top} taking smooth varieties into the category of complex manifolds.
Maximum modulus principle. If f : U \rightarrow \mathbb{C} is a holomorphic function on an open and connected subset U \subset \mathbb{C}^{n} such that |f| has a local maximum, then f is constant on the whole U.
Proof. Let p \in U be a point where |f| attains a local maximum. Since U is connected, we can make the statement local by a property of holomorphic functions on U discussed in a previous posting. Namely, it is enough to find an open neighborhood U_{\epsilon} \ni p in U such that f is constant on U_{0}. We do this in two steps.
Step 1: Reduction to the case n = 1. Take \epsilon > 0 small enough so that U_{\epsilon} := D_{\epsilon}(p_{1}) \times \cdots \times D_{\epsilon}(p_{n}) \subset U. Of course, we note that p = (p_{1}, \dots, p_{n}) \in U_{\epsilon}. We assume that the statement is proven for the case of n = 1 and use this to show that f : U_{\epsilon} \rightarrow \mathbb{C} is constant. Take any z \in U_{\epsilon}, and we would like to prove that f(z) = f(p). Take \delta > 0 small enough so that p + \delta (z - p) \in U_{\epsilon}. Consider the map L_{p,z} : D_{1+\delta}(0) \rightarrow U_{\epsilon} given by w \mapsto (1-w)p + wz, which gives us L_{p,z}(0) = p and L_{p,z}(1) = z. By the special case n = 1, we see that the composition f \circ L_{p,z} : D_{1+\delta}(0) \rightarrow \mathbb{C} is constant, so f(p) = f(L_{p,z}(0)) = f(L_{p,z}(1)) = f(z). Hence, we now focus on the spacial case n = 1.
Step 2: The case n = 1. Let \epsilon > 0 be small enough so that \overline{D_{\epsilon}(p)} \subset U and |f(p)| is maximum in \overline{D_{\epsilon}(p)}. By Cauchy's formula (and taking w = p + \epsilon e^{2\pi i \theta}), we have \begin{align*} f(p) &= \frac{1}{2\pi i} \int_{\partial D_{\epsilon}(p)} \frac{f(w)}{w - p}dw \\ &= \frac{1}{2\pi i}\int_{0}^{1} \frac{f(p + \epsilon e^{2\pi i \theta})}{\epsilon e^{2\pi i \theta}} \epsilon 2\pi e^{2\pi i \theta} d\theta \\ &= \int_{0}^{1} f(p + \epsilon e^{2\pi i \theta}) d\theta \end{align*}. Thus, we have |f(p)| \leq \int_{0}^{1} |f(p + \epsilon e^{2\pi i \theta})| d\theta \leq \int_{0}^{1} |f(p)| d\theta = |f(p)|. Therefore, we have |f(p)| = \int_{0}^{1} |f(p + \epsilon e^{2\pi i \theta})| d\theta. This implies that we have \int_{0}^{1} |f(p) - f(p + \epsilon e^{2\pi i \theta})| d\theta = \int_{0}^{1} |f(p)| - |f(p + \epsilon e^{2\pi i \theta})| d\theta = 0, so continuity of f ensures that we must have f(p) = f(p + \epsilon e^{2\pi \theta}) for all 0 \leq \theta \leq 1. This shows that f is constant on D_{\epsilon}(0), which is enough to conclude that f is constant on U, because U is connected. \Box
Remark. As a corollary, given a connected complex manifold X, if f is a global section of \mathscr{O}_{X} such that |f| attains a local maximum, the section f must be constant on whole X. In particular, if X is a compact connected complex manifold, we have \Gamma(X, \mathscr{O}_{X}) = \mathbb{C}.
Of course, we are using that given a connected complex manifold X, if f : X \rightarrow \mathbb{C} is a global section that vanishes on a nonempty open subset of X, then f = 0. This follows from a similar argument in the last proposition of this previous posting.
Universal property. Let \iota : Y \hookrightarrow X be a complex closed submanifold. Given any holomorphic g : Z \rightarrow X such that g(Z) \subset Y, there is a unique holomorphic map g' : Z \rightarrow Y such that \iota \circ g' = g.
Personal remark. The above remark just seems to say that we can restrict the target of any holomorphic map to a closed submanifold containing the image. Perhaps there are better philosophical interpretations of this, but I am not able to come up with any at the moment.
Maximum rank Jacobian. We now study a criterion that tells us whether given holomorphic functions cut out closed submanifold. This should remind us the definition of smoothness of a variety over a field (Vakil 12.2.6).
Proposition. Let 0 \leq r \leq n. If U \subset \mathbb{C}^{n} is an open subset and f_{1}, \dots, f_{r} \in \mathscr{O}_{\mathbb{C}^{n}}(U), where the rank of the Jacobian matrix \left[\frac{\partial f_{i}}{\partial z_{j}} \right]_{\substack{1 \leq i \leq r \\ 1 \leq j \leq n}} has rank r everywhere in U. Then Z = \{z \in U : f_{1}(z) = \cdots = f_{r}(z) = 0\} \hookrightarrow U defines a closed submanifold of codimension r.
Proof. Given p \in Z, rearranging the orders of f_{1}, \dots, f_{r} if necessary, we may assume that \det\left[\frac{\partial f_{i}}{\partial z_{j}}(p) \right]_{1 \leq i, j \leq r} \neq 0. Consider \varphi : U \rightarrow \mathbb{C}^{n} defined by \varphi(z) = (f_{1}(z), \dots, f_{r}(z), z_{r+1}, \dots, z_{n}). Then \det\left[\frac{\partial \varphi_{i}}{\partial z_{j}}(p) \right]_{1 \leq i, j \leq n} = \det\left[\frac{\partial f_{i}}{\partial z_{j}}(p) \right]_{1 \leq i, j \leq r} \neq 0, so by the Inverse Function Theorem, we see \varphi is biholomorphic in some neighborhood of p, which gives a desired holomorphic chart map near p. \Box
Complex manifolds from smooth complex varieties. Let X be a smooth variety over \mathbb{C} of pure dimension n. Let U \subset X be an affine open subset and U \hookrightarrow \mathbb{A}^{N} a closed immersion. Denote r = N - n, and consider an affine open cover \mathbb{A}^{N} = \bigcup_{\alpha \in I}V_{\alpha}. If V_{\alpha} \cap U is nonempty, then V_{\alpha} \cap U \hookrightarrow V_{\alpha} is a closed subscheme cut out by r sections f_{1}, \dots, f_{r} \in \mathscr{O}_{V_{\alpha}}(U \cap V_{\alpha}) such that \left[\frac{\partial f_{i}}{\partial z_{j}} \right]_{\substack{1 \leq i \leq r \\ 1 \leq j \leq n}} has rank r everywhere in U, by the definition of smoothness. Applying the previous proposition, we see that the each V_{\alpha}(\mathbb{C}) \cap U(\mathbb{C}) \hookrightarrow V_{\alpha}(\mathbb{C}) defines a complex submanifold of codimension r, or equivalently, dimension n. The resulting transition maps are holomorphic, because sections are given as rational maps without singularities in their domain. This defines a complex manifold structure on X^{\mathrm{an}} = X(\mathbb{C}).
Remark. Note that if we had any number of sections in \mathscr{O}_{V_{\alpha}}(V_{\alpha} \cap U) cutting out V_{\alpha} \cap U \hookrightarrow V_{\alpha} bounds the codimension r from above by Krull's Height Theorem. The fact that we can have precisely r sections is the part of the definition of smoothness. I think that the last part of the previous paragraph can be seen by formal gluing as well. Moreover, also note that at least for any smooth variety X over \mathbb{C}, we may say write \mathscr{O}_{X}^{\mathrm{an}} := \mathscr{O}_{X^{\mathrm{an}}}, and think of it as if the structure sheaf \mathscr{O}_{X} is "analytified".
Analytification functor on smooth varieties. Now we note that any morphism for f : X \rightarrow Y of varieties smooth over \mathbb{C}, the induced map f^{\mathrm{an}} : X^{\mathrm{an}} \rightarrow Y^{\mathrm{an}} on complex manifolds is holomorphic. If I am not mistaken, checking this is surprisingly easy: for any affine open V \subset Y, a section \mathscr{O}_{Y^{\mathrm{an}}}(V) is given by analytifying a \mathbb{C}-scheme map V \rightarrow \mathbb{A}^{1}. The pullback of such a section in \mathscr{O}_{X^{\mathrm{an}}}(f^{-1}(V)) is given by analytifying f^{-1}(V) \rightarrow V \rightarrow \mathbb{A}^{1}, which is holomorphic because (locally) rational maps are holomorphic. This phenomenon can be seen as the analytification functor \textbf{Var}_{\mathbb{C}} \rightarrow \textbf{Top} taking smooth varieties into the category of complex manifolds.
Maximum modulus principle. If f : U \rightarrow \mathbb{C} is a holomorphic function on an open and connected subset U \subset \mathbb{C}^{n} such that |f| has a local maximum, then f is constant on the whole U.
Proof. Let p \in U be a point where |f| attains a local maximum. Since U is connected, we can make the statement local by a property of holomorphic functions on U discussed in a previous posting. Namely, it is enough to find an open neighborhood U_{\epsilon} \ni p in U such that f is constant on U_{0}. We do this in two steps.
Step 1: Reduction to the case n = 1. Take \epsilon > 0 small enough so that U_{\epsilon} := D_{\epsilon}(p_{1}) \times \cdots \times D_{\epsilon}(p_{n}) \subset U. Of course, we note that p = (p_{1}, \dots, p_{n}) \in U_{\epsilon}. We assume that the statement is proven for the case of n = 1 and use this to show that f : U_{\epsilon} \rightarrow \mathbb{C} is constant. Take any z \in U_{\epsilon}, and we would like to prove that f(z) = f(p). Take \delta > 0 small enough so that p + \delta (z - p) \in U_{\epsilon}. Consider the map L_{p,z} : D_{1+\delta}(0) \rightarrow U_{\epsilon} given by w \mapsto (1-w)p + wz, which gives us L_{p,z}(0) = p and L_{p,z}(1) = z. By the special case n = 1, we see that the composition f \circ L_{p,z} : D_{1+\delta}(0) \rightarrow \mathbb{C} is constant, so f(p) = f(L_{p,z}(0)) = f(L_{p,z}(1)) = f(z). Hence, we now focus on the spacial case n = 1.
Step 2: The case n = 1. Let \epsilon > 0 be small enough so that \overline{D_{\epsilon}(p)} \subset U and |f(p)| is maximum in \overline{D_{\epsilon}(p)}. By Cauchy's formula (and taking w = p + \epsilon e^{2\pi i \theta}), we have \begin{align*} f(p) &= \frac{1}{2\pi i} \int_{\partial D_{\epsilon}(p)} \frac{f(w)}{w - p}dw \\ &= \frac{1}{2\pi i}\int_{0}^{1} \frac{f(p + \epsilon e^{2\pi i \theta})}{\epsilon e^{2\pi i \theta}} \epsilon 2\pi e^{2\pi i \theta} d\theta \\ &= \int_{0}^{1} f(p + \epsilon e^{2\pi i \theta}) d\theta \end{align*}. Thus, we have |f(p)| \leq \int_{0}^{1} |f(p + \epsilon e^{2\pi i \theta})| d\theta \leq \int_{0}^{1} |f(p)| d\theta = |f(p)|. Therefore, we have |f(p)| = \int_{0}^{1} |f(p + \epsilon e^{2\pi i \theta})| d\theta. This implies that we have \int_{0}^{1} |f(p) - f(p + \epsilon e^{2\pi i \theta})| d\theta = \int_{0}^{1} |f(p)| - |f(p + \epsilon e^{2\pi i \theta})| d\theta = 0, so continuity of f ensures that we must have f(p) = f(p + \epsilon e^{2\pi \theta}) for all 0 \leq \theta \leq 1. This shows that f is constant on D_{\epsilon}(0), which is enough to conclude that f is constant on U, because U is connected. \Box
Remark. As a corollary, given a connected complex manifold X, if f is a global section of \mathscr{O}_{X} such that |f| attains a local maximum, the section f must be constant on whole X. In particular, if X is a compact connected complex manifold, we have \Gamma(X, \mathscr{O}_{X}) = \mathbb{C}.
Of course, we are using that given a connected complex manifold X, if f : X \rightarrow \mathbb{C} is a global section that vanishes on a nonempty open subset of X, then f = 0. This follows from a similar argument in the last proposition of this previous posting.
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