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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