[proofplan] We prove the analytic subset is algebraic by passing from its analytic ideal sheaf to an algebraic coherent ideal sheaf on projective space. The key inputs are coherence of the ideal sheaf of a closed analytic subset and Serre's GAGA comparison theorem for coherent sheaves on complex projective space. Once the analytic ideal is identified with the analytification of an algebraic ideal sheaf, finite generation after twisting by a sufficiently ample line bundle produces finitely many homogeneous polynomials whose common projective zero locus has exactly the same analytic ideal, hence exactly the same underlying subset. [/proofplan] [step:Attach the coherent analytic ideal sheaf of the closed analytic subset] Let $X^{\mathrm{an}}$ denote the complex analytic space associated to $X := \mathbb{P}^N_{\mathbb{C}}$, and regard $A \subset X^{\mathrm{an}}$ as a closed analytic subset. Define the analytic ideal sheaf \begin{align*} \mathcal{I}_A^{\mathrm{an}} \subset \mathcal{O}_{X^{\mathrm{an}}} \end{align*} by declaring, for every [open set](/page/Open%20Set) $U \subset X^{\mathrm{an}}$, that $\mathcal{I}_A^{\mathrm{an}}(U)$ is the ideal of holomorphic functions $f: U \to \mathbb{C}$ such that $f|_{U \cap A} = 0$. By the [Coherence of Analytic Ideal Sheaves](/theorems/???), applied to the closed complex analytic subset $A \subset X^{\mathrm{an}}$, the sheaf $\mathcal{I}_A^{\mathrm{an}}$ is a coherent analytic ideal sheaf. [guided] We first translate the geometric condition "$A$ is a closed analytic subset" into sheaf language. Let $X := \mathbb{P}^N_{\mathbb{C}}$ be complex projective space as an algebraic variety, and let $X^{\mathrm{an}}$ be its associated complex analytic space. Since the theorem assumes $A \subset \mathbb{P}^N$ is closed and complex analytic, we may regard $A$ as a closed analytic subset of $X^{\mathrm{an}}$. Define the analytic ideal sheaf \begin{align*} \mathcal{I}_A^{\mathrm{an}} \subset \mathcal{O}_{X^{\mathrm{an}}} \end{align*} by the rule that for each open set $U \subset X^{\mathrm{an}}$, the set $\mathcal{I}_A^{\mathrm{an}}(U)$ consists of all holomorphic functions $f: U \to \mathbb{C}$ satisfying $f|_{U \cap A} = 0$. This sheaf records exactly the local holomorphic equations cutting out $A$. We apply the [Coherence of Analytic Ideal Sheaves](/theorems/???). Its hypothesis is that the subset under consideration is a closed complex analytic subset of a complex analytic space. Here the ambient analytic space is $X^{\mathrm{an}}$, and the subset is precisely the given closed complex analytic subset $A$. Therefore $\mathcal{I}_A^{\mathrm{an}}$ is a coherent analytic ideal sheaf on $X^{\mathrm{an}}$. This coherence is the finite-generation input needed before applying GAGA. [/guided] [/step] [step:Algebraize the analytic ideal sheaf by GAGA] The variety $X = \mathbb{P}^N_{\mathbb{C}}$ is projective over $\mathbb{C}$, hence proper over $\mathbb{C}$. Since $\mathcal{I}_A^{\mathrm{an}}$ is coherent on $X^{\mathrm{an}}$, [Serre GAGA for Coherent Sheaves](/theorems/???) gives a coherent algebraic ideal sheaf \begin{align*} \mathcal{I} \subset \mathcal{O}_X \end{align*} such that its analytification satisfies \begin{align*} \mathcal{I}^{\mathrm{an}} = \mathcal{I}_A^{\mathrm{an}} \end{align*} as ideal sheaves in $\mathcal{O}_{X^{\mathrm{an}}}$. [guided] The analytic equations for $A$ are now packaged into the coherent analytic sheaf $\mathcal{I}_A^{\mathrm{an}}$. To convert those holomorphic equations into algebraic equations, we use GAGA. The theorem [Serre GAGA for Coherent Sheaves](/theorems/???) applies to a projective complex variety. Its geometric hypothesis is satisfied because $X = \mathbb{P}^N_{\mathbb{C}}$ is projective over $\mathbb{C}$, and projective varieties over $\mathbb{C}$ are proper over $\mathbb{C}$. Its sheaf-theoretic hypothesis is satisfied because the previous step proved that $\mathcal{I}_A^{\mathrm{an}}$ is a coherent analytic sheaf on $X^{\mathrm{an}}$. GAGA therefore produces a coherent algebraic sheaf $\mathcal{I}$ on $X$ whose analytification is $\mathcal{I}_A^{\mathrm{an}}$. Since $\mathcal{I}_A^{\mathrm{an}}$ is an ideal subsheaf of $\mathcal{O}_{X^{\mathrm{an}}}$, the fully faithful part of GAGA identifies it with the analytification of an algebraic ideal subsheaf \begin{align*} \mathcal{I} \subset \mathcal{O}_X. \end{align*} Thus \begin{align*} \mathcal{I}^{\mathrm{an}} = \mathcal{I}_A^{\mathrm{an}}. \end{align*} This equality means that the algebraic ideal sheaf $\mathcal{I}$ and the analytic ideal sheaf of $A$ define the same local equations after analytification. [/guided] [/step] [step:Choose finitely many homogeneous generators after twisting] By [Serre's Twisting Theorem](/theorems/???), applied to the coherent sheaf $\mathcal{I}$ on $X = \mathbb{P}^N_{\mathbb{C}}$, there exists an integer $d \geq 1$ such that $\mathcal{I}(d)$ is generated by finitely many global sections. Choose global sections \begin{align*} s_1, \dots, s_r \in H^0(X, \mathcal{I}(d)) \end{align*} that generate $\mathcal{I}(d)$ as an $\mathcal{O}_X$-module. Under the standard identification \begin{align*} H^0(X, \mathcal{O}_X(d)) \cong \mathbb{C}[Z_0, \dots, Z_N]_d, \end{align*} each $s_j$ is represented by a homogeneous polynomial \begin{align*} P_j \in \mathbb{C}[Z_0, \dots, Z_N]_d. \end{align*} Because $s_j$ is a section of the subsheaf $\mathcal{I}(d) \subset \mathcal{O}_X(d)$, the polynomial $P_j$ belongs locally to the ideal sheaf $\mathcal{I}$ after undoing the twist. [guided] The algebraic ideal sheaf $\mathcal{I}$ is coherent, but the theorem asks for finitely many homogeneous polynomials. The bridge from coherent sheaves to homogeneous polynomials is twisting by $\mathcal{O}_X(d)$. We apply [Serre's Twisting Theorem](/theorems/???). Its hypotheses are satisfied because $X = \mathbb{P}^N_{\mathbb{C}}$ is projective space and $\mathcal{I}$ is a coherent algebraic sheaf on $X$. The theorem gives an integer $d \geq 1$ for which $\mathcal{I}(d)$ is globally generated. Since global sections of a coherent sheaf over projective space form a finite-dimensional complex [vector space](/page/Vector%20Space), we may choose finitely many sections \begin{align*} s_1, \dots, s_r \in H^0(X, \mathcal{I}(d)) \end{align*} that generate $\mathcal{I}(d)$ as an $\mathcal{O}_X$-module. The line bundle $\mathcal{O}_X(d)$ has global sections identified with homogeneous degree-$d$ polynomials in the homogeneous coordinates. More precisely, \begin{align*} H^0(X, \mathcal{O}_X(d)) \cong \mathbb{C}[Z_0, \dots, Z_N]_d. \end{align*} Since $\mathcal{I}(d)$ is a subsheaf of $\mathcal{O}_X(d)$, each chosen section $s_j$ corresponds to a homogeneous polynomial \begin{align*} P_j \in \mathbb{C}[Z_0, \dots, Z_N]_d. \end{align*} The condition $s_j \in H^0(X, \mathcal{I}(d))$ says that, locally on $X$, the section $P_j$ lies in the ideal sheaf $\mathcal{I}$ after the twist by $\mathcal{O}_X(d)$ is trivialized. [/guided] [/step] [step:Identify the zero locus of the generators with the analytic set] Let \begin{align*} B := \{[Z_0: \dots : Z_N] \in X : P_1(Z) = \cdots = P_r(Z) = 0\}. \end{align*} Since $s_1, \dots, s_r$ generate $\mathcal{I}(d)$, they generate the same local ideal as $\mathcal{I}$ after trivializing $\mathcal{O}_X(d)$ on any standard affine chart. Hence the algebraic closed subset cut out by $P_1, \dots, P_r$ is precisely the closed subscheme defined by $\mathcal{I}$, at least on underlying sets. Analytifying and using $\mathcal{I}^{\mathrm{an}} = \mathcal{I}_A^{\mathrm{an}}$, the analytic zero set of these polynomials is the zero set of $\mathcal{I}_A^{\mathrm{an}}$, which is $A$. Therefore $B = A$ as subsets of $\mathbb{P}^N$. [guided] Define the candidate algebraic set by \begin{align*} B := \{[Z_0: \dots : Z_N] \in X : P_1(Z) = \cdots = P_r(Z) = 0\}. \end{align*} We must prove that this set is exactly $A$. Choose a standard affine chart $U_i := \{Z_i \neq 0\} \subset X$. On $U_i$, the line bundle $\mathcal{O}_X(d)$ is trivialized by $Z_i^d$. Under this trivialization, the section $s_j$ corresponds to the regular function \begin{align*} \frac{P_j}{Z_i^d}: U_i \to \mathbb{C}. \end{align*} Because $s_1, \dots, s_r$ generate $\mathcal{I}(d)$ as an $\mathcal{O}_X$-module, the functions $P_j/Z_i^d$ generate the ideal sheaf $\mathcal{I}|_{U_i}$ as an $\mathcal{O}_{U_i}$-module. Since $Z_i^d$ is nowhere zero on $U_i$, the common vanishing of the functions $P_j/Z_i^d$ on $U_i$ is the same as the common vanishing of the homogeneous polynomials $P_j$ on $U_i$. Thus, on every standard affine chart $U_i$, the set $B \cap U_i$ is the closed subset defined by the ideal sheaf $\mathcal{I}|_{U_i}$. The charts $U_0, \dots, U_N$ cover $X$, so $B$ is the underlying closed subset defined by $\mathcal{I}$ on all of $X$. Now analytify. The analytification of the ideal sheaf defining $B$ is $\mathcal{I}^{\mathrm{an}}$, and by the GAGA step \begin{align*} \mathcal{I}^{\mathrm{an}} = \mathcal{I}_A^{\mathrm{an}}. \end{align*} The zero set of $\mathcal{I}_A^{\mathrm{an}}$ is $A$ by the definition of the ideal sheaf of a closed analytic subset. Hence the analytic zero set of $P_1, \dots, P_r$ is $A$. Therefore \begin{align*} A = \{[Z_0: \dots : Z_N] \in \mathbb{P}^N : P_1(Z) = \cdots = P_r(Z) = 0\}. \end{align*} This is the desired finite homogeneous system of equations. [/guided] [/step]