[proofplan] We prove the two implications separately. For the forward direction we use the analytic Kodaira projective embedding theorem in its non-algebraic form: a positive holomorphic line bundle on a compact complex manifold has a sufficiently high tensor power whose global holomorphic sections define a holomorphic embedding into complex projective space. The compact image is closed, and [Chow's theorem](/theorems/3886) then upgrades this closed complex analytic submanifold of projective space to a smooth projective algebraic manifold. Conversely, if $X$ is biholomorphic to a smooth projective algebraic manifold, the hyperplane line bundle on projective space restricts to a positive holomorphic line bundle, and pulling it back along the biholomorphism preserves positivity. [/proofplan] [step:Use a positive line bundle to embed $X$ into projective space] Assume first that $X$ carries a [positive holomorphic line bundle](/page/Positive%20Holomorphic%20Line%20Bundle). Let $L \to X$ denote such a line bundle, and let $h$ be a [Hermitian metric](/page/Hermitian%20Metric) on $L$ whose [Chern curvature form](/page/Chern%20Curvature%20Form) is positive. We invoke the analytic Kodaira projective embedding theorem, used here only in the following separate form: if a compact complex manifold carries a positive holomorphic line bundle $L$, then for some tensor power $L^{\otimes m}$ its global holomorphic sections give a holomorphic embedding into projective space. Its hypotheses are satisfied because $X$ is compact and complex and $L$ is positive. Hence there exist an integer $m \in \mathbb{N}$ and holomorphic sections $s_0,\dots,s_N \in H^0(X,L^{\otimes m})$ such that the map \begin{align*} \Phi_m: X &\to \mathbb{P}^N \\ x &\mapsto [s_0(x):\cdots:s_N(x)] \end{align*} is a holomorphic embedding. Thus $\Phi_m$ is a biholomorphism from $X$ onto the complex submanifold $Y := \Phi_m(X) \subset \mathbb{P}^N$. [guided] Assume first that $X$ carries a [positive holomorphic line bundle](/page/Positive%20Holomorphic%20Line%20Bundle). This means that there is a holomorphic line bundle $L \to X$ and a [Hermitian metric](/page/Hermitian%20Metric) $h$ on $L$ whose [Chern curvature form](/page/Chern%20Curvature%20Form) is positive. The point of positivity is that it is precisely the input for Kodaira's analytic projective embedding theorem, not the Kodaira-Chow characterisation currently being proved. We apply the analytic Kodaira projective embedding theorem in its separate embedding form: if a compact complex manifold carries a positive holomorphic line bundle $L$, then some tensor power $L^{\otimes m}$ is generated by finitely many global holomorphic sections that separate points and tangent vectors, and therefore these sections define a holomorphic embedding into projective space. Its hypotheses are satisfied: compactness of $X$ is part of the theorem statement, $X$ is complex by hypothesis, and $L$ is positive by the present assumption. Therefore there are an integer $m \in \mathbb{N}$ and holomorphic sections $s_0,\dots,s_N \in H^0(X,L^{\otimes m})$ such that the associated projective map \begin{align*} \Phi_m: X &\to \mathbb{P}^N \\ x &\mapsto [s_0(x):\cdots:s_N(x)] \end{align*} is a holomorphic embedding. The target $\mathbb{P}^N$ is complex projective space, and the expression is well-defined because the sections supplied by Kodaira have no common zero. Since $\Phi_m$ is an embedding, it is a biholomorphism from $X$ onto its image $Y := \Phi_m(X) \subset \mathbb{P}^N$, and $Y$ is a complex submanifold of $\mathbb{P}^N$. [/guided] [/step] [step:Apply Chow's theorem to identify the image as algebraic] Because $X$ is compact and $\Phi_m$ is continuous, $Y = \Phi_m(X)$ is compact. Since $\mathbb{P}^N$ is Hausdorff, $Y$ is closed in $\mathbb{P}^N$. The set $Y$ is therefore a closed complex submanifold of projective space. By Chow's theorem, every closed complex analytic submanifold of $\mathbb{P}^N$ is an algebraic subvariety of $\mathbb{P}^N$. Hence $Y$ is a smooth complex projective algebraic manifold, and $X$ is biholomorphic to $Y$ through $\Phi_m$. [guided] We have constructed a holomorphic embedding $\Phi_m: X \to \mathbb{P}^N$ and denoted its image by $Y := \Phi_m(X)$. To conclude that $X$ is projective algebraic, it is not enough that $Y$ is a complex submanifold of projective space; we must know that $Y$ is cut out algebraically. First, $Y$ is compact because $X$ is compact and $\Phi_m$ is continuous. Since complex projective space $\mathbb{P}^N$ is Hausdorff, every compact subset is closed, so $Y$ is closed in $\mathbb{P}^N$. From the previous step, $Y$ is also a complex submanifold. Thus $Y$ is a closed complex analytic submanifold of projective space. We now apply Chow's theorem. The hypotheses are satisfied because $Y \subset \mathbb{P}^N$ is closed and complex analytic. Chow's theorem gives that $Y$ is an algebraic subvariety of $\mathbb{P}^N$. Since $Y$ is already a complex submanifold, it has no singular points as an algebraic variety, so it is a smooth complex projective algebraic manifold. The embedding $\Phi_m$ is a biholomorphism from $X$ onto $Y$, which proves the first implication. [/guided] [/step] [step:Pull back the hyperplane bundle from a projective model] Conversely, assume that $X$ is biholomorphic to a smooth complex projective algebraic manifold. Thus there exist an integer $N \in \mathbb{N}$, a smooth algebraic submanifold $Y \subset \mathbb{P}^N$, and a biholomorphic map \begin{align*} f: X &\to Y. \end{align*} Let $\mathcal{O}_{\mathbb{P}^N}(1) \to \mathbb{P}^N$ denote the [hyperplane line bundle](/page/Hyperplane%20Line%20Bundle) with its [Fubini--Study Hermitian metric](/page/Fubini-Study%20Metric). Its [Chern curvature form](/page/Chern%20Curvature%20Form) is positive, so the restricted holomorphic line bundle \begin{align*} \mathcal{O}_Y(1) := \mathcal{O}_{\mathbb{P}^N}(1)|_Y \to Y \end{align*} is positive. Define the pullback line bundle \begin{align*} M := f^*\mathcal{O}_Y(1) \to X. \end{align*} Because $f$ is biholomorphic, pulling back the Fubini--Study metric on $\mathcal{O}_Y(1)$ gives a Hermitian metric on $M$ whose curvature form is the pullback by $f$ of a positive form. Pullback by a biholomorphism preserves positivity of $(1,1)$-forms, so $M$ is a positive holomorphic line bundle on $X$. [guided] Now assume that $X$ is biholomorphic to a smooth complex projective algebraic manifold. By unpacking the hypothesis, there are an integer $N \in \mathbb{N}$, a smooth algebraic submanifold $Y \subset \mathbb{P}^N$, and a biholomorphic map \begin{align*} f: X &\to Y. \end{align*} The natural positive line bundle on projective space is the [hyperplane line bundle](/page/Hyperplane%20Line%20Bundle). Let $\mathcal{O}_{\mathbb{P}^N}(1) \to \mathbb{P}^N$ denote the hyperplane line bundle equipped with its [Fubini--Study Hermitian metric](/page/Fubini-Study%20Metric). The curvature form of this metric is the Fubini--Study Kähler form, which is positive. Restricting a positive Hermitian holomorphic line bundle to a complex submanifold preserves positivity, because the curvature form restricts to a positive $(1,1)$-form on every complex tangent space. Therefore \begin{align*} \mathcal{O}_Y(1) := \mathcal{O}_{\mathbb{P}^N}(1)|_Y \to Y \end{align*} is a positive holomorphic line bundle on $Y$. We transport this line bundle from $Y$ back to $X$ through the biholomorphism. Define \begin{align*} M := f^*\mathcal{O}_Y(1) \to X. \end{align*} This is a holomorphic line bundle because $f$ is holomorphic. The Fubini--Study metric on $\mathcal{O}_Y(1)$ pulls back to a Hermitian metric on $M$, and the curvature of the pullback metric is the pullback under $f$ of the curvature of $\mathcal{O}_Y(1)$. Since $f$ is a biholomorphism, its differential maps each nonzero complex tangent vector on $X$ to a nonzero complex tangent vector on $Y$; hence the pullback of a positive $(1,1)$-form is again positive. Thus $M$ is a positive holomorphic line bundle on $X$. [/guided] [/step] [step:Combine the two implications to obtain the equivalence] The first two steps prove that the existence of a positive holomorphic line bundle on $X$ implies that $X$ is biholomorphic to a smooth complex projective algebraic manifold. The third step proves the converse implication. Therefore the two stated conditions are equivalent. [/step]