Theorem 1 (Chow)   Every complex analytic projective variety is algebraic.

Proof. Suppose that we have a complex analytic variety $X \in {\mathbb P}^n$ . It is not hard to show that that $\sigma^{-1}(X)$ is a complex analytic subvariety of ${\mathbb C}^{n+1} \setminus \{0\}.$ By the theorem of Remmert-Stein the set $V = \sigma^{-1}(X) \cup \{0\}$ is a subvariety of ${\mathbb C}^{n+1}.$ Furthermore $V$ is a complex cone, that is if $z = (z_1,\ldots,z_{n+1}) \in V,$ then $t z \in V$ for all $t \in {\mathbb C}.$

Final step is to show that if a complex analytic subvariety $V \subset {\mathbb C}^{n+1}$ is a complex cone, then it is given by the vanishing of finitely many homogeneous polynomials. Take a finite set of defining functions of $V$ near the origin. I.e. take $f_1,\ldots,f_k$ defined in some open ball $B = B(0,\epsilon),$ such that in $B \cap V = \{ z \in B \mid f_1(z) = \cdots = f_k(z) = 0 \}.$ We can suppose that $\epsilon$ is small enough that the power series for $f_j$ converges in $B$ for all $j.$ Expand $f_j$ in a power series near the origin and group together homogeneous terms as $f_j = \sum_{m=0}^\infty f_{jm}$ , where $f_{jm}$ is a homogeneous polynomial of degree $m.$ For $t \in {\mathbb C}$ we write \begin{equation*} f_j(t z) = \sum_{m=0}^\infty f_{jm}(tz) = \sum_{m=0}^\infty t^m f_{jm}(z) \end{equation*}For a fixed $z \in V$ we know that $f_{j}(t z) = 0$ for all $\lvert t \rvert < 1,$ hence we have a power series in one variable that is identically zero, and so all coefficients are zero. Thus $f_{jm}$ vanishes on $V \cap B$ and hence on $V.$ It follows that $V$ is defined by a family of homogeneous polynomials. Since the ring of polynomials is Noetherian we need only finitely many, and we are done.