Chow theorem in $\mathbb{C}^2$ I have the following question the answer to which I cannot find in the literature (but it must have been studied):
Suppose that $M\subset\mathbb{C}^2$ is a real surface which may locally be written as the zero set of a holomorphic function. What conditions are required on $M$ so that $M$ may be written as the restriction of a projective algebraic curve $C\subset\mathbb{CP}^2$ to $\mathbb{C}^2\subset\mathbb{CP}^2$ (where we identify $\mathbb{C}^2$ with $\{[z,w,1]\in \mathbb{CP}^2| z,w \in \mathbb{C}\}$)?
Clearly some conditions are necessary for example by considering $M=\{(z,e^z)\in \mathbb{C}^2|z\in\mathbf{C}\}$. I expect something like finite Euler characteristic, finite density at infinity, is this enough? Is there some standard reference for this? Or is there some sneaky way to use the Chow theorem?
 A: Various types of sufficient and necessary conditions for an analytic set in $\mathbb{C}^n, n \geq 2$ to be algebraic have been proved. In addition to Bishop's paper already mentioned in the comments, there is also the following volume-related criterion:
If $M$ is a pure $k$-dimensional complex analytic subset in the space $\mathbb{C}^n$, $0<k<n$, then $M$ is algebraic if and only if the function $V(r)r^{−2k}$ is bounded, where $V(r)$ is the `area' of the intersection $M_r=M\cap\{z\in C^n:\|z\|\leq r\}$, understood as the integral over $M_r$ of certain canonical differential form.
This was proved in 
MR0166393 
Stoll, Wilhelm
The growth of the area of a transcendental analytic set. I, II.
Math. Ann. 156 1964 144–170
A different viewpoint has been offered in the papers 
MR0219750  Rudin W. A geometric criterion for algebraic varieties, J. Math. Mech. 17 (1968), 671-683 and 
MR0294698  Sadullaev, A. A criterion for the algebraicity of analytic sets. (Russian) Funckional. Anal. i Priložen. 6 (1972), no. 1, 85–86.
From these papers, the following criterion arises: An analytic set $M \subset \mathbb{C}^n$ of constant dimension $k$, $0<k<n$, is algebraic if and only if there are vector subspaces $X,Y \subset \mathbb{C}^n$ such that ${\rm dim }X=k$, ${\rm dim Y }=n-k$,  $\mathbb{C}^n=X\oplus Y$, and some $C,s >0$ such that $M \subset \{(x,y): x \in X, y \in Y, \|y\| \leq C(1+\|x\|^s)\}$.
Sadullaev also considered algebraicity in terms of growth of certain plurisubharmonic functions. His (and also Rudin's)  approach was extended in the following paper:
MR1756571  Zeriahi, Ahmed A criterion of algebraicity for Lelong classes and analytic sets. Acta Math. 184 (2000), no. 1, 113–143
Recently, a volume-based approach to algebraicity of analytic subsets of $\mathbb{C}^n$ was reviwed via the use of amoebas:
MR3393363  Madani, Farid; Nisse, Mounir Analytic varieties with finite volume amoebas are algebraic. J. Reine Angew. Math. 706 (2015), 67–81. 
