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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?

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  • $\begingroup$ Isn't it related to en.wikipedia.org/wiki/Kodaira_embedding_theorem? $\endgroup$ Aug 17, 2018 at 12:48
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    $\begingroup$ Bishop proves that if the the projective volume of an analytic set in $\mathbb{C}^n$ is finite, the set is algebraic (and the degree is bounded by the volume). It's in "Partially analytic spaces", Amer. J. Math, 1961. You can also find it in Chirka's book.. $\endgroup$ Aug 17, 2018 at 14:19

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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.

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