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A compact complex manifold $X$ which admits an embedding into $P^n(\mathbb{C})$ (for some $n$) is called a projective algebraic manifold. And by a theorem of Chow, every complex submanifold $V$ of $P^n(\mathbb{C})$ is actually an algebraic submanifold. However, how to determine if the embedded complex submanifold is an irreducible algebraic set?

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  • $\begingroup$ If $X$ is connected its ring of meromorphic functions $M(X)$ is a field. Choose the embedding such that $X' = X$ minus a few points is affine and open in $X$. $\mathbb{C}[X']$ its ring of polynomials (inherited from $A^n(\mathbb{C})$) is a subring of $M(X)$ thus an integral domain and $X'$ is an affine variety. $\endgroup$ – reuns Jan 17 at 18:26
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Assuming you mean embedding to mean injective on points, this question has nothing to do with the embedding. For a smooth locally noetherian scheme over a field, irreducible components are connected components - these conditions imply that all the local rings of the variety are regular local rings, which in turn implies that each point is on exactly one irreducible component (by the correspondence between minimal primes of the local ring and irreducible components containing the point). So $X$ is irreducible iff it's connected.

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