Is a projective algebraic manifold irreducible algebraic set in $P^n$?

Question

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?

Answer 1

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.