Irreducible analytic subvariety of a complex manifold

An analytic subvariety is a closed subset locally defined by finitely many holomorphic functions. It is irreducible if it is not the union of two proper closed analytic subsets. An irreducible analytic subvariety of complex codimension $k$ in an $n$-dimensional complex manifold has pure complex dimension $n-k$. Global irreducibility does not forbid multiple local branches at a singular point.