 Complex analytic hypersurface

A complex analytic hypersurface in an $n$-dimensional complex manifold is a closed analytic subset of pure complex dimension $n-1$. Locally it is the zero set of one reduced holomorphic function $f$. This local single-equation property follows from the factoriality of the ambient regular local ring.^[ref]

A complex analytic hypersurface need not be a complex submanifold. In ${\mathbb{C}}^2$, $z_1=0$ defines a smooth example, whereas $z_1{z}_2=0$ is the union of two complex axes meeting at a singular point. For a reduced local equation, smoothness at a point is equivalent to $df\ne 0$ there, by the holomorphic implicit function theorem.

Irreducibility means that the hypersurface is not the union of two proper closed analytic subsets. It is an additional condition, not another name for every complex analytic hypersurface; the two-axis example is reducible.