 Irreducible complex analytic hypersurface

An irreducible complex analytic hypersurface in a complex manifold is a complex analytic hypersurface that cannot be written as the union of two proper closed analytic subsets. It has pure complex codimension one. Irreducibility is a condition on decomposition into analytic pieces, rather than a requirement of smoothness.

In ${\mathbb{C}}^2$, the hypersurface $xy=0$ is reducible: it is the union of the two coordinate axes. The cusp $y^2=x^3$ is irreducible but singular at the origin. Its equation is irreducible as a convergent germ of a holomorphic function there, while both first derivatives of $y^2-x^3$ vanish at zero. Thus irreducibility does not force the holomorphic implicit function theorem to apply.

Global irreducibility and irreducibility of every local germ must be distinguished in general. Local factorization of a reduced defining equation records its analytic branches, while global irreducibility concerns a decomposition of the entire hypersurface. Such hypersurfaces provide the prime components used to describe complex analytic divisors.