= 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>.
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