= 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>.<Complex Analytic and Differential Geometry by Jean-Pierre Demailly>[Chapter II, Theorem (6.6), printed p.106]{ref}
A <complex analytic hypersurface> need not be a <complex submanifold>. In $\mathbb C^2$, $z_1=0$ defines a smooth example, whereas $z_1z_2=0$ is the <union> of two complex axes meeting at a singular <point>. For a reduced local equation, smoothness at a <point> is equivalent to $df\ne0$ 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.
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