Complex analytic hypersurface

A complex analytic hypersurface in an $n$-dimensional complex manifold is a closed analytic subset of pure complex codimension one. It is irreducible when it is not the union of two proper closed analytic subsets.

A local defining function for a complex analytic hypersurface $Y$ near $x$ is a holomorphic function $f$ on a neighbourhood $U$ such that $Y\cap U=\{f=0\}$. It may be chosen reduced, with each local irreducible factor occurring once. The local ring ${\mathcal{O}}_{X,x}$ of a complex manifold is a regular local ring and hence a unique factorization domain; each height-one prime of the hypersurface germ is principal, and the product of generators for its finitely many local branches gives $f$.

A divisor on a complex manifold is a locally finite formal sumwhere the $Y$ are irreducible complex analytic hypersurfaces and $n_Y\in \mathbb{Z}$. Products of powers of local defining functions give local meromorphic equations $f_{\alpha }$ for $D$ whose ratios $f_{\alpha }/f_{\beta }$ are nowhere-vanishing holomorphic functions.

If $f_{\alpha }$ are local meromorphic equations for a divisor on a complex manifold $D$, glue holomorphic frames $e_{\alpha }$ byThe resulting holomorphic line bundle is denoted $[D]$. The expressions $f_{\alpha }{e}_{\alpha }$ glue to its canonical meromorphic section, whose divisor is $D$.