= Solution
An <irreducible analytic subvariety of a complex manifold> is a closed subset locally defined by finitely many <holomorphic functions> which cannot be expressed as the union of two proper closed analytic subsets. In a complex $n$-manifold, codimension $k$ means pure complex dimension $n-k$. A <divisor on a complex manifold> is a locally finite formal integer combination
$$
D=\sum_\nu m_\nu Y_\nu
$$
of irreducible analytic hypersurfaces. Compactness makes the set of components in this sum finite. Negative coefficients are permitted; a local equation for such a <divisor on a complex manifold> is generally meromorphic rather than holomorphic.
We use these precise local facts. In <holomorphic coordinates> at $x$, the local ring is $\mathcal O_{X,x}\cong\mathbb C\{z_1,\ldots,z_n\}$, a Noetherian <regular local ring> and a <unique factorization domain>. Each analytic hypersurface germ has finitely many irreducible local branches; their prime ideals have height one and are generated by irreducible holomorphic germs. A holomorphic germ is a unit exactly when its value at $x$ is nonzero. Consequently a nonzero meromorphic germ with order zero on every local hypersurface branch is a holomorphic unit, by cancelling its irreducible factors in numerator and denominator. These are the auxiliary local-ring properties required below.
After shrinking a neighbourhood $U_i$ around each point, choose local branch equations $h_{i\nu a}$, and put
$$
f_i=\prod_{\nu,a}h_{i\nu a}^{m_\nu}.
$$
Its zero and pole orders give exactly the local <divisor on a complex manifold> $D|_{U_i}$. A globally irreducible hypersurface may have several branches locally; all of them must be included with the indicated multiplicity. On an overlap, $f_i/f_j$ has zero order along every branch, so the local-ring facts show that it extends to a nowhere-zero <holomorphic function>. These germwise extensions patch uniquely to a holomorphic unit on the overlap. This describes a local defining function of $D$ and proves why multiplicities, not just zero sets, matter.
Set $g_{ij}=f_i/f_j$ and glue line-bundle frames by $e_j=g_{ij}e_i$. The identities $g_{ij}g_{jk}=g_{ik}$ give a <holomorphic line bundle associated to a divisor>, denoted $[D]$. Equivalently, the frame $e_i$ can be represented by $1/f_i$ in the sheaf of meromorphic functions, so $[D]=\mathcal O_X(D)$. If alternative equations are $f'_i=u_i f_i$, where $u_i$ are holomorphic units, the corresponding frames are identified by $e'_i\mapsto u_i^{-1}e_i$. This respects the changed transitions. Refining the cover gives the same identifications, proving that \b[$[D]$ is well-defined up to holomorphic bundle isomorphism]. The expressions $f_i e_i$ glue to its canonical meromorphic section, whose divisor is $D$.
For two <divisors on a complex manifold> use a common cover with equations $f_i$ and $h_i$. Their sum has equation $f_i h_i$, and the <tensor product of vector bundles> has frame transitions
$$
\frac{f_i}{f_j}\frac{h_i}{h_j}=\frac{f_i h_i}{f_j h_j}.
$$
Sending the sum-divisor frame to the tensor product of the two frames gives
$$
\boxed{[D_1+D_2]\cong[D_1]\otimes[D_2].}
$$
This also yields $[-D]\cong[D]^*$.
For a nonsingular <complex analytic hypersurface> $Y$, choose reduced local equations $f_i$ with $df_i$ nonzero along $Y$. The <holomorphic conormal bundle> is
$$
N^*_{Y/X}=\ker(\Omega_X^1|_Y\longrightarrow\Omega_Y^1),
$$
whose fibre is the annihilator of $T^{1,0}Y$. It has local frame $df_i|_Y$. Here this restriction is as a covector in the ambient cotangent bundle along $Y$, not its pullback to $\Omega_Y^1$, where it would vanish. The line bundle $[-Y]$ has local frame $a_i=f_i$, and if $f_j=u_{ji}f_i$, then
$$
a_j=u_{ji}a_i,\qquad
df_j|_Y=u_{ji}|_Y\,df_i|_Y,
$$
since $f_i\,du_{ji}$ vanishes along $Y$. The frame identifications $a_i|_Y\mapsto df_i|_Y$ therefore patch and are fibrewise nonzero. This proves the <conormal line of a smooth analytic hypersurface> formula
$$
\boxed{[-Y]|_Y\cong N^*_{Y/X}.}
$$
Equivalently it identifies $\mathcal O_X(-Y)|_Y=\mathcal I_Y/\mathcal I_Y^2$ with the analytic conormal sheaf.
Back to article page
