Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 1 / c / Solution

On each member of a cover, choose a reduced local defining function of a complex analytic hypersurface $f_i$ for $Y$. On overlaps $f_i=g_{ij}{f}_j$ for $g_{ij}\in {\mathcal{O}}_X^{\ast }$. Since $f_j{|}_Y=0$,Each $d{f}_i$ annihilates $TY$ and is nonzero in the normal direction because $Y$ is smooth. If $e_i=g_{ij}^{-1}{e}_j$ are the frames of the holomorphic line bundle associated to a divisor $\mathcal{O}(Y)$, thenis a well-defined nowhere-zero holomorphic map of line bundles. Thus the normal bundle of a smooth analytic hypersurface is