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

A complex manifold of complex dimension $n$ is a Hausdorff, second-countable topological manifold with charts to open subsets of ${\mathbb{C}}^n$ whose transition maps are biholomorphic. A rank-$r$ holomorphic vector bundle is a complex vector bundle with local trivializations $E{|}_U\cong U\times {\mathbb{C}}^r$ whose transition matrices are holomorphic maps to ${\mathrm{GL}}_r(\mathbb{C})$.

In a holomorphic coordinate chart, $T^{1,0}X$ has frame $\partial /\partial z^1,\dots ,\partial /\partial z^n$. Under a holomorphic coordinate change $w=w(z)$, the chain rule givesThe Jacobian is an invertible matrix of holomorphic functions. These frames therefore make $T^{1,0}X$ the holomorphic tangent bundle.

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

The tangent-normal short exact sequencegives $\det (TX{|}_Y)\cong \det (TY)\otimes N_{Y/X}$. Dualizing and using the canonical bundle,Since $N_{Y/X}\cong \mathcal{O}(Y){|}_Y$, rearrangement proves the Adjunction formula