Poincaré residue along a smooth hypersurface (${\mathrm{Res}}_Y((dz/z)\wedge \beta )=\beta {|}_Y$)

If a meromorphic top form has at most a simple pole along a smooth complex analytic hypersurface $Y=\{z=0\}$, write $\alpha =(dz/z)\wedge \beta$ with $\beta$ holomorphic. Its residue is $\beta {|}_Y$, a holomorphic section of the canonical bundle $K_Y$. A change of defining coordinate $t=az$, with $a$ holomorphic and nonzero, changes $dt/t$ by the holomorphic form $da/a$ and leaves this restriction invariant. Placing the normal factor first fixes the sign convention.

For a smooth hypersurface $Y=\{f=0\}\subset {\mathbb{CP}}^n$ of degree $n+1$, let $E=\sum z_i{\partial }_{z_i}$ and $\Omega =d{z}_0\wedge \cdots \wedge d{z}_n$. The meromorphic form $({\iota }_E\Omega )/f$ descends to projective space because its scaling weights cancel and it is horizontal. Its Poincaré residue is a nowhere-zero section of $K_Y$. On the chart $z_j=1$, with remaining coordinates $t_1,\dots ,t_n$ in original-index order and $F=f{|}_{z_j=1}$, it is $(-1{)}^{j+r-1}(d{t}_1\wedge \cdots \widehat{d{t}_r}\cdots \wedge d{t}_n)/F_{t_r}$ wherever $F_{t_r}\ne 0$. Thus $K_Y\cong {\mathcal{O}}_Y$.