 Poincaré residue along a smooth hypersurface

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.