 Solution

This is the Poincaré residue of a meromorphic top form with a simple pole. Write $z=z_1$ and locally $\alpha =(dz/z)\wedge \beta$, with $\beta$ a holomorphic $(n-1)$-form. The proposed residue is the pullback of $\beta$ to $Y$.

First it does not depend on the choice of $\beta$: if $dz\wedge (\beta -{\beta }^{\prime })=0$, then at each point the difference has a factor $dz$, and its pullback to $Y$ is zero. Now change the local defining coordinate to $t=az$, where $a$ is a nowhere-zero holomorphic function. Such a factor exists because both coordinates define the same smooth complex analytic hypersurface. If $\alpha =(dt/t)\wedge \eta$, thenAt points of $Y$, this is an equality in the ambient exterior-power fibre, and the last term is zero. It follows that $dz\wedge (\beta -\eta )=0$ there, so their pullbacks to $Y$ agree. This also covers changes of the tangential coordinates.

The restrictions are holomorphic top forms on $Y$, and their agreement on overlaps makes them a global section of the canonical bundle. Thus the residue is independent of the adapted coordinates:The ambient-fibre comparison is important: pulling an ambient $n$-form directly back to the $(n-1)$-dimensional hypersurface would give zero and would not prove the required independence.