= Poincaré residue along a smooth hypersurface
{c}
{title2=$\operatorname{Res}_Y((dz/z)\wedge\beta)=\beta|_Y$}
= Poincaré residue
{c}
{synonym}
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.
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