= Residue trivialization for a degree n+1 projective hypersurface
{title2=$K_Y\cong\mathcal O_Y$}
For a smooth hypersurface $Y=\{f=0\}\subset\mathbb{CP}^n$ of degree $n+1$, let $E=\sum z_i\partial_{z_i}$ and $\Omega=dz_0\wedge\cdots\wedge dz_n$. 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 $t_1,\ldots,t_n$ in original-index order and $F=f|_{z_j=1}$, it is $(-1)^{j+r-1}(dt_1\wedge\cdots\widehat{dt_r}\cdots\wedge dt_n)/F_{t_r}$ wherever $F_{t_r}\ne0$. Thus $K_Y\cong\mathcal O_Y$.
Back to article page
