= Normal bundle of a smooth analytic hypersurface
{title2=$N_{Y/X}\cong\mathcal O(Y)|_Y$}
For a smooth <complex analytic hypersurface> $Y\subset X$, its <holomorphic normal bundle> is naturally the restriction of its <holomorphic line bundle associated to a divisor>:
$$
N_{Y/X}\cong\mathcal O(Y)|_Y.
$$
If reduced local defining functions satisfy $f_i=g_{ij}f_j$, then $df_i|_Y=g_{ij}df_j|_Y$. Pairing a normal vector with $df_i$ and using the reciprocal transition functions of $\mathcal O(Y)$ gives the isomorphism.
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