The adjunction formula states that if $Y \subset M$ is a smooth analytic hypersurface then we have an isomorphism $N^*_Y \simeq [-Y]|_Y$, where $N_Y$ is the normal bundle of $Y$ and $[-Y]$ the line bundle associated to the divisor $-Y$.
I wonder if there exists a more 'general' adjunction formula. More precisely, now let the smooth subvariety $Y\subset X$ be the zero set of a holomorphic section $s\in H^0(X,E)$ of a smooth Hermitian holomorphic vector bundle $E$ of rank $m\ge 2$. Do we have $$N_Y\simeq E|_Y?$$
Any suggestion and reference would be appreciated. Thanks a lot.