 Holomorphic conormal bundle

For a complex submanifold $Y\subset X$, the holomorphic conormal bundle is the annihilator of $T^{1,0}Y$ inside ${\Omega }_X^1{|}_Y$, or equivalently the dual of the holomorphic normal bundle. For a smooth analytic hypersurface with local equation $f$, it is generated by $df{|}_Y$. It is also the locally free analytic sheaf ${\mathcal{I}}_Y/{\mathcal{I}}_Y^2$.