Let $Y$ be a smooth submanifold of codimension $r$ in a complex manifold $X$. By virtue of the adjunction formula, we always have the isomorphic
$$K_Y\simeq (K_X\otimes \textrm{det } N_Y)\,|\,_Y.$$

Recall that in the case where $r=1$, i.e., $Y$ is an effective divisor, one gets
$$N_Y\simeq \mathcal O(Y)\,|\,_Y.$$
And thus 
$$N_Y^*\simeq \mathcal O(-Y)\,|\,_Y\subseteq \iota^*\mathcal O_X$$
where $\iota:Y\hookrightarrow X$ is the natural inclusion.

Here is my question:
can one give a criterion (or, provide some examples) to guarantee that the following inclusion holds **(especially in the case where $r\geq 2$)**?

$$\textrm{det } N_Y\subseteq\mathcal O_Y.$$

Thanks a lot.