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

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 
$$N_Y^*\simeq \mathcal O(-Y){\,|\,}_Y\subseteq \mathcal O_Y$$
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 **(in the case where $r\geq 2$)**?

$$\det N_Y^*\subseteq\mathcal O_Y.$$

Thanks a lot.

 EDIT: I have corrected the question, thanks the comments of [abx](https://mathoverflow.net/questions/441904/determinant-of-the-conormal-bundle#comment1140228_441904) and [Jason Starr](https://mathoverflow.net/questions/441904/determinant-of-the-conormal-bundle#comment1140230_441904)!