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 $$N_Y^*\simeq \mathcal O(-Y)\,|\,_Y\subseteq \mathcal O_Y$$ 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 $r\geq 2$)?

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

Thanks a lot.

EDIT: I have corrected the question, thanks the comments of abx and Starr!

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 and Jason Starr!

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.

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 $$N_Y^*\simeq \mathcal O(-Y)\,|\,_Y\subseteq \mathcal O_Y$$ 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 $r\geq 2$)?

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

Thanks a lot.

EDIT: I have corrected the question, thanks the comments of abx and Starr!