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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!

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  • $\begingroup$ Don't you want $\ \det N^*_Y\subset \mathscr{O}_Y$? This is what we have for $r=1$. $\endgroup$
    – abx
    Commented Mar 1, 2023 at 17:04
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    $\begingroup$ The pullback to $Y$ of the natural inclusion $\mathcal{O}(-Y)\hookrightarrow \mathcal{O}_X$ is a zero homomorphism. So you do not have any (natural) inclusion of $N_Y^*$ into $\iota^*\mathcal{O}_X$, not even in codimension $1$. $\endgroup$
    – Jason Starr
    Commented Mar 1, 2023 at 17:07
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    $\begingroup$ Dear @JasonStarr, thanks for your correction! I have corrected the question. $\endgroup$
    – Invariance
    Commented Mar 1, 2023 at 17:30
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    $\begingroup$ One case where you have such an inclusion is when $X$ is a compact homogeneous manifold: then $T_X$ is globally generated, hence so are $N_Y$ and $\det N_Y$, so picking a section of $\det N_Y$ gives an injection $\det N_Y^*\hookrightarrow \mathscr{O}_Y$ (non canonical, as Jason points out). $\endgroup$
    – abx
    Commented Mar 1, 2023 at 17:47
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    $\begingroup$ I feel like I heard Winkelmann and Campana talk about some other cases related to Gromov's h-principle. Maybe these are cases of compact homogeneous manifolds . . . $\endgroup$
    – Jason Starr
    Commented Mar 1, 2023 at 19:27

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