I'm reading the Huybrechts's Complex Geometry, p.71, Prop.2.2.17 and there is some point that makes me somewhat confused :
I can't understand the underlined statement. How can we use Example 2.2.4, viii) for obtaining the Adjunction formula? Here Example 2.2.4, viii) states that "for a short exact sequence of holomorphic vector bundles $ 0 \to E \to F\to G\to 0$, $\operatorname{det}(F) \cong \operatorname{det}(E)\otimes \operatorname{det}(G)$."
As the above image (there $\mathcal{T}_Y$, $\mathcal{T}_X$ are holomorphic tangent bundle of $Y$, $X$ respectively and $\mathcal{T}_{X}|_{Y}$ is the restriction of $\mathcal{T}_X$ to $Y$), we have the normal bundle sequence :
$$0\to \mathcal{T}_Y \to \mathcal{T}_X|_Y\to \mathcal{N}_{Y/X}\to0$$
As in the above image, in Lemma 2.2.15, he states that "If $Y\subset X$ is a complex submanifold, then there is a canonical injection $\mathcal{T}_Y \subseteq \mathcal{T}_{X}|_{Y}$"
How can we use the viii) of Example 2.2.4 ? And $\mathcal{T}_Y \subseteq \mathcal{T}_{X}|_{Y}$ is correct form? If we can use viii) of Example 2.2.4, then from the form of statement of Proposition 2.2.17, a priori(?), it looks that $\mathcal{T}_{X}|_Y \subseteq \mathcal{T}_Y$ is more correct form.
Is it true? Anyone helps?
