Normal bundle of a hyperplane section Let $Y\subset \mathbb{P}^n$ be a smooth projective variety and let $H$ be a smooth hypersurface in $\mathbb{P}^n$ such that $Z=Y\cap H$ is smooth. How are the normal bundles of the various embeddings of $Z$ related?
More precisely, does either of the following hold?
(1) $N_{Z|H}=N_{Y|\mathbb{P}^n}|_Z$? 
(2) $N_{Z|\mathbb{P}^n}=N_{Y|\mathbb{P}^n}|_Z\oplus N_{H|\mathbb{P}^n}|_Z$? 
Any hints or references would be greatly appreciated!
 A: $\require{AMScd} \newcommand{\P}{\Bbb P^n}$Preliminary comment: Your title "hyperplane section" is more specific than you wrote in the body of the question. It holds for any smooth hypersurface $H$ (or one that is smooth where it intersects $Y$). In fact, the result holds more generally for $Z=Y\cap V$ for any smooth subvarieties $Y,V$.
OK, so here we go. First, by definition (in the algebraic category), the normal bundle of a submanifold (smooth subvariety) $Y\subset X$ is given by $N_{Y/X} = TX\big|_Y/TY$.
Next, a crucial observation is that transversality of two submanifolds (smooth subvarieties) $H,Y\subset X$ is the condition that at each point $x\in H\cap Y$ we have $T_x H + T_x Y = T_x X$. This is equivalent to saying that $T_x H \to T_x X/T_x Y = N_x(Y/X)$ is surjective.
So, now we make a ridiculous commutative diagram that puts $Z=H\cap Y$ in its proper place.
$$\begin{CD}
@. 0  @. 0 @. 0 @. \\
@. @VVV @VVV @VVV @. \\
0 @>>> TZ @>>> TY\big|_Z @>>> N_{Z/Y} @>>> 0 \\
@. @VVV @VVV @V{\phi}VV @. \\
0 @>>> TH\big|_Z @>{\alpha}>> T\P\big|_Z @>>> N_{H/\P}\big|_Z @>>> 0 \\
@. @VVV @V{\beta}VV @VVV @. \\
0 @>>> N_{Z/H} @>{\iota}>> N_{Y/\P}\big|_Z @>>> ? @>>> 0 \\
@. @VVV @VVV @VVV @.\\
  @. 0 @. 0 @. 0 @.
\end{CD}$$
Now, by our transversality hypothesis, $\beta\circ\alpha$ is surjective, and so $\iota$ is surjective; i.e., $\iota$ is an isomorphism. This means that we must have $?=0$. We're done, by symmetry, but, having come this far, we apply the so-called Nine Lemma and conclude that $\phi$ is an isomorphism as well. This completes the proof of (1).  
Item (2) follows from another application of the Nine Lemma.
$$\begin{CD}
@. 0  @. 0 @. 0 @. \\
@. @VVV @VVV @VVV @. \\
0 @>>> TZ @>>> TY\big|_Z @>>> N_{Z/Y} @>>> 0 \\
@. @| @VVV @V{\phi}VV @. \\
0 @>>> TZ @>>> T\P\big|_Z @>>> N_{Z/\P}\big|_Z @>>> 0 \\
@. @VVV @VVV @VVV @. \\
0 @>>> 0 @>>> N_{Y/\P}\big|_Z @>>> K @>>> 0 \\
@. @VVV @VVV @VVV @.\\
  @. 0 @. 0 @. 0 @.
\end{CD}$$
Letting $K=\text{coker}(\phi)$, the Nine Lemma tells us that $K\cong N_{Y/\P}\big|_Z$. But we already know that $N_{Y/\P}\big|_Z\cong N_{Z/H}$, and so we have the short exact sequence
$$0\to N_{Z/Y} \to N_{Z/\P} \to N_{Z/H} \to 0\,.$$
But this exact sequence splits, since we could swap the rôles of $Y$ and $H$. Thus, as desired, $N_{Z/\P}\cong N_{Z/Y}\oplus N_{Z/H}$. Nowhere in this argument did we use anything about $H$'s being a hypersurface (i.e., $N_{H/\P}$'s being a line bundle).
