Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N\_{H,X}$ the normal bundle of $H$ in $X$.


If $\dim(H) = 1$, that is $H$ is a line, then $N\_{H,X}$ splits as a sum of line bundles. Does there exists an example with $\dim(H) \geq 2$ in which $N\_{H,X}$ does not split as a sum of line bundles?


Thank you.



Normal bundle of a linear subspace
For instance, if $X$ is a smooth 4-dimensional quadric in $\mathbb{P}^5$ and $H = \mathbb{P}^2$, the normal bundle fits into the exact sequence
$$
0 \to N\_{H/X} \to \mathcal{O}(1)^{\oplus 3} \to \mathcal{O}(2) \to 0,
$$
which implies that
$$
N\_{H/X} \cong \Omega\_H(2).
$$
In particular, it does not split.


