On the normal bundle of submanifolds of $\mathbb{P}^n$

Let $X$ be a submanifold of dimension $d\geq 2$ of the complex projective space $\mathbb{P}^n$. We prove results of the following type.i) If $X$ is irregular and $n=2d$, then the normal bundle $N_{X\vert\mathbb{P}^n}$ is indecomposable. ii) If $X$ is irregular, $d\geq 3$ and $n=2d+1$, then $N_{X\vert\mathbb{P}^n}$ is not the direct sum of two vector bundles of rank $\geq 2$. iii) If $d\geq 3$, $n=2d-1$ and $N_{X\vert\mathbb{P}^n}$ is decomposable, then the natural restriction map $\mathrm{Pic}(\mathbb{P}^n)\to\mathrm{Pic}(X)$ is an isomorphism (and, in particular, if $X=\mathbb{P}^{d-1}\times\mathbb{P}^1$ is embedded Segre in $\mathbb{P}^{2d-1}$, then $N_{X\vert\mathbb{P}^{2d-1}}$ is indecomposable). iv) Let $n\leq 2d$ and $d\geq 3$, and assume that $N_{X\vert\mathbb{P}^n}$ is a direct sum of line bundles; if $n=2d$ assume furthermore that $X$ is simply connected and $\mathscr O_X(1)$ is not divisible in $\mathrm{Pic}(X)$. Then $X$ is a complete intersection. These results follow from Theorem 2.1 below together with Le Potier's vanishing theorem. The last statement also uses a criterion of Faltings for complete intersection. In the case when $n<2d$ this fact was proved by M. Schneider in 1990 in a completely different way.