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

Abstract

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 $N_{X|\mathbb P^n}$ is indecomposable. ii) If $X$ is irregular, $d\geq 3$ and $n=2d+1$ then $N_{X|\mathbb P^n}$ is not the direct sum of two vector bundles of rank $\geq 2$. iii) If $d\geq 3$, $n=2d-1$ and $N_{X|\mathbb P^n}$ is decomposable then the natural restriction map $\Pic(\mathbb P^n)\to\Pic(X)$ is an isomorphism (and in particular, if $X=\mathbb P^{d-1}\times\mathbb P^1$ embedded Segre in $\mathbb P^{2d-1}$ then $N_{X|\mathbb P^{2d-1}}$ is indecomposable). iv) Let $n\leq 2d$ and $d\geq 3$, and assume that $N_{X|\mathbb P^n}$ 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 $\Pic(X)$. Then $X$ is a complete intersection. These results follow from Theorem \ref{exact5} below together with Le Potier 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.