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

Abstract

Let X be a submanifold of dimension d > 2 of the complex projective space P. We prove results of the following type.i) If X is irregular and n = 2d, then the normal bundle N X|P n is indecomposable. ii) If X is irregular, d > 3 and n = 2d +1, then N X|P n is not the direct sum of two vector bundles of rank > 2. iii) If d > 3, n = 2d- 1 and N X|P n is decomposable, then the natural restriction map Pic(P n ) → Pic(X) is an isomorphism (and, in particular, if X = P d-1 x P 1 is embedded Segre in P 2d-1 , then N X|p 2d-1 is indecomposable). iv) Let n ≤ 2d and d > 3, and assume that N X|P n is a direct sum of line bundles; if n = 2d assume furthermore that X is simply connected and O x (1) is not divisible in 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.