On the normal bundle of submanifolds of Pn

Let X be a submanifold of dimension d >= 2 of the complex projective space P-n. We prove results of the following type. i) If X is irregular and n = 2d, then the normal bundle N-X|Pn is indecomposable. ii) If X is irregular, d >= 3 and n = 2d+1, then N-X|Pn is not the direct sum of two vector bundles of rank >= 2. iii) If d >= 3, n = 2d-1 and N-X|Pn is decomposable, then the natural restriction map Pic(P-n)-> Pic(X) is an isomorphism ( and, in particular, if X=Pd-1 x P-1 is embedded Segre in P2d-1, then NX|P2d-1 is indecomposable). iv) Let n <= 2d and d >= 3, and assume that N-X|Pn 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.