On the normal bundle of submanifolds of P^n

Let X be a submanifold of dimension d ! 2 of the complex projective space Pn. We prove results of the following type.i) If X is irregular and n = 2d, then the normal bundle NX|Pn is indecomposable. ii) If X is irregular, d ! 3 and n = 2d+1, then NX|Pn is not the direct sum of two vector bundles of rank ! 2. iii) If d ! 3, n = 2d−1 and NX|Pn is decomposable, then the natural restriction map Pic(Pn) # Pic(X) is an isomorphism (and, in particular, if X = Pd−1 × P1 is embedded Segre in P2d−1, then NX|P2d−1 is indecomposable). iv) Let n % 2d and d ! 3, and assume that NX|Pn is a direct sum of line bundles; if n = 2d assume furthermore that X is simply connected and OX(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.