A pr 2 00 8 Splitting type , global sections and Chern classes for vector bundles on P N Cristina Bertone - Margherita Roggero Dipartimento di Matematica dell ’ Università Via Carlo Alberto 10 10123

In this paper we compare any vector bundle (or, more generally, torsion free sheaf) F on P N and the free vector bundle ⊕ n i=1 O P N (b i) having the same rank and splitting type. We show that the first one has always " less " global sections, while it has a higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of F. As a consequence we obtain a direct, easy and more general proof of the " Horrocks' splitting criterion " , also holding for torsion free sheaves, and lower bounds for the Chern classes c i of F. Especially, we prove for rank n sheaves whose splitting type has no gaps (for instance semistable vector bundles) on P N the following formula: (n − 1)c 2 1 − 2nc 2 < 1 12 n 2 (n 2 − 1) which generalizes Schwartzenberger inequality c 2 1 − 4c 2 ≤ 0 for semistable rank 2 vector bundles on P 3 , giving a partial answer to Problem 1.4.1 of [6] (see also [5]).