Splitting type, global sections and Chern classes for torsion free sheaves on P^N

Abstract

In this paper we compare a torsion free sheaf \FF on \PP^N and the free vector bundle \oplus_{i=1}^n\OPN(b_i) having 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 \FF. 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(\FF(t)) of twists of \FF, only depending on some numerical invariants of \FF. Especially, we prove for rank $n$ torsion free sheaves on \PP^N, whose splitting type has no gap (i.e. $b_i\geq b_{i+1}\geq b_i-1$ for every $i=1, ...,n-1$), the following formula for the discriminant: \Delta(\FF):=2nc_2-(n-1)c_1^2\geq -{1/12}n^2(n^2-1) Finally in the case of rank $n$ reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes c_3(\FF(t)), ..., c_n(\FF(t)), for the dimension of the cohomology modules H^i\FF(t) and for the Castelnuovo-Mumford regularity of \FF; these polynomial bounds only depend only on c_1(\FF), c_2(\FF), the splitting type of \FF and $t$.Comment: Final version, 15 page