SPLITTING TYPE, GLOBAL SECTIONS AND CHERN CLASSES FOR TORSION FREE SHEAVES ON PN

Abstract

In this paper we compare a torsion free sheaf F on <TEX>$P^N$</TEX> and the free vector bundle <TEX>$\oplus^n_{i=1}O_{P^N}(b_i)$</TEX> 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 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 <TEX>$c_i$</TEX>(F(t)) of twists of F, only depending on some numerical invariants of F. Especially, we prove for rank n torsion free sheaves on <TEX>$P^N$</TEX>, whose splitting type has no gap (i.e., <TEX>$b_i{\geq}b_{i+1}{\geq}b_i-1$</TEX> 1 for every i = 1,<TEX>$\ldots$</TEX>,n-1), the following formula for the discriminant: <TEX>$$\Delta(F):=2_{nc_2}-(n-1)c^2_1\geq-\frac{1}{12}n^2(n^2-1)$$</TEX>. Finally in the case of rank n reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes <TEX>$c_3$</TEX>(F(t)),<TEX>$\ldots$</TEX>,<TEX>$c_n$</TEX>(F(t)) for the dimension of the cohomology modules <TEX>$H^iF(t)$</TEX> and for the Castelnuovo-Mumford regularity of F; these polynomial bounds only depend only on <TEX>$c_1(F)$</TEX>, <TEX>$c_2(F)$</TEX>, the splitting type of F and t.