Journal of the
Korean Mathematical Society JKMS

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

Keywords: torsion free sheaf, Chern classes, discriminant

MSC numbers: 14F05, 14C17, 14Jxx