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

TL;DR

This paper compares torsion free sheaves on projective space with split vector bundles, establishing bounds on their Chern classes, global sections, and cohomology, and providing a generalized proof of Horrocks' splitting criterion.

Contribution

It introduces bounds for Chern classes and global sections of torsion free sheaves, extending Horrocks' criterion to a broader class of sheaves and deriving explicit inequalities.

Findings

Torsion free sheaves have fewer global sections than split bundles with same rank.

Bounds for Chern classes depend on maximal free subsheaves.

Discriminant formula for sheaves with no gap in splitting type.

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 $t$.