M. Roggero: Splitting type, global sections  and Chern classes for  vector bundles on P^N.

Abstract. The talk presents some results contained in two joined papers with Cristina Bertone.

It is well known that the Chern classes c_r of a rank n vector bundle  on P^N, generated by global sections, are non negative if  r is lower than or equal to the rank  n and vanish otherwise. Both papers deal with the following question:

does the above  result hold also  for  reflexive or  torsion free sheaves?

We give a negative answer for  c_r  when r is higher than 3, while on the contrary we obtain a positive answer for the first three Chern classes c₁, c₂, c₃ of reflexive (or torsion free) sheaves of any rank that satisfy some hypotheses weaker than the classical one “generated by global sections” .

More precisely in [1] we obtain the positivity of the three Chern classes of a reflexive sheaf F as a consequence of lower bounds for them given by functions depending on the maximal free subsheaves  of F.

In [2] we compare a torsion free sheaf F on P^N and the free vector bundle 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 can obtain an easy and more general proof of “ Horrocks' splitting criterion”,  also holding for torsion free sheaves, and lower bounds for the  Chern classes of F.  Especially, we prove  an upper-bound  for the discriminant  of any rank n torsion free sheaf whose splitting type has no gaps (for instance a semistable vector bundle)  which, for rank 2 semistable reflexive sheaves, is the well known Schwartzenberger inequality.

[1]  Bertone-Roggero: Positivity of Chern classes for Reflexive Sheaves in Pⁿ. arXiv:0709.3218v1

[2]  Bertone-Roggero: Splitting type, global sections and Chern classes for vector bundles on P^N. arXiv:0804.2985