Positivity of Chern Classes for Reflexive Sheaves on P^N

TL;DR

This paper investigates the positivity of Chern classes for reflexive sheaves on projective space, showing that higher Chern numbers can be negative, but lower ones remain positive under certain conditions, with classifications of extremal cases.

Contribution

It extends known positivity results from vector bundles to reflexive sheaves, providing bounds and classifications for Chern classes in this broader context.

Findings

Chern numbers c_i with i≥4 can be arbitrarily negative for reflexive sheaves.

Chern classes c_i with i≤3 are positive under weaker hypotheses.

Complete classification of sheaves reaching extremal Chern class bounds.

Abstract

It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on $\PP^N$, generated by global sections, are non-negative if $i\leq n$ and vanish otherwise. This paper deals with the following question: does the above result hold for the wider class of reflexive sheaves? We show that the Chern numbers $c_i$ with $i\geq 4$ can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for $i\leq 3$ we show positivity of the $c_i$ with weaker hypothesis. We obtain lower bounds for $c_1$, $c_2$ and $c_3$ for every reflexive sheaf $\FF$ which is generated by $H^0\FF$ on some non-empty open subset and completely classify sheaves for which either of them reach the minimum allowed, or some value close to it.