Positivity of Chern Classes for Reflexive Sheaves on PN

It is well known that the Chern classes c i of a rank n vector bundle on P N , generated by global sections, are non-negative if i ≤ 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 ≥ 4 can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for i ≤ 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 $${\mathcal {F}}$$ which is generated by $${H^0\mathcal {F}}$$ 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.