It is well known that the Chern classes ci of a rank n vector bundle on PN, 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 ci with i ≥ 4 can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for i ≤ 3 we show positivity of the ci with weaker hypothesis. We obtain lower bounds for c1, c2 and c3 for every reflexive sheaf \documentclass[12pt]{minimal}
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