Positivity of Chern Classes for Reflexive Sheaves on P^N

Details Positivity of Chern Classes for Reflexive Sheaves on P^N Positivity of Chern Classes for Reflexive Sheaves on P^N

2007-09-20

arxiv.org/abs/0709.3218v2

Geom. Dedicata 142, 121-138 (2009)

Mathematics

Algebraic Geometry

16 pages, no figures

Scientific paper

10.1007/s10711-009-9362-5

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.

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