Semistable sheaves on homogeneous spaces and abelian varieties

Abstract

In this paper we prove that semistable sheaves with zero Chern classes on homogeneous spaces are trivial and semistable sheaves on abelian varieties with zero Chern classes are filtered by line bundles numerically equivalent to zero. The method consists in reducing mod p and then showing that the Frobenius morphism preserves semistability on the above class of varieties. For technical reasons, we have to assume boundedness of semistable sheaves in char p.