  We introduce a variant of global generation for coherent sheaves on abelian
varieties which, under certain circumstances, implies ampleness. This extends a
criterion of Debarre asserting that a continuously globally generated coherent
sheaf on an abelian variety is ample. We apply this to show the ampleness of
certain sheaves, which we call naive Fourier-Mukai-Poincar\'e transforms, and
to study the structure of GV sheaves. In turn, one of these applications allows
to extend the classical existence and connectedness results of Brill-Noether
theory to a wider context, e.g. singular curves equipped with a suitable
morphism to an abelian variety. Another application is a general inequality of
Brill-Noether type involving the Euler characteristic and the homological
dimension.
