Generation and ampleness of coherent sheaves on abelian varieties, with application to Brill-Noether theory

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.