  We investigate the relation between the localization of generalized Wannier
bases and the topological properties of two-dimensional gapped quantum systems
of independent electrons in a disordered background, including magnetic fields,
as in the case of Chern insulators and quantum Hall systems. We prove that the
existence of a well-localized generalized Wannier basis for the Fermi
projection implies the vanishing of the Chern character, which is proportional
to the Hall conductivity in the linear response regime. Moreover, we state a
localization dichotomy conjecture for general non-periodic gapped quantum
systems.
