  We study the localization properties of the equal-time electron Green's
function in a Chern insulator in an arbitrary dimension and with an arbitrary
number of bands. We prove that the Green's function cannot decay
super-exponentially if the Hamiltonian is finite-range and the quantum Hall
response is nonzero. For a general band Hamiltonian (possibly infinite-range),
we prove that the Green's function cannot be finite-range if the quantum Hall
response is nonzero. The proofs use methods of algebraic geometry.
