  Geometric techniques have played an important role in the seventies, for the
study of the spectrum of many-body Schr\"odinger operators. In this paper we
provide a formalism which also allows to study nonlinear systems. We start by
defining a weak topology on many-body states, which appropriately describes the
physical behavior of the system in the case of lack of compactness, that is
when some particles are lost at infinity. We provide several important
properties of this topology and use them to provide a simple proof of the
famous HVZ theorem in the repulsive case. In a second step we recall the method
of geometric localization in Fock space as proposed by Derezi\'nski and
G\'erard, and we relate this tool to our weak topology. We then provide several
applications. We start by studying the so-called finite-rank approximation
which consists in imposing that the many-body wavefunction can be expanded
using finitely many one-body functions. We thereby emphasize geometric
properties of Hartree-Fock states and prove nonlinear versions of the HVZ
theorem, in the spirit of works of Friesecke. In the last section we study
translation-invariant many-body systems comprising a nonlinear term, which
effectively describes the interactions with a second system. As an example, we
prove the existence of the multi-polaron in the Pekar-Tomasevich approximation,
for certain values of the coupling constant.
