  In this note, we consider how the bundle geometry of field space interplays
with the covariant phase space methods so as to allow to write results of some
generality on the presymplectic structure of invariant gauge theories coupled
to matter. We obtain in particular the generic form of Noether charges
associated with field-independent and field-dependent gauge parameters, as well
as their Poisson bracket. We also provide the general field-dependent gauge
transformations of the presymplectic potential and 2-form, which clearly
highlight the problem posed by boundaries in generic situations. We then
conduct a comparative analysis of two strategies recently considered to evade
the boundary problem and associate a modified symplectic structure to a gauge
theory over a bounded regions: namely the use of edge modes on the one hand,
and of variational connections on the other. To do so, we first try to give the
clearest geometric account of both, showing in particular that edge modes are a
special case of differential geometric tool of gauge symmetry reduction known
as the "dressing field method". Applications to Yang-Mills theory and General
Relativity reproduce or generalise several results of the recent literature.
