  ADHM invariants are equivariant virtual invariants of moduli spaces of
twisted cyclic representations of the ADHM quiver in the abelian category of
coherent sheaves of a smooth complex projective curve X. The goal of the
present paper is to present a generalization of this construction employing a
more general stability condition which depends on a real parameter. This yields
a chamber structure in the ADHM theory of curves, residual ADHM invariants
being defined by equivariant virtual integration in each chamber. Wallcrossing
results and applications to local stable pair invariants will be presented in
the second part of this work.
