  The goal of this paper is to generalize Demailly's asymptotic holomorphic
Morse inequalities to the case of a covering manifold of a compact manifold. We
shall obtain estimates which involve Atiyah's ``normalized dimension'' of the
square integrable harmonic spaces. The techniques used are those of Shubin who
gave a proof for the usual Morse inequalities in the presence of a group action
relying on Witten ideas. As a consequence we obtain estimates for the dimension
of the square integrable holomorphic sections of the pull-back of a line bundle
on the base manifold under some mild hypothesis for the curvature.
