Strong q-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces

The main result is that, for any projective compact analytic subset A of dimension q>0 in a reduced complex space X, there is a neighborhood U of A such that, for any covering space Z of X in which the lifting B of A has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a smooth exhaustion function on Z which is strongly q-convex on the lifting of U outside a uniform neighborhood of the q-dimensional compact irreducible components B.