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

The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space ${Upsiloncolonwidehat Xto X}The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space Ucolon[^(X)]? X{Upsiloncolonwidehat Xto X} in which [^(Y)] o U^-1(Y){widehat YequivUpsilon^{-1}(Y)} has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C ^∞ exhaustion function j{varphi} on [^(X)]{widehat X} which is strongly q-convex on [^(W)]=U^-1(W){widehatOmega=Upsilon^{-1}(Omega)} outside a uniform neighborhood of the q-dimensional compact irreducible components of [^(Y)]{widehat Y}.