Stein complements in compact K\"ahler manifolds

Given a projective or compact K\"ahler manifold X and a (smooth) hypersurface Y, we study conditions under which $X \setminus Y$ could be Stein. We apply this in particular to the case when X is the projectivization of the so-called canonical extension of the tangent bundle $T_M$ of a projective manifold M with Y being the projectivization of $T_M$ itself.