Stein complements in compact Kähler manifolds

Given a projective or compact Kähler manifold X and a (smooth) hypersurface Y, we study conditions under which \(X {\setminus } Y\) $X\backslash Y$ 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 \(T_M\) $T_M$ of a projective manifold M with Y being the projectivization of \(T_M\) $T_M$ itself.