The authors study complex submanifolds in indefinite Kählerian space forms with large index of relative nullity. It is shown that if the ambient space is flat and the index of relative nullity is \(\geq n-1\) then te Kählerian submanifold has a cylindrical structure. In particular, if the relative nullity foliation is non-degenerate then the submanifold is the product of a complex curve and a complex vector space. In addition if the ambient space has positive constant holomorphic sectional curvature, the submanifold is geodesically complete and the index of the relative nullity is greater than the signature of the metric of the ambient space form, then the submanifold is shown to be totally geodesic.