  We study the conditions under which an almost Hermitian structure $(G,J)$ of
general natural lift type on the cotangent bundle $T^*M$ of a Riemannian
manifold $(M,g)$ is K\" ahlerian. First, we obtain the algebraic conditions
under which the manifold $(T^*M,G,J)$ is almost Hermitian. Next we get the
integrability conditions for the almost complex structure $J$, then the
conditions under which the associated 2-form is closed. The manifold
$(T^*M,G,J)$ is K\" ahlerian iff it is almost Kahlerian and the almost complex
structure $J$ is integrable. It follows that the family of Kahlerian structures
of above type on $T^*M$ depends on three essential parameters (one is a certain
proportionality factor, the other two are parameters involved in the definition
of $J$).
