  We analyse with the algebraic, regularisation independent, cohomological
B.R.S. methods, the renormalisability of torsionless N=2 supersymmetric
non-linear $\si$ models built on compact K\"ahler spaces. Surprisingly enough
with respect to the common wisdom, we obtain an anomaly candidate, when the
Hodge number $h^{3,0}$ of the target space manifold is different from zero :
this occurs in particular in the Calabi-Yau case. On the contrary, in the
compact homogeneous K\"ahler case, the anomaly candidate disappears.
