  We analyse with the algebraic, regularisation independant, cohomological
B.R.S. methods, the renormalisability of torsionless N=2 and N= 4
supersymmetric non-linear $\si$ models built on K\"ahler spaces. Surprisingly
enough with respect to the common wisdom, in the case of N=2 supersymmetry, we
obtain an anomaly candidate, at least in the compact K\"ahler Ricci-flat case.
If its coefficient does differ from zero, such anomaly would imply the breaking
of global N=2 supersymmetry and get into trouble some schemes of superstring
compactification as such non-linear $\si$ models offer candidates for the
superstring vacuum state. In the compact homogeneous K\"ahler case, as
expected, the anomaly candidate disappears. The same phenomena occurs when one
enforces N=4 supersymmetry : in that case, we obtain the first rigorous proof
of the expected all-orders renormalisability -`` in the space of metrics"- of
the corresponding non-linear $\si$ models. PAR/LPTHE/94-11
