Kahler geometry and the renormalization of supersymmetricσmodels

It is shown using general arguments based on Kahler geometry that supersymmetric $\sigma$ models defined on manifolds which are (A) Ricci-flat or (B) locally symmetric have the following ultraviolet properties. Class A theories are finite at one- and two-loop order. Class B theories, which include the $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ and $\mathrm{C}{\mathrm{P}}^{n-1\mathrm{}}$ models, (i) are one-loop divergent, (ii) but have no two-loop divergences. The geometrical argument can be extended to higher order with new complications which are discussed. It seems probable that class B theories have no higher-loop ultraviolet divergences and quite possible that class A theories are entirely ultraviolet finite.