Kahler geometry and the renormalization of supersymmetric models

It is shown using general arguments based on Kahler geometry that supersymmetric 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 O(n) and CPn-1 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.