First-order supersymmetric sigma models and target space geometry

We study the conditions under which N=(1,1) generalized sigma models support an extension to N=(2,2). The enhanced supersymmetry is related to the target space complex geometry. Concentrating on a simple situation, related to Poisson sigma models, we develop a language that may help us analyze more complicated models in the future. In particular, we uncover a geometrical framework which contains generalized complex geometry as a special case.