On the sigma-model of deformed special geometry

We discuss the deformed sigma-model that arises when considering four-dimensional N=2 abelian vector multiplets in the presence of an arbitrary chiral background field. In addition, we allow for a class of deformations of special geometry by non-holomorphic terms. We analyze the geometry of the sigma-model in terms of intrinsic torsion classes. We show that, generically, the deformed geometry is non-Kähler. We illustrate our findings with an example. We also express the deformed sigma-model in terms of the Hesse potential that underlies the real formulation of special geometry