I will consider four dimensional supersymmetric field theories on compact manifolds M and discuss constraints on the dependence of their partition functions Z_M (and other supersymmetric observables) on the geometry of M. For N=1 theories with a U(1) R symmetry, supersymmetry requires M to be a complex manifold with a Hermitian metric. I will show that the introduction of twisted variables allows to understand how Z_M depends on the geometry and the parameters entering the Lagrangian. I will then use these results to relate supersymmetric observables for theories in diverse dimensions.