We compute the supersymmetric partition function of an N=1 chiral multiplet coupled to an external Abelian gauge field on complex manifolds with T^2xS^2 topology. The result is locally holomorphic in the complex structure moduli of T^2xS^2. This computation illustrates in a simple example some recently obtained constraints on the parameter dependence of supersymmetric partition functions.

We also devise a simple method to compute the chiral multiplet partition function on any four-manifold M4 preserving two supercharges of opposite chiralities, via supersymmetric localization. In the case of M4=S^3xS^1, we provide a path integral derivation of the previously known result, the elliptic gamma function, which emphasizes its dependence on the S^3xS^1 complex structure moduli.