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We compute the supersymmetric partition function of an $\mathcal{N}=1$ chiralmultiplet coupled to an external Abelian gauge field on complex manifolds with$T^2 \times S^2$ topology. The result is locally holomorphic in the complexstructure moduli of $T^2\times S^2$. This computation illustrates in a simpleexample some recently obtained constraints on the parameter dependence ofsupersymmetric partition functions.  We also devise a simple method to compute the chiral multiplet partitionfunction on any four-manifold $\mathcal{M}_4$ preserving two supercharges ofopposite chiralities, via supersymmetric localization. In the case of$\mathcal{M}_4=S^3\times S^1$, we provide a path integral derivation of thepreviously known result, the elliptic gamma function, which emphasizes itsdependence on the $S^3 \times S^1$ complex structure moduli.

The $\mathcal{N}=1$ Chiral Multiplet on $T^2\times S^2$ and Supersymmetric Localization