  We consider supersymmetric field theories on compact manifolds M and obtain
constraints on the parameter dependence of their partition functions Z_M. Our
primary focus is the dependence of Z_M on the geometry of M, as well as
background gauge fields that couple to continuous flavor symmetries. For N=1
theories with a U(1)_R symmetry in four dimensions, M must be a complex
manifold with a Hermitian metric. We find that Z_M is independent of the metric
and depends holomorphically on the complex structure moduli. Background gauge
fields define holomorphic vector bundles over M and Z_M is a holomorphic
function of the corresponding bundle moduli. We also carry out a parallel
analysis for three-dimensional N=2 theories with a U(1)_R symmetry, where the
necessary geometric structure on M is a transversely holomorphic foliation
(THF) with a transversely Hermitian metric. Again, we find that Z_M is
independent of the metric and depends holomorphically on the moduli of the THF.
We discuss several applications, including manifolds diffeomorphic to S^3 x S^1
or S^2 x S^1, which are related to supersymmetric indices, and manifolds
diffeomorphic to S^3 (squashed spheres). In examples where Z_M has been
calculated explicitly, our results explain many of its observed properties.
