  We compute the ${\cal N}=2$ supersymmetric partition function of a gauge
theory on a four-dimensional compact toric manifold via equivariant
localization. The result is given by a piecewise constant function of the
K\"ahler form with jumps along the walls where the gauge symmetry gets
enhanced. The partition function on such manifolds is written as a sum over the
residues of a product of partition functions on $\mathbb{C}^2$. The evaluation
of these residues is greatly simplified by using an "abstruse duality" that
relates the residues at the poles of the one-loop and instanton parts of the
$\mathbb{C}^2$ partition function. As particular cases, our formulae compute
the $SU(2)$ and $SU(3)$ {\it equivariant} Donaldson invariants of
$\mathbb{P}^2$ and $\mathbb{F}_n$ and in the non-equivariant limit reproduce
the results obtained via wall-crossing and blow up methods in the $SU(2)$ case.
Finally, we show that the $U(1)$ self-dual connections induce an anomalous
dependence on the gauge coupling, which turns out to satisfy a $\mathcal{N}=2$
analog of the $\mathcal{N}=4$ holomorphic anomaly equations.
