  Recent years have witnessed lots of progress in the computation of
supersymmetric partition functions of SCFTs on curved manifolds via
localization. The twisted partition function on product manifolds of the form
$S^1 \times \Sigma_g$, where $\Sigma_g$ is a two-dimensional Riemann surface,
is of particular relevance due to its role in the microstate counting for
magnetic static AdS$_4$ black holes realizing the topological twist. We review
here supergravity solutions having as conformal boundary more general 3d
manifolds. We first focus on solutions (AdS-Taub-NUT and AdS-Taub-Bolt) having
as boundary a circle bundle over $\Sigma_g$, showing the matching of their
on-shell action with the large $N$ limit of the partition function of the dual
CFT. We then discuss some recent results for a challenging example, which
involves the refinement by angular momentum. The gravitational backgrounds in
this case are rotating supersymmetric AdS$_4$ black holes. We review the
salient features of two different classes of such solutions in theories of
supergravity with uplift in M-theory, and comment on the current status of
their entropy counting in the dual CFT.
