We use holographic renormalization of minimal $\mathcalN=2$ gauged
supergravity in order to derive the general form of the quantum Ward identities
for 3d $\mathcalN=2$ and 4d $\mathcalN=1$ superconformal theories on
general curved backgrounds, including an arbitrary fermionic source for the
supercurrent. The Ward identities for 4d $\mathcalN=1$ theories contain both
bosonic and fermionic global anomalies, which we determine explicitly up to
quadratic order in the supercurrent source. The Ward identities we derive apply
to any superconformal theory, independently of whether it admits a holographic
dual, except for the specific values of the $a$ and $c$ anomaly coefficients,
which are equal due to our starting point of a two-derivative bulk supergravity
theory. In the case of 4d $\mathcalN=1$ superconformal theories, we show that
the fermionic anomalies lead to an anomalous transformation of the supercurrent
under rigid supersymmetry on backgrounds admitting Killing spinors, even if all
anomalies are numerically zero on such backgrounds. The anomalous
transformation of the supercurrent under rigid supersymmetry leads to an
obstruction to the $Q$-exactness of the stress tensor in supersymmetric vacua,
and may have implications for the applicability of localization techniques. We
use this obstruction to the $Q$-exactness of the stress tensor in order to
resolve a number of apparent paradoxes relating to the supersymmetric Casimir
energy, the BPS condition for supsersymmetric vacua, and the compatibility of
holographic renormalization with supersymmetry, that were presented in the
literature.
