Rotating black holes in the FI-gauged $N=2$, $D=4$ $\overline{\mathbb{C}\text{P}}^n$ model

We construct supersymmetric black holes with rotation or NUT charge for the $\overline{\mathbb{C}\text{P}}^n$- and the $\text{t}^3$ model of $N=2$, $D=4$ $\text{U}(1)$ FI-gauged supergravity. The solutions preserve 2 real supercharges, which are doubled for their near-horizon geometry. For the $\overline{\mathbb{C}\text{P}}^n$ model we also present a generalization to the nonextremal case, which turns out to be characterized by a Carter-Pleba\'nski-type metric, and has $n+3$ independent parameters, corresponding to mass, angular momentum as well as $n+1$ magnetic charges. We discuss the thermodynamics of these solutions, obtain a Christodoulou-Ruffini mass formula, and shew that they obey a first law of thermodynamics and that the product of horizon areas depends on the angular momentum and the magnetic charges only. At least some of the BPS black holes that we obtain may become instrumental for future microscopic entropy computations involving a supersymmetric index.