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We study compact gravitating $Q$-ball, $Q$-shell solutions in a sigma modelwith the target space $\mathbb{C}P^N$. Models with odd integer $N$ and suitablepotential can be parameterized by $N$-th complex scalar fields and they supportcompact solutions. A coupling with gravity allows for harboring of theSchwarzschild black holes for the $Q$-shell solutions. The energy of thesolutions behaves as $E\sim |Q|^{5/6}$, where $Q$ stands for the $U(1)$ Noethercharge, for both the gravitating and the black hole solutions.Notabledifference from the solutions of the flat space is that upper bound of $|Q|$appears when the coupling with gravity is stronger. The maximal value of $|Q|$quickly reduces for larger coupling constant. It may give us a useful hint ofhow a star forms its shape with a certain finite number of particles.

Gravitating compact $Q$-ball and $Q$-shell solutions in the $\mathbb{C}P^N$ nonlinear sigma model