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We use the coupled cluster method to high orders of approximation in order tocalculate the ground-state phase diagram of the XXZ spin-$s$ kagomeantiferromagnet with easy-plane anisotropy, i.e. the anisotropy parameter$\Delta$ varies between $\Delta=1$ (isotropic Heisenberg model) and $\Delta=0$($XY$ model). We find that for the extreme quantum case $s=1/2$ the groundstate is magnetically disordered in the entire region $0 \le \Delta \le 1$. For$s=1$ the ground state is disordered for $0.818 < \Delta \le 1$, it exhibits$\sqrt{3}\times\sqrt{3}$ magnetic long-range order for $0.281 < \Delta <0.818$,and $q=0$ magnetic long-range order for $0 \le \Delta < 0.281$. We confirm therecent result of Chernyshev and Zhitomirsky (Phys. Rev. Lett. 113, 237202(2014)) that the selection of the ground state by quantum fluctuations isdifferent for small $\Delta$ ($XY$ limit) and for $\Delta$ close to one(Heisenberg limit), i.e., $q=0$ magnetic order is favored over$\sqrt{3}\times\sqrt{3}$ for $0\le \Delta <\Delta_c$ and vice versa for$\Delta_c < \Delta \le 1$. We calculate $\Delta_c$ as a function of the spinquantum number $s$.

The ground-state phase diagram of the XXZ spin-s kagome antiferromagnet: A coupled-cluster study