  We report a study of the low-lying states of deformed $^{21}$Ne within the
framework of quantum-number projected generator coordinate method (PGCM),
starting from a chiral two-nucleon-plus-three-nucleon (NN+3N) interaction. The
wave functions of states are constructed as a linear combination of a set of
axially-deformed Hartree-Fock-Bogliubov (HFB) wave functions with different
quadrupole deformations. These HFB wave functions are projected onto different
angular momenta and the correct neutron and proton numbers for $^{21}$Ne. The
results of calculations based on the effective Hamiltonians derived by
normal-ordering the 3N interaction with respect to three different reference
states, including the quantum-number projected HFB wave functions for
$^{20}$Ne, $^{22}$Ne, and an ensemble of them with equal weights, are compared.
This study serves as a key step towards ab initio calculations of odd-mass
deformed nuclei with the in-medium GCM.
