  We investigate the properties of the nearest-neighbor singlet pairing and the
emergence of d-wave superconductivity in the doped honeycomb lattice
considering the limit of large interactions and the $t-J_1-J_2$ model. First,
by applying a renormalized mean-field procedure as well as slave-boson theories
which account for the proximity to the Mott insulating state, we confirm the
emergence of d-wave superconductivity in agreement with earlier works. We show
that a small but finite $J_2$ spin coupling between next-nearest neighbors
stabilizes d-wave symmetry compared to the extended s-wave scenario. At small
hole doping, to minimize energy and to gap the whole Fermi surface or all the
Dirac points, the superconducting ground state is characterized by a $d+id$
singlet pairing assigned to one valley and a $d-id$ singlet pairing to the
other, which then preserves time-reversal symmetry. The slightly doped
situation is distinct from the heavily doped case (around 3/8 and 5/8 filling)
supporting a pure chiral $d+id$ symmetry and breaking time-reversal symmetry.
Then, we apply the functional Renormalization Group and we study in more detail
the competition between antiferromagnetism and superconductivity in the
vicinity of half-filling. We discuss possible applications to
strongly-correlated compounds with Copper hexagonal planes such as
In$_3$Cu$_{2}$VO$_9$. Our findings are also relevant to the understanding of
exotic superfluidity with cold atoms.
