A microscopic theory of the electronic spectrum and of superconductivity within the t-J model on the honeycomb lattice is developed. We derive the equations for the normal and anomalous Green functions in terms of the Hubbard operators by applying the projection technique. Superconducting pairing of d + id-type mediated by the antiferromagnetic exchange is found. The superconducting Tc as a function of hole doping exhibits a two-peak structure related to the van Hove singularities of the density of states for the two-band t-J model. At half-filling and for large enough values of the exchange coupling, gapless superconductivity may occur. For small doping the coexistence of antiferromagnetic order and superconductivity is suggested. It is shown that the s-wave pairing is prohibited, since it violates the constraint of no-double-occupancy.
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