  Based on a microscopic theoretical study, we show that novel
superconductivity is induced by carrier doping in layered perovskite Ir oxides
where a strong spin-orbit coupling causes an effective total angular momentum
J_{eff}=1/2 Mott insulator. Using a variational Monte Carlo method, we find an
unconventional superconducting state in the ground state phase diagram of a
t_{2g} three-orbital Hubbard model on the square lattice. This superconducting
state is characterized by a d_{x^2-y^2}-wave "pseudospin singlet" formed by the
J_{eff}=1/2 Kramers doublet, which thus contains inter-orbital as well as both
singlet and triplet components of t_{2g} electrons. The superconducting state
is found stable only by electron doping, but not by hole doping, for the case
of carrier doped Sr_2IrO_4. We also study an effective single-orbital Hubbard
model to discuss the similarities to high-T_c cuprate superconductors and the
multi-orbital effects.
