  Kondo-lattice theory for the t-J model and a phenomenological theory based on
it are developed to study superconductivity in the vicinity of the Mott
metal-insulator transition. Since the quenching of magnetic moments by
single-site quantum spin fluctuations or the Kondo effect is reduced by the
opening of a superconducting gap, spin density wave (SDW) can appear in a
superconducting state and the Knight shift can deviate from the Yosida function
to be small. The electron-phonon interaction, which arises from the modulation
of the superexchange interaction by phonons, is crucial in the coexistence of
superconductivity and SDW. It is proposed that the coexistence of
superconductivity, a double-Q SDW, a double-Q lattice distortion, and a
double-Q charge density wave induced by the SDW rather than the lattice
distortion, is responsible for the checkerboard structure and the
zero-temperature pseudo-gap observed in under-doped cuprate superconductors.
