  Despite the intense theoretical and experimental effort, an understanding of
the superconducting pairing mechanism of the high-temperature superconductors,
leading to an unprecedented high transition temperature $T_c$, is still
lacking. Starting from the $t$-$J$ model, we present a microscopic approach to
the physical properties of the superconducting phase at moderate hole-doping in
the framework of a novel renormalization scheme, called PRM. Our microscopic
approach allows us to explain the experimental findings in the underdoped as
well as in the optimal hole doping regime. In good agreement with experiments,
we find no superconducting solutions for very small hole doping. In the
superconducting phase, the order parameter turns out to have d-wave symmetry
with a coherence length of a few lattice constants. The spectral function,
obtained from angle-resolved photoemission spectroscopy (ARPES) along the Fermi
surface, is also in good agreement with experiment: The spectra display
peak-like structures which are caused alone by coherent excitations in a small
range around the Fermi energy.
