  High-Tc superconducting materials (HTSC) have anomalous properties such as
the pseudo-gap or spin-gap etc., in Hall coefficient, 1/T1T and the density of
states etc. First including effects of strong on-site repulsion between
d-electrons at Cu-sites, we obtain quasi-particles with super-exchange
interaction Js, whose band width tends to zero, i.e., the system goes to
insulator, as the hole-doping rate tends to zero. The quasi-particles
correspond to Zhang-Rice singlet states. Js larger than the band width combined
with the 2-dimensional character of the system induces strong
antiferro-magnetic (AF) and superconducting (SC) fluctuations. We treat effects
of the AF ones in the FLEX approximation and those of the SC ones in the
self-consistent t-matrix approximation to show that both fluctuations in the
under-doped region start to increase at T0 as T decreases from T>>Tc, the AF
ones dominate the SC ones at T>Tsg, while SC ones dominate at T<Tsg. This
cross-over of the fluctuations causes the anomalous phenomena in the
under-doped region. We also obtain the phase diagram of HTSC consistent to one
observed in experiments.
