  We discuss the physics of the high temperature superconductivity in hole
doped copper oxide ceramics in the pseudogap region. Starting from an effective
reduced Hamiltonian relevant to the dynamics of holes injected into the copper
oxide layers proposed in a previous paper, we determine the superconductive
condensate wavefunction. We show that the low-lying elementary condensate
excitations are analogous to the rotons in superfluid $^4He$. We argue that the
rotons-like excitations account for the specific heat anomaly at the critical
temperature. We discuss and compare with experimental observations the London
penetration length, the Abrikosov vortices, the upper and lower critical
magnetic fields, and the critical current density. We give arguments to explain
the origin of the Fermi arcs and Fermi pockets. We investigate the nodal gap in
the cuprate superconductors and discuss both the doping and temperature
dependence of the nodal gap. We suggest that the nodal gap is responsible for
the doping dependence of the so-called nodal Fermi velocity detected in angle
resolved photoemission spectroscopy studies. We discuss the thermodynamics of
the nodal quasielectron liquid and their role in the low temperature specific
heat. We propose that the ubiquitous presence of charge density wave in hole
doped cuprate superconductors in the pseudogap region originates from
instabilities of the nodal quasielectrons driven by the interaction with the
planar $CuO_2$ lattice. We investigate the doping dependence of the charge
density wave gap and the competition between charge order and
superconductivity. We discuss the effects of external magnetic fields on the
charge density wave gap and elucidate the interplay between charge density wave
and Abrikosov vortices. Finally, we examine the physics underlying quantum
oscillations in the pseudogap region.
