This paper deduces that the particular electronic structure of cuprate superconductors confines Cooper pairs to be first formed in the antinodal region which is far from the Fermi surface, and these pairs are incoherent and result in the pseudogap state. With the change of doping or temperature, some pairs are formed in the nodal region which locates the Fermi surface, and these pairs are coherent and lead to superconductivity. Thus the coexistence of the pseudogap and the superconducting gap is explained when the two kinds of gaps are not all on the Fermi surface. It also shows that the symmetry of the pseudogap and the superconducting gap are determined by the electronic structure, and non-s wave symmetry gap favours the high-temperature superconductivity. Why the high-temperature superconductivity occurs in the metal region near the Mott metal-insulator transition is also explained.

Considering the electron states both inside and outside the solid, we derive a formula of photoemission intensity. A general theoretical way to determine electronic structures of solids from ARPES experiments is outlined. It is shown that the results from ARPES experiments cannot be understood until these results have been made consistent with a theoretical calculation.

The competition between singlet and triplet su- perconductivity is examined in consideration of correla- tions on an extended Hubbard model. It is shown that the triplet superconductivity may not be included in the com- mon Hubbard model since the strong correlation favors the singlet superconductivity, and thus the triplet superconduc- tivity should be induced by the electron–phonon interaction and the ferromagnetic exchange interaction.We also present a superconducting qualification with which magnetism is unbeneficial to superconductivity.