  Studies on the interplay between the charge order and the $d$-wave
superconductivity in the copper-oxide high $T_{\rm c}$ superconductors are
reviewed with a special emphasis on the exploration based on the unconventional
concept of the electron fractionalization and its consequences supported by
solutions of high-accuracy quantum many-body solvers. Severe competitions
between the superconducting states and the charge inhomogeneity including the
charge/spin striped states revealed by the quantum many-body solvers are first
addressed for the Hubbard models and then for the {\it ab initio} Hamiltonians
of the cuprates derived without adjustable parameters to represent the
low-energy physics of the cuprates. The charge inhomogeneity and
superconductivity are born out of the same mother, namely, the carrier
attraction arising from the strong Coulomb repulsion near the Mott insulator
(Mottness) and accompanied electron fractionalization. The same mother makes
the severe competition of the two brothers inevitable. The electron
fractionalization has a remarkable consequences on the mechanism of the
superconductivity. Recent explorations motivated by the concept of the
fractionalization and their consequences on experimental observations in
energy-momentum resolved spectroscopic measurements including the angle
resolved photoemission spectroscopy (ARPES) and the resonant inelastic X-ray
spectroscopy (RIXS) are overviewed, with future vision for the integrated
spectroscopy to challenge the long-standing difficulties in the cuprates as
well as in other strongly correlated matter in general.
