  We study the Holstein-Hubbard model at half filling to explore ordered phases
including su- perconductivity (SC), antiferromagnetism (AF), and charge order
(CO) in situations where the electron-electron and electron-phonon interactions
are strong (~ electronic bandwidth). The model is solved in the dynamical
mean-field approximation using a continuous-time quantum Monte Carlo impurity
solver. We determine the superconducting transition temperature (Tc) and the SC
order parameter and show that the phonon-induced retardation or the strong
Coulomb interaction leads to a significant reduction and shift of the Tc dome
if one interprets the system as having an effective static interaction Ueff
given by the Hubbard U reduced by the phonon-mediated attraction in the static
limit. This behavior is analyzed by comparison to an effective static model in
the polaron representation with a reduced bandwidth. We also determine the
finite-temperature phase diagram including AF and CO. In the moderate-coupling
regime, there is a hysteretic region of AF and CO around Ueff = 0, while the
two phases are separated by a paramagnetic metal in the weak-coupling regime
and a paramagnetic insulator in the strong coupling regime.
