  Recent high pressure experiments discovered abnormal double-dome
superconductivities in the newly-synthesized kagome materials $A$V$_3$Sb$_5$
($A$ = K, Rb, Cs), which also host abundant emergent quantum phenomena such as
charge density wave (CDW), anomalous Hall effect, nontrivial topological
property, etc. In this work, by using first-principles electronic structure
calculations, we have studied the CDW state, superconductivity, and topological
property in CsV$_3$Sb$_5$ under pressures ($<$ 50 GPa). Based on the
electron-phonon coupling theory, our calculated superconducting $T_\text{c}$s
are consistent with the observed ones in the second superconducting dome at
high pressure, but are much higher than the measured values at low pressure.
The further calculations including the Hubbard U indicate that with modest
electron-electron correlation the magnetism on the V atoms exists at low
pressure and diminishes gradually at high pressure. We thus propose that the
experimentally observed superconductivity in CsV$_3$Sb$_5$ at ambient/low
pressures may still belong to the conventional Bardeen-Cooper-Schrieffer (BCS)
type but is partially suppressed by the V magnetism, while the
superconductivity under high pressure is fully conventional without invoking
the magnetism. We also predict that there are a second weak CDW state and
topological phase transitions in CsV$_3$Sb$_5$ under pressures. Our theoretical
assertion calls for future experimental examination.
