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.
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