  Kagome metals $A$V$_3$Sb$_5$ ($A=$ K, Rb, and Cs) exhibit intriguing
superconductivity below $0.9 \sim 2.5 $ K, a charge density wave (CDW)
transition around $80\sim 100 $ K, and $\mathbb{Z}_{2}$ topological surface
states. The nature of the CDW phase and its relation to superconductivity
remains elusive. In this work, we investigate the electronic and structural
properties of CDW by first-principles calculations. We reveal an inverse Star
of David deformation as the $2\times2\times2$ CDW ground state of the kagome
lattice. The kagome lattice shows softening breathing-phonon modes, indicating
the structural instability. However, electrons play an essential role in the
CDW transition via Fermi surface nesting and van Hove singularity. The inverse
Star of David structure agrees with recent experiments by scanning tunneling
microscopy (STM). The CDW phase inherits the nontrivial $\mathbb{Z}_{2}$-type
topological band structure. Further, we find that the electron-phonon coupling
is too weak to account for the superconductivity $T_c$ in all three materials.
It implies the existence of unconventional pairing of these kagome metals. Our
results provide essential knowledge toward understanding the superconductivity
and topology in kagome metals.
