  Kagome metals $A$V$_3$Sb$_5$ ($A=$ K, Rb, and Cs) exhibit intriguing
superconductivity below $0.9 \sim 2.5 $ K with possibly $\mathbb{Z}_{2}$
topological surface states, after a charge density wave (CDW) transition around
$80\sim 100 $ K. The nature of the CDW phase and its relation to the
superconductivity remain elusive. In this work, we investigate the electronic
and structural properties of the CDW phase by first-principles simulations. We
reveal an inverse Star of David structure as the $2\times2$ CDW ground state of
the kagome lattice. The CDW transition is driven by the Peierls instability
characterized by the softening of a breathing phonon mode in the kagome
lattice. The simulated scanning tunneling microscopy (STM) images of the
inverse Star of David structure agree with recent experiments. The nontrivial
$\mathbb{Z}_{2}$-type topological band structure remains the same after the CDW
transition. Further, we evaluate the electron-phonon coupling in the CDW phase
and find it is too weak to rationalize the superconductivity 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.
