  Starting from a low-energy model for the band dispersion of the $2 \times 2$
charge-ordered phase of the kagome metals $A$V$_3$Sb$_5$ ($A=$ K, Rb, Cs), we
show that nematicity can develop in this state driven either by $4 \times 4$
charge fluctuations preemptive of a $1 \times 4$ charge order (CO), or by an
actual zero momentum $d$-wave charge Pomeranchuk instability (PI). We perform
an analysis that starts from a Kohn-Luttinger theory in the particle-hole
sector, which allows us to establish a criterion for the development of an
attractive nematic channel near the onset of the $1 \times 4$ CO and near the
$d$-wave charge PI, respectively. We derive an effective charge-fermion model
for the $d$-wave PI with a nematic susceptibility given via a random phase
approximation (RPA) summation. By contrast, for the finite momentum CO, the RPA
scheme breaks down and needs to be improved upon by including Aslamazov-Larkin
contributions to the nematic pairing vertex. We then move to the derivation of
the Ginzburg-Landau potentials for the $4 \times 4$ CO and for the $d$-wave PI,
and we obtain the corresponding expression for the nematic susceptibility at
the nematic transition temperature T $ \sim \text{T}_\text{nem}$ in both cases.
Our work establishes a relation between the nematicity observed in some of the
iron-based superconductors, where the nematic phase might be driven by spin
fluctuations, and the vanadium-based kagome metals, where charge fluctuations
likely induce nematicity. The two microscopic mechanisms we propose for the
stabilization of the nematic state in $A$V$_3$Sb$_5$ are distinguishable by
diffusive scattering experiments, meaning that it is possible to gauge which of
the two theories, if any, is the most likely to describe this phase. Both
mechanisms might also be relevant for the recently discovered titanium-based
family $A$Ti$_3$Sb$_5$.
