  As the first magnetic kagome material to exhibit the charge density wave
(CDW) order, FeGe has attracted much attention in recent studies. Similar to
AV$_{3}$Sb$_{5}$ (A = K, Cs, Rb), FeGe exhibits the CDW pattern with an
in-plane 2$\times $2 structure and the existence of van Hove singularities
(vHSs) near the Fermi level. However, sharply different from AV$_{3}$Sb$_{5}$
which has phonon instability at $M$ point, all the theoretically calculated
phonon frequencies in FeGe remain positive. Here, we perform a comprehensive
study of the band structures, Fermi surfaces and nesting function of FeGe
through first-principles calculations. Surprisingly, we find that the maximum
of nesting function is at $K$ point instead of $M$ point. Two Fermi pockets
with Fe-$d_{xz}$ and Fe-$d_{x^{2}-y^{2}}$/$d_{xy}$ orbital characters have
large contribution to the Fermi nesting, which evolve significantly with
$k_{z}$, indicating the highly three-dimensional (3D) feature of FeGe in
contrast to AV$_{3}$Sb$_{5}$. Meanwhile, the vHSs are close to the Fermi
surface only in a small $k_{z}$ range, and does not play a leading role in
nesting function. Considering the effect of local Coulomb interaction, we
reveal that the Fermi level eigenstates nested by vector $K$ are mainly
distributed from unequal sublattice occupancy, thus the instability at $K$
point is significantly suppressed. Meanwhile, the wave functions nested by
vector $M$ have many ingredients located at the same Fe site, thus the
instability at $M$ point is enhanced. This indicates that the electron
correlation, rather than electron-phonon interaction, plays a key role in the
CDW transition at $M$ point.
