  Here we report a combined study of low-temperature scanning tunneling
microscopy (STM) and dynamical mean-field theory (DMFT) on PdCrO$_2$, a
delafossite metal with an antiferromagnetic order below ~37.5 K. First, on the
CrO$_2$-terminated polar surface we detect a gap-like feature both below and
above the N\'eel temperature. The DMFT calculations indicate that this gap is
opened due to the strong correlations of Cr-3d electrons, suggesting the hidden
Mott nature of the gap. Then, we observe two kinds of Pd-terminated polar
surfaces. One is a well-ordered Pd surface with the
Fermi-surface-nesting-induced incommensurate charge modulation, while the other
one is a reconstructed Pd surface with the individual nano-scale non-periodic
domain structures. On the well-ordered Pd surface, the interference between the
incommensurate charge modulation and the atomic lattice forms the periodic
moir\'e pattern. Our results provide important microscopic information for
fully understanding the correlated electronic properties of this class of
materials.
