  In this work, we explored the effect of the fuzzy dark matter (FDM) (or wave
dark matter) halo on a supermassive black hole (SMBH). Such a dark matter
introduces a soliton core density profile, and we treat it ideally as a
spherical distribution that surrounds the SMBH located at its center. In this
direction, we obtained a new metric due to the union of the black hole and dark
matter spacetime geometries. We applied the solution to the two known SMBH -
Sgr. A* and M87* and used the empirical data for the shadow diameter by EHT to
constrain the soliton core radius $r_\text{c}$ given some values of the boson
mass $m_\text{b}$. Then, we examine the behavior of the shadow radius based on
such constraints and relative to a static observer. We found that different
shadow sizes are perceived at regions $r_\text{obs}<r_\text{c}$ and
$r_\text{obs}>r_\text{c}$, and the deviation is greater for values
$m_\text{b}<10^{-22}$ eV. Concerning the shadow behavior, we have also analyzed
the effect of the soliton profile on the thin-accretion disk. Soliton dark
matter effects manifest through the varying luminosity near the event horizon.
We also analyzed the weak deflection angle and the produced Einstein rings due
to soliton effects. We found considerable deviation, better than the shadow
size deviation, for the light source near the SMBH with impact parameters
comparable to the soliton core. Our results suggest the possible experimental
detection of soliton dark matter effects using an SMBH at the galactic centers.
