  Scalar-field dark matter (SFDM) halos exhibit a core-envelope structure with
soliton-like cores and CDM-like envelopes. Simulations without self-interaction
(free-field case) report a core-halo mass relation $M_c\propto M_{h}^{\beta}$,
with either $\beta=1/3$ or $\beta=5/9$, which can be understood if core and
halo obey certain energy or velocity scalings. We extend the core-halo mass
relations to include SFDM with self-interaction (SI), either repulsive or
attractive, and investigate its implications for the gravitational instability
and collapse of solitonic cores, leading to supermassive black hole (SMBH)
formation. For SFDM parameters that make $\sim$ Kpc-sized cores and CDM-like
structure formation on large scales but suppressed on small scales, cores are
stable for all galactic halos of interest, from the free-field to the repulsive
SI limit. For attractive SI, however, halos masses $M_h\sim (10^{10}-10^{12})
M_\odot$ have cores that collapse to SMBHs with $M_{SMBH}\sim 10^{6}-10^8
M_\odot$, as observations seem to require, while smaller-mass halos have stable
cores, for particle masses $m\simeq (2.14\times 10^{-22}-9.9\times
10^{-20})\rm{eV}/c^2$, if the free-field has $\beta=1/3$, or $m = 2.23\times
10^{-21}-1.7\times 10^{-18}\rm{eV}/c^2$, if $\beta=5/9$. For free-field and
repulsive cases, however, if previous constraints on particle parameters are
relaxed to allow much smaller (sub-galactic scale) cores, then halos can also
form SMBHs, for the same range of halo and BH masses, as long as $\beta = 5/9$
is correct for the free-field. In that case, structure formation in SFDM would
be largely indistinguishable from Cold Dark Matter (CDM). Such SFDM models
might not resolve the small-scale structure problems of CDM, but they would
explain the formation of SMBHs quite naturally. Since CDM, itself, has not yet
been ruled out, such SFDM models must also be viable (Abbreviated).
