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Scalar-field dark matter (SFDM) halos exhibit a core-envelope structure withsoliton-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 andhalo obey certain energy or velocity scalings. We extend the core-halo massrelations to include SFDM with self-interaction (SI), either repulsive orattractive, and investigate its implications for the gravitational instabilityand collapse of solitonic cores, leading to supermassive black hole (SMBH)formation. For SFDM parameters that make $\sim$ Kpc-sized cores and CDM-likestructure formation on large scales but suppressed on small scales, cores arestable for all galactic halos of interest, from the free-field to the repulsiveSI limit. For attractive SI, however, halos masses $M_h\sim (10^{10}-10^{12})M_\odot$ have cores that collapse to SMBHs with $M_{SMBH}\sim 10^{6}-10^8M_\odot$, as observations seem to require, while smaller-mass halos have stablecores, for particle masses $m\simeq (2.14\times 10^{-22}-9.9\times10^{-20})\rm{eV}/c^2$, if the free-field has $\beta=1/3$, or $m = 2.23\times10^{-21}-1.7\times 10^{-18}\rm{eV}/c^2$, if $\beta=5/9$. For free-field andrepulsive cases, however, if previous constraints on particle parameters arerelaxed to allow much smaller (sub-galactic scale) cores, then halos can alsoform 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 wouldbe largely indistinguishable from Cold Dark Matter (CDM). Such SFDM modelsmight not resolve the small-scale structure problems of CDM, but they wouldexplain the formation of SMBHs quite naturally. Since CDM, itself, has not yetbeen ruled out, such SFDM models must also be viable (Abbreviated).

Core-Halo Mass Relation in Scalar Field Dark Matter Models and its Consequences for the Formation of Supermassive Black Holes