Ultramassive black holes in the most massive galaxies: $M_{\rm
  BH}-\sigma$ versus $M_{\rm BH}-R_{\rm b}$
  [Abridged] We investigate the nature of the relations between black hole (BH)
mass ($M_{\rm BH}$) and the central velocity dispersion ($\sigma$) and, for
core-S\'ersic galaxies, the size of the depleted core ($R_{\rm b}$). Our sample
of 144 galaxies with dynamically determined $M_{\rm BH}$ encompasses 24
core-S\'ersic galaxies, thought to be products of gas-poor mergers, and
reliably identified based on high-resolution HST imaging. For core-S\'ersic
galaxies -- i.e., combining normal-core ($R_{\rm b} < 0.5 $ kpc) and large-core
galaxies ($R_{\rm b} \gtrsim 0.5$ kpc), we find that $M_{\rm BH}$ correlates
remarkably well with $R_{\rm b}$ such that $M_{\rm BH} \propto R_{\rm b}^{1.20
\pm 0.14}$ (rms scatter in log $M_{\rm BH}$ of $\Delta_{\rm rms} \sim 0.29$
dex), confirming previous works on the same galaxies except three new ones.
Separating the sample into S\'ersic, normal-core and large-core galaxies, we
find that S\'ersic and normal-core galaxies jointly define a single log-linear
$M_{\rm BH}-\sigma$ relation $M_{\rm BH} \propto \sigma^{ 4.88 \pm 0.29}$ with
$\Delta_{\rm rms} \sim 0.47$ dex, however, at the high-mass end large-core
galaxies (four with measured $M_{\rm BH}$) are offset upward from this relation
by ($2.5-4) \times \sigma_{\rm s}$, explaining the previously reported
steepening of the $M_{\rm BH}-\sigma$ relation for massive galaxies. Large-core
spheroids have magnitudes $M_{V} \le -23.50$ mag, half-light radii Re $>$ 10
kpc and are extremely massive $M_{*} \ge 10^{12}M_{\odot}$. Furthermore, these
spheroids tend to host ultramassive BHs ($M_{\rm BH} \ge 10^{10}M_{\odot}$)
tightly connected with their $R_{\rm b}$ rather than $\sigma$. The less popular
$M_{\rm BH}-R_{\rm b}$ relation exhibits $\sim$ 62% less scatter in log $M_{\rm
BH}$ than the $M_{\rm BH}- \sigma$ relations.
