  Growing evidence indicate supermassive black holes (SMBHs) in a mass range of
$M_{\rm BH}$$\sim 106-10^{10}M_{\odot}$ lurking in central stellar bulges of
galaxies.Extensive observations reveal fairly tight power laws of $M_{\rm BH}$
versus the mean stellar velocity dispersion $\sigma$ of the host stellar
bulge.Together with evidence for correlations between $M_{\rm BH}$ and other
properties of host bulges, the dynamic evolution of a bulge and the formation
of a central SMBH should be linked. In this Letter, we reproduce the empirical
$M_{\rm BH}-\sigma$ power laws based on our recent theoretical analyses (Lou &
Wang; Wang & Lou; Lou, Jiang & Jin) for a self-similar general polytropic
quasi-static dynamic evolution of bulges with self-gravity and spherical
symmetry and present a sensible criterion of forming a central SMBH. The key
result is $M_{\rm BH}={\cal L}\sigma^{1/(1-n)}$ where $2/3<n<1$ and ${\cal L}$
is a proportional coefficient characteristic of different classes of host
bulges. By fitting and comparing several empirical $M_{\rm BH}-\sigma$ power
laws, we conclude that SMBHs and galactic bulges grow and evolve in a coeval
manner and most likely there exist several classes of galactic bulge systems in
quasi-static self-similar evolution and that to mix them together can lead to
an unrealistic fitting. Based on our bulge-SMBH model, we provide explanations
for intrinsic scatter in the relation and a unified scenario for the formation
and evolution of SMBHs in different classes of host bulges.
