  Growing evidence indicate supermassive black holes (SMBHs) in the mass range
of $M_{\rm BH}$$\sim 10^6-10^{10}M_{\odot}$ lurking in central bulges of many
galaxies. Extensive observations reveal fairly tight power laws of $M_{\rm BH}$
versus the mean stellar velocity dispersion $\sigma$ of the host bulge. The
dynamic evolution of a bulge and the formation of a central SMBH should be
physically linked by various observational clues. In this contribution, we
reproduce the empirical $M_{\rm BH}-\sigma$ power laws based on a self-similar
general polytropic quasi-static bulge evolution and a sensible criterion of
forming a SMBH surrounding the central density singularity of a general
singular polytropic sphere (SPS) \cite{loujiang2008}. Other properties of host
bulges and central SMBHs are also examined. Based on our model, we discuss the
intrinsic scatter of the $M_{\rm BH}-\sigma$ relation and a scenario for the
evolution of SMBHs in different host bulges.
