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Growing evidence indicate supermassive black holes (SMBHs) in the mass rangeof $M_{\rm BH}$$\sim 10^6-10^{10}M_{\odot}$ lurking in central bulges of manygalaxies. Extensive observations reveal fairly tight power laws of $M_{\rm BH}$versus the mean stellar velocity dispersion $\sigma$ of the host bulge. Thedynamic evolution of a bulge and the formation of a central SMBH should bephysically linked by various observational clues. In this contribution, wereproduce the empirical $M_{\rm BH}-\sigma$ power laws based on a self-similargeneral polytropic quasi-static bulge evolution and a sensible criterion offorming a SMBH surrounding the central density singularity of a generalsingular polytropic sphere (SPS) \cite{loujiang2008}. Other properties of hostbulges and central SMBHs are also examined. Based on our model, we discuss theintrinsic scatter of the $M_{\rm BH}-\sigma$ relation and a scenario for theevolution of SMBHs in different host bulges.

Mass and Mean Velocity Dispersion Relations for Supermassive Black Holes in Galactic Bulges