We investigate the correlations between the black hole mass $M_{BH}$, the velocity dispersion $sigma$, the bulge mass $M_{Bu}$, the bulge average spherical density $rho_h$ and its spherical half mass radius $r_h$, constructing a database of 97 galaxies (31 core ellipticals, 17 power-law ellipticals, 30 classical bulges, 19 pseudo bulges) by joining 72 galaxies from the literature to 25 galaxies observed during our recent SINFONI black hole survey. For the first time we discuss the full error covariance matrix. We analyse the well known $M_{BH}-sigma$ and $M_{BH}-M_{Bu}$ relations and establish the existence of statistically significant correlations between $M_{Bu}$ and $r_h$ and anti-correlations between $M_{Bu}$ and $rho_h$. We establish five significant bivariate correlations ($M_{BH}-sigma-rho_h$, $M_{BH}-sigma-r_h$, $M_{BH}-M_{Bu}-sigma$, $M_{BH}-M_{Bu}-rho_h$, $M_{BH}-M_{Bu}-r_h$) that predict $M_{BH}$ of 77 core and power-law ellipticals and classical bulges with measured and intrinsic scatter as small as $approx 0.36$ dex and $approx 0.33$ dex respectively, or 0.26 dex when the subsample of 45 galaxies defined by Kormendy and Ho (2013) is considered. In contrast, pseudo bulges have systematically lower $M_{BH}$, but approach the predictions of all the above relations at spherical densities $rho_hge 10^{10} M_odot/kpc^3$ or scale lengths $r_hle 1$ kpc. These findings fit in a scenario of co-evolution of BH and classical-bulge masses, where core ellipticals are the product of dry mergers of power-law bulges and power-law Es and bulges the result of (early) gas-rich mergers and of disk galaxies. In contrast, the (secular) growth of BHs is decoupled from the growth of their pseudo bulge hosts, except when (gas) densities are high enough to trigger the feedback mechanism responsible for the existence of the correlations between $M_{BH}$ and galaxy structural parameters.
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