  It has been recently established that, if the nonlinear relationship between
the overdensity perturbations and the curvature perturbations are taken into
account, non-Gaussianity is introduced in the overdensity statistics which
alters the expected primordial black hole abundance. This is explored by using
the nonlinear relationship between the overdensities and curvature
perturbations up to second order where a negative skewness and positive
kurtosis aims at lowering and increasing the abundance while an abundance
comparable to Gaussian perturbations is obtained by adjusting the amplitude of
the curvature power spectrum. The effects of the nonvanishing skewness and
kurtosis are studied using a toy model Dirac delta and lognormal curvature
power spectra as well as one obtained from an $\alpha-$attractor model capable
of primordial black hole production. Finally, the nonlinear calculations using
Press-Schechter are compared with peaks theory.
