  This work studies the Cauchy problem for the energy-critical inhomogeneous
Hartree equation with inverse square potential $$i\partial_t u-\mathcal
K_\lambda u=\pm |x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u, \quad
\mathcal K_\lambda=-\Delta+\frac\lambda{|x|^2}$$ in the energy space
$H_\lambda^1:=\{f\in L^2,\quad\sqrt{\mathcal{K}_\lambda}f\in L^2\}$. In this
paper, we develop a well-posedness theory and investigate the blow-up of
solutions in $H_\lambda^1$. Furthermore we present a dichotomy between energy
bounded and non-global existence of solutions under the ground state threshold.
To this end, we use Caffarelli-Kohn-Nirenberg weighted interpolation
inequalities and some equivalent norms considering $\mathcal K_\lambda$, which
make it possible to control the non-linearity involving the singularity
$|x|^{-\tau}$ as well as the inverse square potential. The novelty here is the
investigation of the energy critical regime which remains still open and the
challenge is to deal with three technical problems: a non-local source term, an
inhomogeneous singular term $|\cdot|^{-\tau}$, and the presence of an inverse
square potential.
