  We consider $L^2$-critical focusing nonlinear Schroedinger equations with
Hartree type nonlinearity $$i \pr_t u = -\DD u - \big (\Phi \ast |u|^2 \big) u
\quad {in $\RR^4$},$$ where $\Phi(x)$ is a perturbation of the convolution
kernel $|x|^{-2}$. Despite the lack of pseudo conformal invariance for this
equation, we prove the existence of critical mass finite-time blowup solutions
$u(t,x)$ that exhibit the pseudo-conformal blowup rate $$ \| \nabla u(t)
\|_{L^2_x} \sim \frac{1}{|t|} \quad {as} \quad t \nearrow 0 . $$ Furthermore,
we prove the finite-codimensional stability of this conformal blow up, by
extending the nonlinear wave operator construction by Bourgain and Wang (see
\cite{Bourgain+Wang1997}) to $L^2$-critical Hartree NLS.
