  In this paper we are interested in the following critical Hartree equation
\begin{equation*} \begin{cases} -\Delta u
=\displaystyle{\Big(\int_{\Omega}\frac{u^{2_{\mu}^\ast}
(\xi)}{|x-\xi|^{\mu}}d\xi\Big)u^{2_{\mu}^\ast-1}}+\varepsilon u
,~~~\text{in}~\Omega,\\
u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}~\partial\Omega,
\end{cases} \end{equation*} where $N\geq4$, $0<\mu\leq4$, $\varepsilon>0$ is a
small parameter, $\Omega$ is a bounded domain in $\mathbb{R}^N$, and
$2_{\mu}^\ast=\frac{2N-\mu}{N-2}$ is the critical exponent in the sense of the
Hardy-Littlewood-Sobolev inequality. By establishing various versions of local
Pohozaev identities and applying blow-up analysis, we first investigate the
location of the blow-up points for single bubbling solutions to above the
Hartree equation. Next we prove the local uniqueness of the blow-up solutions
that concentrates at the non-degenerate critical point of the Robin function
for $\varepsilon$ small.
