$\DeclareMathOperator\Aut{Aut}$Let $X$ and $Y$ be two complex manifolds of dimension $n$, $n\geq 2$. Denote by $\Aut(X)$ and $\Aut(Y)$ the group of bi-holomorphisms of $X$ and $Y$, respectively. Suppose the symmetric group on $n$-symbols $S_n$ is contained in both $\Aut(X)$ and $\Aut(Y)$ such that
$X/S_n$ and $Y/S_n$ are complex manifolds of dimension $n$;
$X/S_n$ and $Y/S_n$ are bi-holomorphic.
Question. Is it true that $X$ and $Y$ are bi-holomorphic?
