# Quotients of complex manifolds by symmetric group

$$\DeclareMathOperator\Aut{Aut}$$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

1. $$X/S_n$$ and $$Y/S_n$$ are complex manifolds of dimension $$n$$;

2. $$X/S_n$$ and $$Y/S_n$$ are bi-holomorphic.

Question. Is it true that $$X$$ and $$Y$$ are bi-holomorphic?

Take two smooth hyperelliptic curves, which are not biholomorphic. Then, both have a degree $$2$$ morphism to $$\mathbb{P}^1$$. This is map may be seen as quotienting by involutions of the curves, I.e. an action of $$S_{2}$$.

Note that by taking different genus they don't have to be even homeomorphic.

Edit, I noticed you required that the manifolds have dimension at least 2,so taking product of the construction with a complex manifold should give the required example.

For a series of examples in dimension $$n=2$$, take a degree $$2k$$ hypersurface $$H_{2k} \subset \mathbb{P}^2$$. Correspondingly, there is a double cover $$X_{2k} \to \mathbb{P}^2$$, branched over $$k$$, so $$X_{2k}/\mathbb{Z}_2 \cong \mathbb{P}^2$$.

If $$k \neq h$$ then $$X_{2k}$$ and $$X_{2h}$$ are not even homeomorphic, for instance because they have different topological Euler number.

• The $n$ in $S_n$ is the same as the dimensions of $X$ and $Y$ respectively. Jul 2 at 13:20
• Ah, ok. Then just take $n=2$. I will edit the answer. Jul 2 at 13:24
• In dimension $3$, I guess you can take the Galois closure of triple covers branched on hypersurfaces of the form $X_{3k} \subset \mathbb{P}^3$. Jul 2 at 13:28
• I also would like to add that the case that I am in, the complex manifolds $X$ and $Y$ both admit the $n$-dimensional polydisc (the cartesian product of $n$-copies of the unit disc in the complex plane) as a holomorphic universal cover. Any ideas/suggestions in this case? Jul 2 at 13:28
• @vikram Suitable product of curves (of genus $g \geq 2$) endowed with a diagonal $S_n$ action should do the job. You could look at the papers by Catanese and collaborators about varieties isogenous to a product. Jul 2 at 13:29