# A Good Quotient from a Bad Action

## Good Quotients from Bad Actions

### Not all Good Actions are Free

In constructing examples of complex manifolds the transport of structure philosophy is useful. This principle means only doing hard work for a handful of examples and then throwing that well-earned structure around, through maps, to get new creations bearing the same structure.

For example, proving that $(\mathbb{Z},+)$ is associative is an annoying set-theoretic exercise. When proving that $(\mathbb{Z}[i], +)$ is associative we do not start from pure set theory. Rather, we borrow known associativity of $\mathbb{Z}.$ We say that addition of Gaussian integers is defined pointwise

$(n_1+m_1i) + (n_2 + m_2i) = (n_1+m_1) + i(n_2 + m_2)$

and so is associative because it is associative in each coordinate by associativity of integer addition.

The case is similar in topology. Proving that $\pi_1(S^1) \simeq \mathbb{Z}$ is kind of hard. Computing $\pi_1$ of the sphere, torus and figure eight are not as hard; we use tools like van Kampen and Mayer-Vietoris to reduce the problem to computing $\pi_1(S^1).$

Writing down examples of complex manifolds is the same game. Our basic examples are open subsets $U \subset \mathbb{C}^n$. We seek to get new manifolds by taking orbit spaces of a group action. We shall describe some common conditions upon which a quotient space is a Hausdorff complex manifold and then give an example of an action not satisfying these common conditions but whose quotient is nonetheless a Hausdorff complex manifold. This (counter)example teaches us that not all good actions are free.

$\textbf{Definition 1.}$ A group action $G \times X \to X$ is said to be free if $g \cdot x \neq x$ whenever $g \neq 1.$

$\textbf{Definition 2.}$ A group action $G \times X \to X$ is said to be proper if the map $(g,x) \mapsto (g \cdot x,x)$ is proper.

Quotients by free and proper actions are manifolds.

$\textbf{Theorem 3.}$ Let $G \times X \to X$ be a free and proper group action on a manifold $X.$ Then, $X/G$ is a (Hausdorff) complex manifold.

$\textbf{Proof.}$ For all $x,y \in X$ such that $x \notin G \cdot y$ there exist open nieghborhoods $x \in U, y \in V$ such that, for any $g \in G$$U \cap g\cdot V = \empty.$ Now, suppose we have charts $(U_i,\varphi_i)$ that witness $X$ as a complex manifold and such that $g \cdot U_i \cap U_i = \empty$ for all $i$ and for all $1 \neq g \in G.$ Then, $U_i \cong \pi(U_i)$ (where$\pi$ denotes the projection$\pi : X \to X/G.$) We get holomorphic charts

$\pi^{-1} \circ \varphi_i : \pi(U_i) \to \mathbb{C}^n.$

The properness condition ensures the quotient is Hausdorff.

Now, this is just one way to get a complex structure on a quotient. This counterexample comes from D. Huybrechts Complex Geometry p. 60: Consider a complex torus $\mathbb{C}/\Gamma.$ The group $\mathbb{Z}/n\mathbb{Z}$ acts on $\mathbb{C}/\Gamma$ by $z \mapsto -z.$ This action has four fixed points: $0, \tau_1/2,\tau_2/2, (\tau_1+\tau_2)/2$, where $\Gamma = \tau_1\mathbb{Z} + \tau_2\mathbb{Z}.$ A little dexterity with the Weierstrass function shows this quotient is isomorphic to $\mathbb{P}^1.$ This quotient is indeed a complex manifold (goodbut the action is not free (bad).
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