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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