Annotated Definitions

Proper discontinuity

(Credit where due: this is a summary things from MathOverflow 55726 and Math Stack Exchange 1082834.)

Let G be a locally-compact group acting on a locally-compact Hausdorff topological space X (these additional conditions on G and X being certainly satisfied if e.g. G is finitely-generated and X is a proper metric space.)

We say the action is properly discontinuous if for any compact set K \subset X, K \cap gK \neq \varnothing for at most finitely many g \in G. (cf. the action of \mathbb{Z}^n on \mathbb{R}^n by translations, which is discontinuous: for any [sufficiently small] compact set KK \cap gK = \varnothing for every g \in \mathbb{Z}^n; here we allow finite overlap even at arbitrarily small resolution: this allows for finite stabilizers, parabolic elements, etc.)

In particular this implies that every point x \in X has an open neighborhood U s.t. U \cap gU \neq \varnothing for at most finitely many g\ in G. (Take U to sit inside some compact set K.)

Note that the second condition is strictly weaker than the first: consider the \mathbb{Z}-action on the punctured plane given by n \cdot (x,y) = (2^n x, 2^{-n}y). This satisfies the second condition, but not the first—take K to be any closed annulus centered about the origin, for instance.

The main point of imposing these mildly technical conditions when we study “geometric” group actions is this:

This second condition is equivalent to the quotient map X \to X/G being a covering map. The first condition (the definition) serves to ensure that the map G \times X \to X \times X given by (g,x) \mapsto (x,gx) is proper, and hence that the quotient X/G is Hausdorff.

Annotated Definitions

Kähler manifolds

A Kähler metric on a Riemmanian manifold is one which has especially nice (indeed, bemerkenswert) properties, which are usually summarized by saying that it has compatible Riemannian, symplectic, and complex structures. More precisely (and less transparently), a Kähler structure on a Riemannian manifold (M^n, g) is a pair (\Omega, J) where

  • J is a complex structure, i.e. a field of endomorphisms of the tangent bundle TM satisfying J^2 = -\mathrm{id} which is also integrable;
  • \Omega is a symplectic 2-form (i.e. d\Omega = 0);
  • g(X,Y) = g(JX, JY) for all X, Y \in TM (i.e. J makes g a Hermitian metric, or the complex and Riemannian structures play nice with each other), and
  • \Omega(X,Y) = g(JX, Y) (a compatibility condition for the symplectic structure).

For a Kähler metric, the Hermitian metric tensor g is specified by a unique function u, in the sense that g_{\alpha\bar{\beta}} = \frac{\partial^2 u}{\partial z_\alpha \partial \bar{z_\beta}}. This gives rise to simple explicit expression for the Christoffel symbols and the Ricci and curvature tensors, and “a long list of miracles occur then.” (quoted from Moroianu’s notes, which presumably quote from Kähler’s original paper.)

Kähler metrics  may also be characterized, analytically, by the existence of holomorphic normal coordinates around each point.

Some examples of Kähler manifolds (i.e. manifolds which admit Kähler metrics): \mathbb{C}^m with the usual Hermitian metric; any Riemann surface; complex projective space with the Fubini-Study metric; any projective manifold; more examples from algebraic geometry …

Moroianu’s notes contains much more on Kähler manifolds, including elaborations and proofs of many of the assertions above.

Annotated Definitions

The Cross-ratio

The cross-ratio of four points is [z_1, z_2, z_3, z_4] := \frac{z_1 - z_3}{z_1 - z_4} \frac{z_2 - z_4}{z_2 - z_3}.

By taking z_1, z_2, z_3, z_4 \in \mathbb{R} we define the cross-ratio on (ordered) quadruples of points on the real line, and thereby, by interpreting the differences as signed distances, on ordered quadruples of collinear points in Euclidean space. We can extend this definition to the projective line or projective space by allowing z_i = \infty.

By taking z_1, z_2, z_3, z_4 \in \mathbb{C} \cup \{\infty\}, we define the cross-ratio on quadruples of points in the Riemann sphere \hat{\mathbb{C}}. (More generally, we can define the cross-ratio on quadruples of points in any field.)

The cross-ratio is interesting because it is the essential projective invariant (in 2 dimensions); indeed, this might be taken as the more “natural” or “universal” definition, or what Tim Gowers refers to when he says that [about tensor products in particular, but also mathematical objects in general] “exactly how they are defined is not important: what matters is the properties they have.”

More precisely, the cross-ratio is invariant under projective transformations, i.e. (in 2 dimensions) under \mathrm{Aut}(\hat{\mathbb{C}}) \cong \mathrm{PSL}(2, \mathbb{C}) := \mathrm{SL}(2, \mathbb{C}) / \pm I. It is essentially the only projective invariant of ordered quadruples of points, in the sense of a universal property (any projective invariant can be bijectively transformed into the cross-ratio), since \mathrm{PSL}(2, \mathbb{C}) acts simply transitively on ordered triples of points in \hat{\mathbb{C}}. [I feel like I don’t quite understand the details of the argument here yet.]

Furthermore, quadruples of points are the natural choice of objects to  determine (2-8dimensional) projective invariants on: the Euclidean distance between 2 points is invariant under translations or rotations and the ratio of the distances between 3 points is invariant under Euclidean similarities, but \mathrm{PSL}(2, \mathbb{C}) acts transitively on ordered triples of points in \hat{\mathbb{C}}.

This property of invariance of cross-ratio allows us to define hyperbolic distance in terms of the cross-ratio: more precisely, taking the Poincaré half-plane model \mathbb{H} of the hyperbolic plane, we have that d_{\mathbb{H}}(x,y) = \log [x', y, x, y'] where x', y' are the endpoints on \partial\mathbb{H} of the geodesic between x and y (we may replace \mathbb{H} with the Poincaré disc model \mathbb{D} throughout.)

To prove this we conformally map our points to the imaginary axis and treat only this special case: note that \mathrm{Aut} (\mathbb{H}) \subset \mathrm{Aut}(\widehat{\mathbb{C}}) and also that \mathrm{Aut} (\mathbb{H}) preserves the cross-ratio. Now WLOG let x, y lie on the imaginary axis with x = i and y = ai (where a \in \mathbb{R}.) Then d_{\mathbb{H}}(x,y) = a and [x',y,x,y'] = \frac{0-ai}{0-i} \frac{\infty - i}{\infty-ai} = a, as desired.