Riemann Surfaces 1
The definition of Riemann surfaces.
1 Topological Groups
A topological group \(G\) is defined as a group equipped with a topological space structure such that the group multiplication \(G \times G \to G\) and the inversion map \(G \to G\) are continuous.
If \(G\) further admits the structure of a smooth manifold such that both multiplication and inversion are smooth, then \(G\) is referred to as a Lie group.
In the case where \(G\) is a topological group, the left translation map \(L_g: x \mapsto gx\) is a homeomorphism as a topological space and a continuous map. In fact, it constitutes a group isomorphism and a diffeomorphism, as its inverse map is explicitly given by \(L_{g^{-1}}\).
For an element \(x\) in a topological space \(X\), we denote the orbit of \(x\) as \(\text{Orb}(x)\), which represents the set of images of \(x\) under various translations, namely \(Gx\). The stabilizer of \(x\), denoted by \(\text{Stab}(x)\), is the set of elements \(g \in G\) whose corresponding translation action leaves \(x\) invariant, expressed as \[\text{Stab}(x) = \{g \in G \mid gx = x\}\]
By virtue of group theory, it is readily established that \(G / \text{Stab}(x)\) is isomorphic to \(Gx\) via the assignment \(g \cdot \text{Stab}(x) \mapsto gx\). Furthermore, if the action of \(G\) on \(X\) is transitive, we obtain an isomorphism between \(G / \text{Stab}(x)\) and \(X\).
- Example 1: The action of a trivial group on an arbitrary topological space.
- Example 2: Translations of \(\mathbb{R}\) or \(\mathbb{C}\) on themselves, and the scaling action of \(\mathbb{R}^*\) or \(\mathbb{C}^*\) on \(\mathbb{R}\) or \(\mathbb{C}\).
- Example 3: The action of the general linear group \(GL(n, \mathbb{R} \text{ or } \mathbb{C})\) on a topological vector space.
We define \(G \backslash X\) as the set of orbits of \(X\) under the action of \(G\). This set can be endowed with the quotient topology, which is the coarsest topology such that the projection map \(p: X \to G \backslash X\), defined by \(x \mapsto Gx\), is continuous. Under this construction, a subset of \(G \backslash X\) is open if and only if the union of the orbits contained within it is open in \(X\).
It is observed that \(p\) is an open mapping. Indeed, for any open set \(U \subseteq X\), its image \[GU = \bigcup_{g \in G} gU\] is an open set in \(X\), which implies that \(p(U)\) is open in the quotient space \(G \backslash X\) by the definition of the quotient topology.
For a subgroup \(H\) of \(G\), the group \(H\) can act on \(G\) from the left and the right respectively, yielding two coset spaces: the left coset space \(H \backslash G\) and the right coset space \(G / H\).
The quotient space \(G/H\) is Hausdorff if and only if \(H\) is a closed subgroup of \(G\).
To establish the necessity, we consider the identity coset \(eH \in G/H\). Since \(G/H\) is assumed to be Hausdorff, the singleton set \(\{eH\}\) must be closed. By the continuity of the natural projection \(p\), it follows that the preimage \(H = p^{-1}(eH)\) is necessarily a closed subset of \(G\).
Conversely, suppose \(H\) is closed in \(G\). We consider the continuous mapping \(f: G \times G \to G\) defined by \((a, b) \mapsto a^{-1}b\). Since \(H\) is closed, its preimage \(f^{-1}(H) = \{(a, b) \in G \times G \mid a^{-1}b \in H\}\) is a closed subset of the product space \(G \times G\). This preimage corresponds exactly to the equivalence relation defining the cosets, where \((a, b) \in f^{-1}(H)\) if and only if \(aH = bH\). For any two distinct cosets \(aH\) and \(bH\) in \(G/H\), the pair \((a, b)\) does not lie in \(f^{-1}(H)\). Because \(f^{-1}(H)\) is closed, there exists an open neighborhood of \((a, b)\) in \(G \times G\), which we may denote as \(U \times V\), that is disjoint from \(f^{-1}(H)\). Consequently, the images \(p(U)\) and \(p(V)\) in \(G/H\) are disjoint open sets that separate \(aH\) and \(bH\), thereby proving that \(G/H\) is Hausdorff.
Suppose a topological group \(G\) acts continuously and transitively on a topological space \(X\). If both \(G\) and \(X\) are locally compact and \(G\) is second-countable (C2), then the natural bijection \(G / \text{Stab}(x) \to X\) is a homeomorphism for all \(x \in X\).
The proof of this result is omitted here, as it fundamentally relies on the Baire Category Theorem1.
1 Note: The application of the Baire Category Theorem is typically used to show that the orbital map \(p: G \to X\) is an open map, which is the crucial step in proving that the induced map from the quotient space is a homeomorphism.
It is important to observe that this result holds whenever \(G\) is a manifold, particularly in the case where \(G\) is a smooth manifold (i.e., a Lie group). In such contexts, the topological requirements are naturally satisfied, ensuring that the orbit-stabilizer correspondence is not merely a set-theoretic bijection but a structural homeomorphism.
2 Riemann Surfaces
Why do we care about topological groups? Because many fundamental Riemann surfaces (like Tori and Elliptic Curves) are constructed as quotients of Lie groups.
A Riemann surface is a central object of study that can be viewed as a complex manifold of dimension one. However, in the contexts of number theory or algebraic geometry, it carries additional structural interpretations.
We define a Riemann surface as a topological space \(X\) equipped with a complex structure. This complex structure is defined as follows:
Analogous to the construction of differential manifolds, a complex structure is given by an open cover of \(X\) together with a collection of mappings \(\{U_i, \phi_i\}\), where each \(\phi_i\) is a homeomorphism from \(U_i\) to an open subset of the complex plane \(\mathbb{C}\). We require that on the intersection \(U_i \cap U_j\), the transition map \(\phi_j \circ \phi_i^{-1}\) is holomorphic and has a non-vanishing derivative.
2 In a rigorous sense, this data merely specifies a coordinate atlas for \(X\). A true complex structure is defined as an equivalence class of such atlases. However, similar to the convention in differential manifold theory, any single atlas can be uniquely extended to a maximal one (an equivalence class), and thus we shall omit the formal distinction between an atlas and its class.
Furthermore, readers familiar with algebraic geometry will recognize that a Riemann surface is essentially a topological space \(X\) endowed with a sheaf of \(\mathbb{C}\)-algebras, such as the sheaf of holomorphic functions \(\mathcal{O}_X\) or the sheaf of meromorphic functions \(\mathcal{M}_X\).
A mapping \(f: U \to \mathbb{C}\) from an open subset \(U \subseteq X\) is said to be holomorphic if \(f \circ \phi_i^{-1}\) is holomorphic on \(\phi_i(U \cap U_i)\) for every index \(i\). It is easily verified that this condition is independent of the choice of representative for the equivalence class of coordinates, ensuring that the concept is well-defined even under the strictest definitions. The definition of meromorphic mappings follows in a completely analogous manner.
- Open Subsets: Any open subset of the complex plane \(\mathbb{C}\) is naturally a Riemann surface.
- The Riemann Sphere: Consider the unit sphere \(S^2\) defined by \(x^2 + y^2 + z^2 = 1\). A complex structure is induced via stereographic projections:
- \(\psi_N: (x, y, z) \mapsto \frac{x + iy}{1 - z}\) from \(X \setminus \{N\}\) to \(\mathbb{C}\)
- \(\psi_S: (x, y, z) \mapsto \frac{x - iy}{1 + z}\) from \(X \setminus \{S\}\) to \(\mathbb{C}\) where \(N(0, 0, 1)\) and \(S(0, 0, -1)\) denote the North and South poles, respectively.
- Complex Tori: A particularly important class of Riemann surfaces is given by \(\mathbb{R}^2 / \mathbb{Z}^2\). While these are all homeomorphically equivalent to the torus, they admit uncountably many distinct complex structures (parameterized by the upper half-plane).
As with differential manifolds, we define holomorphic maps between Riemann surfaces as follows:
A continuous map \(F: X \to Y\) between two Riemann surfaces is called holomorphic if for every chart \((U, \phi)\) on \(X\) and \((V, \psi)\) on \(Y\) with \(F(U) \subseteq V\), the composition \(\psi \circ F \circ \phi^{-1}\) is a holomorphic function between open sets of \(\mathbb{C}\). If \(F\) is a bijective holomorphic map whose inverse is also holomorphic, it is called a biholomorphism, and \(X\) and \(Y\) are said to be equivalent as Riemann surfaces.
We assume that readers trained in algebraic geometry are more familiar with the language of ringed spaces. Moving forward, we shall adopt the language of \(\mathbb{C}\)-algebra sheaves as much as possible. Let \(X\) be a Riemann surface, and we define the sheaf \(\mathcal{O}\) on \(X\) such that for any open set \(U \subseteq X\): \[\Gamma(U, \mathcal{O}) = \{f: U \to \mathbb{C} \mid f \text{ is holomorphic}\}\] This \(\mathcal{O}\) is referred to as the structure sheaf of \(X\). For any open subset \(V \subseteq X\), the restriction of \(\mathcal{O}\) to \(V\) is denoted by \(\mathcal{O}_V\). 3
3 The verification that \(\mathcal{O}\) constitutes a sheaf is a worthwhile exercise. The “locality” and “gluing” axioms are satisfied because the property of being holomorphic is inherently local. Specifically, if a collection of functions \(\{f_i\}\) is defined on an open cover \(\{U_i\}\) and they agree on overlaps, the uniquely glued function \(f\) is holomorphic because its restriction to each \(U_i\) is holomorphic.
It is crucial to recognize that while a complex structure on \(X\) uniquely determines a \(\mathbb{C}\)-algebra sheaf, the converse requires additional conditions to ensure that the ringed space \((X, \mathcal{O})\) “looks like” a complex manifold locally.
A topological space \(X\) equipped with a sheaf of \(\mathbb{C}\)-algebras \(\mathcal{O}\) corresponds to a Riemann surface if and only if for every point \(p \in X\), there exists an open neighborhood \(U\) of \(p\) such that the restricted ringed space \((U, \mathcal{O}_U)\) is isomorphic to the ringed space \((\mathbb{D}, \mathcal{O}_{\mathbb{D}})\), where \(\mathbb{D}\) is the unit disk in \(\mathbb{C}\) and \(\mathcal{O}_{\mathbb{D}}\) is the standard sheaf of holomorphic functions.
To prove this, we first assume \((X, \mathcal{O})\) is locally isomorphic to the standard model. For each \(p\), let \(\psi: (U, \mathcal{O}_U) \to (\mathbb{D}, \mathcal{O}_{\mathbb{D}})\) be the isomorphism. The underlying homeomorphism \(\phi: U \to \mathbb{D}\) provides a coordinate chart. Given two such charts \((U_i, \phi_i)\) and \((U_j, \phi_j)\), the transition map \(\phi_j \circ \phi_i^{-1}\) must preserve the structure of the sheaf of holomorphic functions. In the language of ringed spaces, this means the pullback of a holomorphic function under the transition map remains holomorphic, which is exactly the condition that the transition map itself is a biholomorphism. Thus, the collection of such charts forms a complex atlas. The converse follows directly from the definition of a holomorphic map between a Riemann surface and \(\mathbb{C}\), which identifies the structure sheaf as the sheaf of functions that are locally representable by power series in any chart.
Consider the group \(G = \mathbb{Z}\) acting on \(\mathbb{C}\) by translations \(z \mapsto z + n\). The quotient space \(X = \mathbb{C} / \mathbb{Z}\) inherits a unique complex structure such that the projection \(p: \mathbb{C} \to X\) is holomorphic. The structure sheaf \(\mathcal{O}_X\) consists of \(\mathbb{Z}\)-periodic holomorphic functions on \(\mathbb{C}\).
Interestingly, the exponential map \(\exp: \mathbb{C} \to \mathbb{C}^\times\) defined by \(z \mapsto e^{2\pi i z}\) induces a biholomorphism between the quotient Riemann surface \(\mathbb{C} / \mathbb{Z}\) and the punctured plane \(\mathbb{C} \setminus \{0\}\). Under this isomorphism, the structure sheaf \(\mathcal{O}_X\) is identified with the sheaf of holomorphic functions on \(\mathbb{C}^\times\). This demonstrates how a complex structure on a quotient topological group can be realized as a concrete algebraic variety.