nLab Riemann surface

Idea

A Riemann surface is a $1$-dimensional algebro-geometric object with good properties. The name ‘surface’ comes from the classical case, which is $1$-dimensional over the complex numbers and therefore $2$-dimensional over the real numbers.

There are several distinct meaning of what is a Riemann surface, and it can be considered in several generalities. But the main definition by far is the classical one.

Definition

Classically, a Riemann surface is a connected complex-$1$-dimensional complex manifold, in the strictest sense of ‘manifold’. In other words, it’s a Hausdorff second countable space $M$ which is locally homeomorphic to the complex plane $\mathbb{C}$ via charts (i.e., homeomorphisms) $\phi_i:U_i \to V_i$ for $U_i \subset M, V_i \subset \mathbb{C}$ open and such that $\phi_j \circ \phi_i^{-1}: V_i \cap V_j \to V_i \cap V_j$ is holomorphic.

It is not necessary to make any assumption about whether there exists a countable base for the topology (second countable) or whether it has a countable dense subset (separable). This is because Tibor Rado (1923) proved that all Riemann surfaces, without such prior assumptions, must necessarily have a countable base. Thus for elegance this condition is customarily not assumed by specialists in Riemann surfaces.

There are generalizations, e.g., over local fields in rigid analytic geometry.

Examples

Evidently an open subspace of a Riemann surface is a Riemann surface. In particular, an open subset of $\mathbb{C}$ is a Riemann surface in a natural manner.

The Riemann sphere $P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}$ or $S^2$ is a Riemann surface with the open sets $U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}$ and the charts

$$\phi_1 =z, \;\phi_2 = \frac{1}{z}.$$

The transition map is $\frac{1}{z}$ and thus holomorphic on $U_1 \cap U_2 = \mathbb{C}^*$.

An important example comes from analytic continuation, which we will briefly sketch below. A function element is a pair $(f,V)$ where $f: V \to \mathbb{C}$ is holomorphic and $V \subset \mathbb{C}$ is an open disk. Two function elements $(f,V), (g,W)$ are said to be direct analytic continuations of each other if $V \cap W \neq \emptyset$ and $f \equiv g$ on $V \cap W$. By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element $\gamma = (f,V)$, we can consider the totality $X$ of all equivalence classes of function elements that can be obtained by continuing $\gamma$ along curves in $\mathbb{C}$. Then $X$ is actually a Riemann surface.

Indeed, we must first put a topology on $X$. If $(g,W) \in X$ with $W=D_r(w_0)$ centered at $w_0$, then let a neighborhood of $g$ be given by all function elements $(g_w, W')$ for $w \in W, W' \subset W$; these form a basis for a suitable topology on $X$. Then the coordinate projections $(g,W) \to w_0$ form appropriate local coordinates. In fact, there is a globally defined map $X \to \mathbb{C}$, whose image in general will be a proper subset of $\mathbb{C}$.

Properties

Basic facts

Since we have local coordinates, we can define a map $f: X \to Y$ of Riemann surfaces to be holomorphic or regular if it is so in local coordinates. In particular, we can define a holomorphic complex function as a holomorphic map $f: X \to \mathbb{C}$; for meromorphicity, this becomes $f: X \to S^2$.

Many of the usual theorems of elementary complex analysis (that is to say, the local ones) transfer immediately to the case of Riemann surfaces. For instance, we can locally get a Laurent expansion, etc.

Theorem

Let $f: X \to Y$ be a regular map. If $X$ is compact and $f$ is nonconstant, then $f$ is surjective and $Y$ compact.

To see this, note that $f(X)$ is compact, and an open subset by the open mapping theorem?, so the result follows by connectedness of $Y$.

Complexified differentials

Since a Riemann surface $X$ is a $2$-dimensional smooth manifold in the usual (real) sense, it is possible to do the usual exterior calculus. We could consider a 1-form to be a section of the (usual) cotangent bundle $T^*(X)$, but it is more natural to take the complexified cotangent bundle $\mathbb{C} \otimes_{\mathbb{R}} T^*(X)$, which we will in the future just abbreviate $T^*(X)$; this should not be confusing since we will only do this when we talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.

If $z = x + i y$ is a local coordinate on $X$, defined say on $U \subset X$, define the (complex) differentials

$$d z = d x + i d y , \;d\bar{z} = d x - i d y.$$

These form a basis for the complexified cotangent space at each point of $U$. There is also a dual basis

$$\frac{\partial}{\partial z } := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right), \; \frac{\partial}{\partial \bar{z} } := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)$$

for the complexified tangent space.

We now claim that we can split the tangent space $T(X) = T^{1,0}(X) + T^{0,1}(X)$, where the former consists of multiples of $\frac{\partial}{\partial z}$ and the latter of multiples of $\frac{\partial}{\partial \bar{z}}$; clearly a similar thing is possible for the cotangent space. This is always possible locally, and a holomorphic map preserves the decomposition. One way to see the last claim quickly is that given $g: U \to \mathbb{C}$ for $U \subset \mathbb{C}$ open and $0 \in U$ (just for convenience), we can write

$$g(z) = g(0) + Az + A' \bar{z} + o(|z|)$$

where $A = \frac{\partial g }{\partial z }(0), A' = \frac{\partial g }{\partial \bar{z} }(0)$, which we will often abbreviate as $g_z(0), g_{\bar{z}}(0)$. If $\psi: U' \to U$ is holomorphic and conformal sending $z_0 \in U' \to 0 \in U$, we have

$$g(\phi(\zeta)) = g(\phi(0)) + A \phi'(z_0)(\zeta-z_0) + A' \overline{ \phi'(z_0)(\zeta-z_0)} + o(|z|);$$

in particular, $\phi$ preserves the decomposition of $T_0(\mathbb{C})$.

Given $f: X \to \mathbb{C}$ smooth, we can consider the projections of the 1-form $df$ onto $T^{1,0}(X)$ and $T^{0,1}(X)$, respectively; these will be called $\partial f, \overline{\partial} f$. Similarly, we define the corresponding operators on 1-forms: to define $\partial \omega$, first project onto $T^{0,1}(M)$ (the reversal is intentional!) and then apply $d$, and vice versa for $\overline{\partial} \omega$.

In particular, if we write in local coordinates $\omega = u d z + v d\bar{z}$, then

$$\partial \omega = d( v d \bar{z}) = v_z d z \wedge d\bar{z},$$

and

$$\overline{\partial} \omega = d( u d z) = u_{\bar{z}} d\bar{z} \wedge d z.$$

To see this, we have tacitly observed that $d v = v_z d z + v_{\bar{z}} d\bar{z}$.

Picard group of holomorphic line bundles

The Picard group of a Riemann surface is the group of holomorphic line bundles in it. Introductions include (Bobenko, section 8).

See also at Narasimhan–Seshadri theorem and at moduli space of connections – Flat connections over a torus.

Central theorems

In the theory of Riemann surfaces, there are several important theorems. Here are two:

• The Riemann-Roch theorem, which analyzes the vector space of meromorphic functions satisfying certain conditions on zeros and poles;

• The uniformization theorem?, which partially classifies Riemann surfaces.

Homotopy type

A compact Riemann surface of genus $g \geq 2$ is a homotopy 1-type. The fundamental groupoid is a Fuchsian group.

(MO discussion)

Branched covers

By the Riemann existence theorem, every connected compact Riemann surface admits the structure of a branched cover of the Riemann sphere. (MO discussion)

Function field analogy

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$ (Spec(Z))	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

Related concepts

• annulus, 2-sphere, trinion

• Teichmüller space

• moduli space of Riemann surfaces

• super Riemann surface

• stable vector bundle

• elliptic curve

• worldsheet

• beta-gamma system

• quadratic differential

References

Historical references:

• Hermann Weyl, Die Idee der Riemannschen Fläche, 1913 (The concept of a Riemann surface) (on the book, by Peter Schreiber, 2013: web)

• Tibor Radó, Über den Begriff der Riemannschen Fläche, Acta Litt. Sci. Szeged, 2 (101-121), 10 (pdf, pdf)

  (proving the triangulation theorem)

Monograph:

• Robert C. Gunning, Lectures on Riemann Surfaces, Princeton University Press (1966) [pdf]

Lecture notes:

• Alexander Bobenko, Compact Riemann Surfaces lecture notes (pdf)

• Eberhard Freitag, Riemann surfaces – Sheaf theory, Riemann Surfaces, Automorphic forms, 2013 (pdf)

• Joachim Wehler, Riemann surfaces 2020 (pdf)