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The fundamental object of study in real differential geometry are the real manifolds: Hausdorff topological spaces ${M = M^n}$ that locally look like open subsets of a Euclidean space ${{\bf R}^n}$, and which can be equipped with an atlas ${(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}}$ of coordinate charts ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$ from open subsets ${U_\alpha}$ covering ${M}$ to open subsets ${V_\alpha}$ in ${{\bf R}^n}$, which are homeomorphisms; in particular, the transition maps ${\tau_{\alpha,\beta}: \phi_\alpha( U_\alpha \cap U_\beta ) \rightarrow \phi_\beta( U_\alpha \cap U_\beta )}$ defined by ${\tau_{\alpha,\beta}: \phi_\beta \circ \phi_\alpha^{-1}}$ are all continuous. (It is also common to impose the requirement that the manifold ${M}$ be second countable, though this will not be important for the current discussion.) A smooth real manifold is a real manifold in which the transition maps are all smooth.

In a similar fashion, the fundamental object of study in complex differential geometry are the complex manifolds, in which the model space is ${{\bf C}^n}$ rather than ${{\bf R}^n}$, and the transition maps ${\tau_{\alpha\beta}}$ are required to be holomorphic (and not merely smooth or continuous). In the real case, the one-dimensional manifolds (curves) are quite simple to understand, particularly if one requires the manifold to be connected; for instance, all compact connected one-dimensional real manifolds are homeomorphic to the unit circle (why?). However, in the complex case, the connected one-dimensional manifolds – the ones that look locally like subsets of ${{\bf C}}$ – are much richer, and are known as Riemann surfaces. For sake of completeness we give the (somewhat lengthy) formal definition:

Definition 1 (Riemann surface) If ${M}$ is a Hausdorff connected topological space, a (one-dimensional complex) atlas is a collection ${(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}}$ of homeomorphisms from open subsets ${(U_\alpha)_{\alpha \in A}}$ of ${M}$ that cover ${M}$ to open subsets ${V_\alpha}$ of the complex numbers ${{\bf C}}$, such that the transition maps ${\tau_{\alpha,\beta}: \phi_\alpha( U_\alpha \cap U_\beta ) \rightarrow \phi_\beta( U_\alpha \cap U_\beta )}$ defined by ${\tau_{\alpha,\beta}: \phi_\beta \circ \phi_\alpha^{-1}}$ are all holomorphic. Here ${A}$ is an arbitrary index set. Two atlases ${(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}}$, ${(\phi'_\beta: U'_\beta \rightarrow V'_\beta)_{\beta \in B}}$ on ${M}$ are said to be equivalent if their union is also an atlas, thus the transition maps ${\phi'_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U'_\beta) \rightarrow \phi'_\beta(U_\alpha \cap U'_\beta)}$ and their inverses are all holomorphic. A Riemann surface is a Hausdorff connected topological space ${M}$ equipped with an equivalence class of one-dimensional complex atlases.

A map ${f: M \rightarrow M'}$ from one Riemann surface ${M}$ to another ${M'}$ is holomorphic if the maps ${\phi'_\beta \circ f \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap f^{-1}(U'_\beta)) \rightarrow {\bf C}}$ are holomorphic for any charts ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$, ${\phi'_\beta: U'_\beta \rightarrow V'_\beta}$ of an atlas of ${M}$ and ${M'}$ respectively; it is not hard to see that this definition does not depend on the choice of atlas. It is also clear that the composition of two holomorphic maps is holomorphic (and in fact the class of Riemann surfaces with their holomorphic maps forms a category).

Here are some basic examples of Riemann surfaces.

Example 2 (Quotients of ${{\bf C}}$) The complex numbers ${{\bf C}}$ clearly form a Riemann surface (using the identity map ${\phi: {\bf C} \rightarrow {\bf C}}$ as the single chart for an atlas). Of course, maps ${f: {\bf C} \rightarrow {\bf C}}$ that are holomorphic in the usual sense will also be holomorphic in the sense of the above definition, and vice versa, so the notion of holomorphicity for Riemann surfaces is compatible with that of holomorphicity for complex maps. More generally, given any discrete additive subgroup ${\Lambda}$ of ${{\bf C}}$, the quotient ${{\bf C}/\Lambda}$ is a Riemann surface. There are an infinite number of possible atlases to use here; one such is to pick a sufficiently small neighbourhood ${U}$ of the origin in ${{\bf C}}$ and take the atlas ${(\phi_\alpha: U_\alpha \rightarrow U)_{\alpha \in {\bf C}/\Lambda}}$ where ${U_\alpha := \alpha+U}$ and ${\phi_\alpha(\alpha+z) := z}$ for all ${z \in U}$. In particular, given any non-real complex number ${\omega}$, the complex torus ${{\bf C} / \langle 1, \omega \rangle}$ formed by quotienting ${{\bf C}}$ by the lattice ${\langle 1, \omega \rangle := \{ n + m \omega: n,m \in {\bf Z}\}}$ is a Riemann surface.

Example 3 Any open connected subset ${U}$ of ${{\bf C}}$ is a Riemann surface. By the Riemann mapping theorem, all simply connected open ${U \subset {\bf C}}$, other than ${{\bf C}}$ itself, are isomorphic (as Riemann surfaces) to the unit disk (or, equivalently, to the upper half-plane).

Example 4 (Riemann sphere) The Riemann sphere ${{\bf C} \cup \{\infty\}}$, as a topological manifold, is the one-point compactification of ${{\bf C}}$. Topologically, this is a sphere and is in particular connected. One can cover the Riemann sphere by the two open sets ${U_1 := {\bf C}}$ and ${U_2 := {\bf C} \cup \{\infty\} \backslash \{0\}}$, and give these two open sets the charts ${\phi_1: U_1 \rightarrow {\bf C}}$ and ${\phi_2: U_2 \rightarrow {\bf C}}$ defined by ${\phi_1(z) := z}$ for ${z \in {\bf C}}$, ${\phi_2(z) := 1/z}$ for ${z \in {\bf C} \backslash \{0\}}$, and ${\phi_2(\infty) := 0}$. This is a complex atlas since the ${1/z}$ is holomorphic on ${{\bf C} \backslash \{0\}}$.

An alternate way of viewing the Riemann sphere is as the projective line ${\mathbf{CP}^1}$. Topologically, this is the punctured complex plane ${{\bf C}^2 \backslash \{(0,0)\}}$ quotiented out by non-zero complex dilations, thus elements of this space are equivalence classes ${[z,w] := \{ (\lambda z, \lambda w): \lambda \in {\bf C} \backslash \{0\}\}}$ with the usual quotient topology. One can cover this space by two open sets ${U_1 := \{ [z,1]: z \in {\bf C} \}}$ and ${U_2: \{ [1,w]: w \in {\bf C} \}}$ and give these two open sets the charts ${\phi: U_1 \rightarrow {\bf C}}$ and ${\phi_2: U_2 \rightarrow {\bf C}}$ defined by ${\phi_1([z,1]) := z}$ for ${z \in {\bf C}}$, ${\phi_2([1,w]) := w}$. This is a complex atlas, basically because ${[z,1] = [1,1/z]}$ for ${z \in {\bf C} \backslash \{0\}}$ and ${1/z}$ is holomorphic on ${{\bf C} \backslash \{0\}}$.

Exercise 5 Verify that the Riemann sphere is isomorphic (as a Riemann surface) to the projective line.

Example 6 (Smooth algebraic plane curves) Let ${P(z_1,z_2,z_3)}$ be a complex polynomial in three variables which is homogeneous of some degree ${d \geq 1}$, thus

$\displaystyle P( \lambda z_1, \lambda z_2, \lambda z_3) = \lambda^d P( z_1, z_2, z_3). \ \ \ \ \ (1)$

Define the complex projective plane ${\mathbf{CP}^2}$ to be the punctured space ${{\bf C}^3 \backslash \{0\}}$ quotiented out by non-zero complex dilations, with the usual quotient topology. (There is another important topology to place here of fundamental importance in algebraic geometry, namely the Zariski topology, but we will ignore this topology here.) This is a compact space, whose elements are equivalence classes ${[z_1,z_2,z_3] := \{ (\lambda z_1, \lambda z_2, \lambda z_3)\}}$. Inside this plane we can define the (projective, degree ${d}$) algebraic curve

$\displaystyle Z(P) := \{ [z_1,z_2,z_3] \in \mathbf{CP}^2: P(z_1,z_2,z_3) = 0 \};$

this is well defined thanks to (1). It is easy to verify that ${Z(P)}$ is a closed subset of ${\mathbf{CP}^2}$ and hence compact; it is non-empty thanks to the fundamental theorem of algebra.

Suppose that ${P}$ is irreducible, which means that it is not the product of polynomials of smaller degree. As we shall show in the appendix, this makes the algebraic curve connected. (Actually, algebraic curves remain connected even in the reducible case, thanks to Bezout’s theorem, but we will not prove that theorem here.) We will in fact make the stronger nonsingularity hypothesis: there is no triple ${(z_1,z_2,z_3) \in {\bf C}^3 \backslash \{(0,0,0)\}}$ such that the four numbers ${P(z_1,z_2,z_3), \frac{\partial}{\partial z_j} P(z_1,z_2,z_3)}$ simultaneously vanish for ${j=1,2,3}$. (This looks like four constraints, but is in fact essentially just three, due to the Euler identity

$\displaystyle \sum_{j=1}^3 z_j \frac{\partial}{\partial z_j} P(z_1,z_2,z_3) = d P(z_1,z_2,z_3)$

that arises from differentiating (1) in ${\lambda}$. The fact that nonsingularity implies irreducibility is another consequence of Bezout’s theorem, which is not proven here.) For instance, the polynomial ${z_1^2 z_3 - z_2^3}$ is irreducible but singular (there is a “cusp” singularity at ${[0,0,1]}$). With this hypothesis, we call the curve ${Z(P)}$ smooth.

Now suppose ${[z_1,z_2,z_3]}$ is a point in ${Z(P)}$; without loss of generality we may take ${z_3}$ non-zero, and then we can normalise ${z_3=1}$. Now one can think of ${P(z_1,z_2,1)}$ as an inhomogeneous polynomial in just two variables ${z_1,z_2}$, and by nondegeneracy we see that the gradient ${(\frac{\partial}{\partial z_1} P(z_1,z_2,1), \frac{\partial}{\partial z_2} P(z_1,z_2,1))}$ is non-zero whenever ${P(z_1,z_2,1)=0}$. By the (complexified) implicit function theorem, this ensures that the affine algebraic curve

$\displaystyle Z(P)_{aff} := \{ (z_1,z_2) \in {\bf C}^2: P(z_1,z_2,1) = 0 \}$

is a Riemann surface in a neighbourhood of ${(z_1,z_2,1)}$; we leave this as an exercise. This can be used to give a coordinate chart for ${Z(P)}$ in a neighbourhood of ${[z_1,z_2,z_3]}$ when ${z_3 \neq 0}$. Similarly when ${z_1,z_2}$ is non-zero. This can be shown to give an atlas on ${Z(P)}$, which (assuming the connectedness claim that we will prove later) gives ${Z(P)}$ the structure of a Riemann surface.

Exercise 7 State and prove a complex version of the implicit function theorem that justifies the above claim that the charts in the above example form an atlas, and an algebraic curve associated to a non-singular polynomial is a Riemann surface.

Exercise 8

• (i) Show that all (irreducible plane projective) algebraic curves of degree ${1}$ are isomorphic to the Riemann sphere. (Hint: reduce to an explicit linear polynomial such as ${z_3}$.)
• (ii) Show that all (irreducible plane projective) algebraic curves of degree ${2}$ are isomorphic to the Riemann sphere. (Hint: to reduce computation, first use some linear algebra to reduce the homogeneous quadratic polynomial to a standard form, such as ${z_1^2+z_2^2+z_3^2}$ or ${z_2 z_3 - z_1^2}$.)

Exercise 9 If ${a,b}$ are complex numbers, show that the projective cubic curve

$\displaystyle \{ [z_1, z_2, z_3]: z_2^2 z_3 = z_1^3 + a z_1 z_3^2 + b z_3^3 \}$

is nonsingular if and only if the discriminant ${-16 (4a^3 + 27b^2)}$ is non-zero. (When this occurs, the curve is called an elliptic curve (in Weierstrass form), which is a fundamentally important example of a Riemann surface in many areas of mathematics, and number theory in particular. One can also define the discriminant for polynomials of higher degree, but we will not do so here.)

A recurring theme in mathematics is that an object ${X}$ is often best studied by understanding spaces of “good” functions on ${X}$. In complex analysis, there are two basic types of good functions:

Definition 10 Let ${X}$ be a Riemann surface. A holomorphic function on ${X}$ is a holomorphic map from ${X}$ to ${{\bf C}}$; the space of all such functions will be denoted ${{\mathcal O}(X)}$. A meromorphic function on ${X}$ is a holomorphic map from ${X}$ to the Riemann sphere ${{\bf C} \cup \{\infty\}}$, that is not identically equal to ${\infty}$; the space of all such functions will be denoted ${M(X)}$.

One can also define holomorphicity and meromorphicity in terms of charts: a function ${f: X \rightarrow {\bf C}}$ is holomorphic if and only if, for any chart ${\phi_\alpha: U_\alpha \rightarrow {\bf C}}$, the map ${f \circ \phi^{-1}_\alpha: \phi_\alpha(U_\alpha) \rightarrow {\bf C}}$ is holomorphic in the usual complex analysis sense; similarly, a function ${f: X \rightarrow {\bf C} \cup \{\infty\}}$ is meromorphic if and only if the preimage ${f^{-1}(\{\infty\})}$ is discrete (otherwise, by analytic continuation and the connectedness of ${X}$, ${f}$ will be identically equal to ${\infty}$) and for any chart ${\phi_\alpha: U_\alpha \rightarrow X}$, the map ${f \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha) \rightarrow {\bf C} \cup \{\infty\}}$ becomes a meromorphic function in the usual complex analysis sense, after removing the discrete set of complex numbers where this map is infinite. One consequence of this alternate definition is that the space ${{\mathcal O}(X)}$ of holomorphic functions is a commutative complex algebra (a complex vector space closed under pointwise multiplication), while the space ${M(X)}$ of meromorphic functions is a complex field (a commutative complex algebra where every non-zero element has an inverse). Another consequence is that one can define the notion of a zero of given order ${k}$, or a pole of order ${k}$, for a holomorphic or meromorphic function, by composing with a chart map and using the usual complex analysis notions there, noting (from the holomorphicity of transition maps and their inverses) that this does not depend on the choice of chart. (However, one cannot similarly define the residue of a meromorphic function on ${X}$ this way, as the residue turns out to be chart-dependent thanks to the chain rule. Residues should instead be applied to meromorphic ${1}$-forms, a concept we will introduce later.) A third consequence is analytic continuation: if two holomorphic or meromorphic functions on ${X}$ agree on a non-empty open set, then they agree everywhere.

On the complex numbers ${{\bf C}}$, there are of course many holomorphic functions and meromorphic functions; for instance any power series with an infinite radius of convergence will give a holomorphic function, and the quotient of any two such functions (with non-zero denominator) will give a meromorphic function. Furthermore, we have extremely wide latitude in how to specify the zeroes of the holomorphic function, or the zeroes and poles of the meromorphic function, thanks to tools such as the Weierstrass factorisation theorem or the Mittag-Leffler theorem (covered in previous quarters).

It turns out, however, that the situation changes dramatically when the Riemann surface ${X}$ is compact, with the holomorphic and meromorphic functions becoming much more rigid. First of all, compactness eliminates all holomorphic functions except for the constants:

Lemma 11 Let ${f \in \mathcal{O}(X)}$ be a holomorphic function on a compact Riemann surface ${X}$. Then ${f}$ is constant.

This result should be seen as a close sibling of Liouville’s theorem that all bounded entire functions are constant. (Indeed, in the case of a complex torus, this lemma is a corollary of Liouville’s theorem.)

Proof: As ${f}$ is continuous and ${X}$ is compact, ${|f(z_0)|}$ must attain a maximum at some point ${z_0 \in X}$. Working in a chart around ${z_0}$ and applying the maximum principle, we conclude that ${f}$ is constant in a neighbourhood of ${z_0}$, and hence is constant everywhere by analytic continuation. $\Box$

This dramatically cuts down the number of possible meromorphic functions – indeed, for an abstract Riemann surface, it is not immediately obvious that there are any non-constant meromorphic functions at all! As the poles are isolated and the surface is compact, a meromorphic function can only have finitely many poles, and if one prescribes the location of the poles and the maximum order at each pole, then we shall see that the space of meromorphic functions is now finite dimensional. The precise dimensions of these spaces are in fact rather interesting, and obey a basic duality law known as the Riemann-Roch theorem. We will give a mostly self-contained proof of the Riemann-Roch theorem in these notes, omitting only some facts about genus and Euler characteristic, as well as construction of certain meromorphic ${1}$-forms (also known as Abelian differentials).

A more detailed study of Riemann surface (and more generally, complex manifolds) can be found for instance in Griffiths and Harris’s “Principles of Algebraic Geometry“.