📐Complex Analysis Unit 11 – Riemann Surfaces

Key Concepts and Definitions

• Riemann surfaces are one-dimensional complex manifolds that allow for a consistent definition of holomorphic functions
• Every Riemann surface is a two-dimensional real manifold equipped with a complex structure
• The complex structure on a Riemann surface is an atlas of charts where transition functions are holomorphic
• Holomorphic functions on a Riemann surface are complex-valued functions that are differentiable in the complex sense
• Meromorphic functions on a Riemann surface are holomorphic functions except for isolated poles
• The genus of a Riemann surface is a topological invariant that measures the number of holes or handles
  • Riemann surfaces of genus 0 are topologically equivalent to the Riemann sphere (complex plane plus a point at infinity)
  • Riemann surfaces of genus 1 are topologically equivalent to a torus

Topology of Riemann Surfaces

• The topology of a Riemann surface determines its global structure and connectivity
• Riemann surfaces are classified topologically by their genus, which counts the number of holes or handles
• The Euler characteristic $\chi$ of a Riemann surface is related to its genus $g$ by the formula $\chi = 2 - 2g$
• Compact Riemann surfaces without boundary are characterized by their genus (sphere, torus, double torus, etc.)
• Non-compact Riemann surfaces include the complex plane, the punctured plane, and the universal cover of a compact Riemann surface
• The fundamental group of a Riemann surface encodes information about its loops and paths
  • The fundamental group of a genus $g$ surface is generated by $2g$ loops with a single relation

Complex Structure and Holomorphic Functions

• A complex structure on a Riemann surface is an atlas of charts where transition functions are holomorphic
• Holomorphic functions on a Riemann surface are locally expressible as power series in the complex coordinate
• The maximum modulus principle states that a non-constant holomorphic function on a compact Riemann surface attains its maximum on the boundary
• Holomorphic functions on a compact Riemann surface are constant by the maximum modulus principle
• The Riemann-Roch theorem relates the dimension of the space of meromorphic functions with prescribed poles to the genus and the degree of the divisor
• Holomorphic 1-forms on a Riemann surface form a vector space of dimension equal to the genus
  • Holomorphic 1-forms are locally of the form $f(z)dz$ where $f$ is holomorphic

Examples and Classifications

• The Riemann sphere $\hat{\mathbb{C}}$ is the simplest example of a Riemann surface, obtained by adding a point at infinity to the complex plane
• Elliptic curves are Riemann surfaces of genus 1, described by an equation of the form $y^2 = x^3 + ax + b$
  • Elliptic curves have a group structure given by the chord-tangent construction
• Hyperelliptic curves are Riemann surfaces described by an equation of the form $y^2 = P(x)$ where $P$ is a polynomial of degree greater than 4
• Compact Riemann surfaces are classified up to biholomorphism by their genus (topological type) and a finite number of moduli parameters
• The moduli space of genus $g$ Riemann surfaces has dimension $3g-3$ for $g \geq 2$
• Riemann surfaces can also be constructed from polygons by identifying edges (pair of pants decomposition)

Covering Spaces and Monodromy

• A covering space of a Riemann surface is another Riemann surface that locally looks like the original surface
• The universal cover of a Riemann surface is a simply connected covering space (usually the complex plane or the disk)
• The deck transformation group of a covering space consists of the biholomorphic self-maps that preserve the covering map
  • The deck transformation group is isomorphic to the fundamental group of the base surface
• Monodromy describes how solutions of differential equations on a Riemann surface behave under analytic continuation along loops
• The monodromy group is a representation of the fundamental group that encodes the branching behavior of a covering space
• Riemann surfaces can be constructed as branched covers of the Riemann sphere, with branch points corresponding to singularities

Meromorphic Functions and Divisors

• Meromorphic functions on a Riemann surface are holomorphic functions except for isolated poles
• The zeros and poles of a meromorphic function define a divisor, which is a formal sum of points with integer coefficients
• The degree of a divisor is the sum of its coefficients, counting zeros positively and poles negatively
• The divisor of a meromorphic function has degree zero by the argument principle
• Two divisors are linearly equivalent if their difference is the divisor of a meromorphic function
• The Riemann-Roch theorem relates the dimension of the space of meromorphic functions with prescribed poles to the genus and the degree of the divisor
  • The Riemann-Roch theorem states that $\dim L(D) - \dim L(K-D) = \deg(D) - g + 1$, where $L(D)$ is the space of meromorphic functions with poles bounded by $D$, $K$ is the canonical divisor, and $g$ is the genus

Differential Forms and Integration

• Holomorphic differential forms on a Riemann surface are locally of the form $f(z)dz$ where $f$ is holomorphic
• The space of holomorphic 1-forms on a genus $g$ surface has dimension $g$
• Meromorphic differential forms are allowed to have poles, and they form a larger space than holomorphic forms
• Integration of differential forms on Riemann surfaces is defined using charts and partitions of unity
• The residue theorem relates the integral of a meromorphic form around a closed curve to the sum of its residues at the enclosed poles
• The period matrix of a Riemann surface is the matrix of integrals of a basis of holomorphic 1-forms over a basis of homology cycles
  • The period matrix is symmetric and has positive definite imaginary part (Riemann bilinear relations)

Applications in Physics and Geometry

• Riemann surfaces arise naturally in the study of algebraic curves and their function fields
• The moduli space of Riemann surfaces is a fundamental object in algebraic geometry and has deep connections with the theory of modular forms
• Riemann surfaces are used to describe the worldsheets of strings in string theory, where the genus corresponds to the number of loops
• The Riemann-Hilbert correspondence relates monodromy representations of the fundamental group to systems of differential equations on Riemann surfaces
• Conformal field theories on Riemann surfaces are important in statistical physics and quantum field theory
• Spectral curves of integrable systems are often Riemann surfaces, and their geometry encodes the dynamics of the system
• The uniformization theorem states that every Riemann surface is the quotient of the Riemann sphere, the complex plane, or the hyperbolic plane by a discrete group of automorphisms