Riemann surfaces

Builds on

Required prerequisites, all the way down: 49

You will learn to

• Construct Riemann surfaces from charts, from algebraic equations such as w² = z³ − z, and as quotients ℂ/Λ.
• Apply the Riemann–Hurwitz formula to compute the genus of branched covers.
• State the uniformisation theorem and the Riemann–Roch theorem and use them in examples.

Definition

• A Riemann surface is a connected one-dimensional complex manifold.
• A holomorphic map between Riemann surfaces is one that is holomorphic in local coordinates.
• The genus of a compact Riemann surface is the genus of the underlying topological surface.

Theorem

• Every non-constant holomorphic map between compact Riemann surfaces is a branched covering of some degree.
• Riemann–Hurwitz: for a degree-n holomorphic map f : X → Y, 2g_X − 2 = n(2g_Y − 2) + Σ_p (e_p − 1).
• Uniformisation: every simply connected Riemann surface is biholomorphic to the Riemann sphere, ℂ or the open unit disc.
• Every compact Riemann surface is biholomorphic to a smooth complex projective algebraic curve.

Common misconceptions

• Thinking all tori are biholomorphic; complex tori ℂ/Λ form a family depending on one complex parameter (the j-invariant) up to biholomorphism, though all are homeomorphic.
• Believing the square root is a function on ℂ ∖ {0}; it becomes single-valued only on a two-sheeted Riemann surface.

Leads to

Optional discoveries

• Arithmetic geometry and elliptic curves A complex elliptic curve is a torus ℂ/Λ; studying the same cubic equation over the rationals is the arithmetic of elliptic curves.
• Complex and Kähler geometry; Hodge theory Riemann surfaces are the one-dimensional complex manifolds; Kähler and Hodge theory study their higher-dimensional analogues.
• ← Complex numbers

Further reading

• Rick Miranda, Algebraic Curves and Riemann Surfaces (1995)
• Otto Forster, Lectures on Riemann Surfaces (1981)