Complex Analysis II, Spring 2024

Course Information

• NTU COOL
• Lectures: Tuesday 13:20 ~ 15:10 and Thursday 13:20 ~ 15:10 at Astro-Math 101.
• Grading scheme:
  • Homework 40%
    Students are asked to do the homework on the blackboard at 14:20 ~ 15:10 on Thursdays.
  • Final Exam 60%
• Main Topics:
  This is a continuation of Complex Analysis (I). The main topic is the theory of Riemann surfaces, including Riemann--Roch theorem, uniformization, Abel-Jacobi theorem, etc.
• References:
  • Simon Donaldson, Riemann surfaces.
  • Farkas and Kra, Riemann surfaces. Second edition.

Lecture summaries and references

• (week 1)
  • holomorphic function
  • implicit function theorem via Cauchy integral
  • analytic continuation and complex ODE
  • classification of surface: Klein bottle ≌ RP² # RP², Klein bottle # RP² ≌ T² # RP²
  reference: Donaldson ch.1 and ch.2.
• (week 2)
  • definition of Riemann surface
  • basic examples
  • projective space and algebraic curve
  • quotients
  reference: Donaldson ch.3.
• (week 3)
  • holomorphic map between Riemann surfaces
  • covering spaces and fundamental groups
  reference: Donaldson ch.4.
• (week 4)
  • monodromy representation of holomorphic map
  • Riemann's existence theorem
  • basic theory of differential forms on oriented surfaces
  reference: Donaldson ch.4 and ch.5.
• (week 5)
  • Poincare duality
  • decomposition of d on Riemann surfaces
  • the Laplace operator and Dirichlet energy
  • Euler characteristic and Poincare-Hopf theorem
  reference: Donaldson ch.5 and ch.6.
• (week 6)
  • zero and poles of meromorphic 1-forms
  • Riemann-Hurwitz formula
  • degree-genus formula
  • cohomology defined by \bar{\partial}
  reference: Donaldson ch.6 and ch.8.
• (week 7)
  • genus 0 Riemann surfaces
  • genus 1 Riemann surfaces
  • Riemann-Roch theorem (restricted version)
  reference: Donaldson ch.8.
• (week 8)
  • completion with respect to the Dirichlet energy and the bounded linear functional
  • Poincare type inequality in 2D
  • Weyl's lemma
  reference: Donaldson ch.9.
• (week 9)
  • solving the Laplace on non-compact surfaces
  • behavior at infinity
  • uniformization theorem
  reference: Donaldson ch.10.
• (week 10)
  • uniformization theorem (continued)
  • Cech cohomology
  • sheaf and sheaf cohomology
  • relation to the existence of meromorphic functions
  reference: Donaldson ch.10 and ch.12.
• (week 11)
  • fine sheaf
  • line bundles and its sheaf cohomology
  • divisors and the corresponding line bundles
  reference: Donaldson ch.12.
• (week 12)
  • Riemann-Roch for divisors
  • Serre duality
  • different interpretations of Riemann-Roch
  reference: Donaldson ch.12.
• (week 13)
  • Jacobian variety
  • Picard group
  • equivalence between line bundles and divisors
  reference: Donaldson ch.12.
• (week 14)
  • Weierstrass points
  • first Chern class in terms of Chern connection

Homework

• See NTU COOL.